{"id":19531,"date":"2025-05-09T16:29:58","date_gmt":"2025-05-09T14:29:58","guid":{"rendered":"https:\/\/pointerpro.com\/?p=19531"},"modified":"2025-11-20T11:55:31","modified_gmt":"2025-11-20T10:55:31","slug":"understanding-standard-deviation-exploring-the-formula-and-its-applications-in-data-analysis","status":"publish","type":"post","link":"https:\/\/pointerpro.com\/blog\/standard-deviation-assessment-data\/","title":{"rendered":"Understanding standard deviation: Exploring the formula and Its applications in data analysis"},"content":{"rendered":"\n<style>\n    #section-IWuHv {\n        padding-top: 2.8rem;\n        padding-bottom: 6rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-IWuHv {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #421f3c;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-IWuHv-container  #421f3c custom-padding \" id=\"section-IWuHv\"\n    style=\"opacity: 100%; background-color: #421f3c;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-10 px-0 custom-padding row\">\n                    \n\n<style>\n    #heading-7AR6M .heading h1 {\n        color: #ffffff;\n        font-size: 3.125rem;\n        line-height: 4rem !important;\n    }\n\n    #heading-7AR6M {\n        margin-top: 0rem;\n        margin-bottom: 2rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-7AR6M {\n            margin-left: 0;\n            margin-right: 0;\n        }\n\n        .single #heading-7AR6M .heading h1 {\n            line-height: 3rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-1-container lazyblock-heading-1-7AR6M  px-0\" id=\"heading-7AR6M\">\n    <div class=\"row heading lazyblock-heading-1 justify-content-center  lazyblock-heading-1-7AR6M\">\n        <div class=\"col-12 col-lg-12 text-center \">\n            <a class=\"heading-1-anchor heading-anchor\" id=\"understanding-standard-deviation-exploring-the-formula-and-its-applications-in-data-analysis\"><\/a>\n            <h1>Understanding standard deviation: Exploring the formula and Its applications in data analysis<\/h1>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-1ARQs6 .heading h2 {\n        color: #FF7418;\n        font-size: 1.25rem;\n        line-height: 1.5rem !important;\n    }\n\n    #heading-1ARQs6 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1ARQs6 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-1ARQs6 date-and-author-container px-0\" id=\"heading-1ARQs6\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-center date-and-author lazyblock-heading-2-1ARQs6\">\n        <div class=\"col-12 col-lg-12 text-center\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"written-may-9-2025-by-jeremy-moser\"><\/a>\n            <h2>Written May 9, 2025, by Jeremy Moser<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-ZkOzLn {\n        padding-top: 2.8rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-ZkOzLn {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-ZkOzLn-container  #ffffff custom-padding \" id=\"section-ZkOzLn\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-6 px-0 custom-padding row\">\n                    \n\n<style>\n    .lazyblock-content-1SJqKx p,\n    .lazyblock-content-1SJqKx p a,\n    .lazyblock-content-1SJqKx h1,\n    .lazyblock-content-1SJqKx h2,\n    .lazyblock-content-1SJqKx h3,\n    .lazyblock-content-1SJqKx h4,\n    .lazyblock-content-1SJqKx h5 {\n        color: ;\n    }\n\n    #content-1SJqKx {\n        margin-top: 0rem;\n        margin-bottom: 1rem;\n    }\n\n    #content-1SJqKx {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-1SJqKx {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-1SJqKx-container  px-0 \"\n     id=\"content-1SJqKx\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-1SJqKx\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><em><span style=\"font-weight: 400;\">Haley, an HR professional, is tasked with analyzing employee satisfaction survey results from hundreds of staff members. The data is overwhelming, and she needs to present insights that are clear, actionable, and easy to digest.\u00a0<\/span><\/em><\/p>\n<p><i><span style=\"font-weight: 400;\">But how can she make sense of all the numbers without getting lost in the noise? This is where the concept of standard deviation becomes invaluable. Data analysts like Haley face the challenge of distilling vast volumes of data into meaningful insights. They need concise, easy-to-compare measures that effectively represent the core characteristics of large <a href=\"https:\/\/pointerpro.com\/blog\/nominal-data-examples\/\" target=\"_blank\" rel=\"noopener\">datasets<\/a>.<\/span><\/i><\/p>\n<p><span style=\"font-weight: 400;\">In this article, we&#8217;ll explore how understanding and applying the standard deviation formula can help you unlock deeper insights and make more informed decisions.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">One fundamental tool in a data analyst\u2019s arsenal is the standard deviation.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-ULfJl {\n        padding-top: 2.8rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-ULfJl {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #f5f5f5;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-ULfJl-container  #f5f5f5 custom-padding \" id=\"section-ULfJl\"\n    style=\"opacity: 100%; background-color: #f5f5f5;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-6 px-0 custom-padding row\">\n                    \n\n<style>\n    #heading-Z8vOoQ .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-Z8vOoQ {\n        margin-top: 0rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z8vOoQ {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-Z8vOoQ date-and-author-container px-0\" id=\"heading-Z8vOoQ\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start date-and-author lazyblock-heading-2-Z8vOoQ\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"what-is-the-standard-deviation\"><\/a>\n            <h2>What is the standard deviation?<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-QuJmK p,\n    .lazyblock-content-QuJmK p a,\n    .lazyblock-content-QuJmK h1,\n    .lazyblock-content-QuJmK h2,\n    .lazyblock-content-QuJmK h3,\n    .lazyblock-content-QuJmK h4,\n    .lazyblock-content-QuJmK h5 {\n        color: ;\n    }\n\n    #content-QuJmK {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-QuJmK {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-QuJmK {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-QuJmK-container  px-0 \"\n     id=\"content-QuJmK\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-QuJmK\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Standard deviation (SD) is defined as the square root of a dataset\u2019s variance. It\u2019s an essential measure in descriptive statistics that shows how spread out the individual data points are from that dataset\u2019s mean (average) value.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">In simpler terms, it quantifies a data point\u2019s average distance from the distribution center.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">For example, in our example of Haley, an HR consultant, let\u2019s say she\u2019s just collected responses from a large training needs assessment. She has a lot of data, but she needs a way to figure out how \u201cspread out\u201d the responses are. That\u2019s where <\/span><b>standard deviation<\/b><span style=\"font-weight: 400;\"> comes in.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">In simple terms, it tells her how much each individual response (like a rating on training topics) differs from the average or &#8220;center&#8221; of the data. If the standard deviation is small, it means most responses are pretty similar and close to the average. But if it\u2019s large, the responses vary a lot, and there\u2019s more diversity in opinions or needs.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">For example, if most employees think the same training programs are necessary, the standard deviation will be small. If some employees think certain skills are essential while others feel they need entirely different training, the standard deviation will be larger.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">So, as an HR consultant, understanding standard deviation helps Haley see how consistent or varied her employees\u2019 training needs are, making it easier to design a program tailored to the group.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-atzB5 .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-atzB5 {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-atzB5 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-atzB5 date-and-author-container px-0\" id=\"heading-atzB5\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start date-and-author lazyblock-heading-2-atzB5\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"what-does-standard-deviation-measure\"><\/a>\n            <h2>What does standard deviation measure?<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z10a0f4 p,\n    .lazyblock-content-Z10a0f4 p a,\n    .lazyblock-content-Z10a0f4 h1,\n    .lazyblock-content-Z10a0f4 h2,\n    .lazyblock-content-Z10a0f4 h3,\n    .lazyblock-content-Z10a0f4 h4,\n    .lazyblock-content-Z10a0f4 h5 {\n        color: ;\n    }\n\n    #content-Z10a0f4 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z10a0f4 {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z10a0f4 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z10a0f4-container  px-0 \"\n     id=\"content-Z10a0f4\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z10a0f4\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Standard deviation is a measure of variability or dispersion within a <\/span><a href=\"https:\/\/pointerpro.com\/blog\/quantitative-vs-qualitative-digital-marketing\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">quantitative dataset<\/span><\/a><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">While sample and population variances also measure this spread, the standard deviation is expressed in the same units as the original data in the statistical population, making it far easier to interpret.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<ul class=\"lazyblocks-icon-list-items\" id=\"icon-list-Z240EKk\">\n            <li class=\"lazyblocks-icon-list-item\">\n            <span class=\"lazyblocks-icon-list-icon\"><img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/bullet-orange-150x150-1.png\" alt=\"bullet orange 150x150 1\"><\/span>\n            <span class=\"lazyblocks-icon-list-text\">\n                <p><b>A high SD<\/b><span style=\"font-weight: 400;\"> indicates data points are widely scattered around the mean, suggesting greater diversity or volatility in the dataset.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Let\u2019s stick with Haley, the HR consultant, and her training needs assessment. Imagine she collects survey data on the employees\u2019 preferences for specific types of training programs, such as leadership, communication, and technical skills.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">After calculating the average rating for each training topic, Haley finds that the <\/span><b>standard deviation for leadership training<\/b><span style=\"font-weight: 400;\"> is <\/span><b>high<\/b><span style=\"font-weight: 400;\">. This means that while some employees rated leadership training as crucial, others felt it wasn\u2019t necessary at all. There\u2019s a significant spread in how people view the importance of leadership training, indicating greater diversity or volatility in the responses.<\/span><\/p>            <\/span>\n                    <\/li>\n            <li class=\"lazyblocks-icon-list-item\">\n            <span class=\"lazyblocks-icon-list-icon\"><img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/bullet-orange-150x150-1.png\" alt=\"bullet orange 150x150 1\"><\/span>\n            <span class=\"lazyblocks-icon-list-text\">\n                <p><b>A low SD <\/b><span style=\"font-weight: 400;\">indicates data points are clustered closely around the mean, suggesting a more uniform and predictable dataset.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Returning to Haley, the HR consultant, let\u2019s say she gathered responses from employees on their training needs. After analyzing the data, she finds that the <\/span><b>standard deviation for technical skills training<\/b><span style=\"font-weight: 400;\"> is <\/span><b>low<\/b><span style=\"font-weight: 400;\">. This indicates that most employees rated technical training programs similarly, with responses clustering around the average score.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">For Haley, this low standard deviation means she can confidently focus on offering a standard technical training program, knowing that most employees will benefit from the same content. There\u2019s less need for customization, as the data shows a shared understanding and agreement about what training is most valuable.<\/span><\/p>            <\/span>\n                    <\/li>\n    <\/ul>\n\n\n<style>\n    .lazyblock-content-Z1onYLc p,\n    .lazyblock-content-Z1onYLc p a,\n    .lazyblock-content-Z1onYLc h1,\n    .lazyblock-content-Z1onYLc h2,\n    .lazyblock-content-Z1onYLc h3,\n    .lazyblock-content-Z1onYLc h4,\n    .lazyblock-content-Z1onYLc h5 {\n        color: ;\n    }\n\n    #content-Z1onYLc {\n        margin-top: -2rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z1onYLc {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1onYLc {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1onYLc-container  px-0 \"\n     id=\"content-Z1onYLc\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1onYLc\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">By understanding \u200cstandard deviation, we gain valuable insights into the consistency, predictability, and risk associated with the data we\u2019re analyzing.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-nRr0V {\n        padding-top: 2.8rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-nRr0V {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-nRr0V-container  #ffffff custom-padding \" id=\"section-nRr0V\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-6 px-0 custom-padding row\">\n                    \n\n<style>\n    #heading-Z1jjDqW .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-Z1jjDqW {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z1jjDqW {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-Z1jjDqW  px-0\" id=\"heading-Z1jjDqW\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-Z1jjDqW\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"sample-vs-population-standard-deviation-in-data-analysis-s-vs-%cf%83\"><\/a>\n            <h2>Sample vs. population standard deviation in data analysis (S vs. \u03c3)<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-jq0Xt p,\n    .lazyblock-content-jq0Xt p a,\n    .lazyblock-content-jq0Xt h1,\n    .lazyblock-content-jq0Xt h2,\n    .lazyblock-content-jq0Xt h3,\n    .lazyblock-content-jq0Xt h4,\n    .lazyblock-content-jq0Xt h5 {\n        color: ;\n    }\n\n    #content-jq0Xt {\n        margin-top: 1rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-jq0Xt {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-jq0Xt {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-jq0Xt-container  px-0 \"\n     id=\"content-jq0Xt\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-jq0Xt\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Standard deviation is a measure of variability or dispersion within a <\/span><a href=\"https:\/\/pointerpro.com\/blog\/quantitative-vs-qualitative-digital-marketing\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">quantitative dataset<\/span><\/a><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">While sample and population variances also measure this spread, the standard deviation is expressed in the same units as the original data in the statistical population, making it far easier to interpret.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<ul class=\"lazyblocks-icon-list-items\" id=\"icon-list-Z20i7GI\">\n            <li class=\"lazyblocks-icon-list-item\">\n            <span class=\"lazyblocks-icon-list-icon\"><img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/bullet-orange-150x150-1.png\" alt=\"bullet orange 150x150 1\"><\/span>\n            <span class=\"lazyblocks-icon-list-text\">\n                <p><span style=\"font-weight: 400;\">When you have complete information about every individual in a group or dataset, you can calculate the <\/span><b>population standard deviation<\/b><span style=\"font-weight: 400;\">, denoted by \u03c3 (the Greek letter sigma).<\/span><\/p>            <\/span>\n                    <\/li>\n            <li class=\"lazyblocks-icon-list-item\">\n            <span class=\"lazyblocks-icon-list-icon\"><img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/bullet-orange-150x150-1.png\" alt=\"bullet orange 150x150 1\"><\/span>\n            <span class=\"lazyblocks-icon-list-text\">\n                <p><span style=\"font-weight: 400;\">When you\u2019re working with a subset of the population and want to estimate the standard deviation of the entire population, you use the <\/span><b>sample standard deviation<\/b><span style=\"font-weight: 400;\"> denoted by S.<\/span><\/p>            <\/span>\n                    <\/li>\n    <\/ul>\n\n\n<style>\n    .lazyblock-content-AyPSL p,\n    .lazyblock-content-AyPSL p a,\n    .lazyblock-content-AyPSL h1,\n    .lazyblock-content-AyPSL h2,\n    .lazyblock-content-AyPSL h3,\n    .lazyblock-content-AyPSL h4,\n    .lazyblock-content-AyPSL h5 {\n        color: ;\n    }\n\n    #content-AyPSL {\n        margin-top: -2rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-AyPSL {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-AyPSL {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-AyPSL-container  px-0 \"\n     id=\"content-AyPSL\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-AyPSL\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">It\u2019s important to note that it\u2019s common for people to use S and \u03c3 interchangeably, but they\u2019re not the same. When someone doesn\u2019t specify which SD they mean, they usually refer to S, even if they use the symbol \u03c3.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-1bvF39 .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-1bvF39 {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1bvF39 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-1bvF39  px-0\" id=\"heading-1bvF39\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-1bvF39\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"standard-deviation-equations\"><\/a>\n            <h2>Standard deviation equations<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZEaMqE p,\n    .lazyblock-content-ZEaMqE p a,\n    .lazyblock-content-ZEaMqE h1,\n    .lazyblock-content-ZEaMqE h2,\n    .lazyblock-content-ZEaMqE h3,\n    .lazyblock-content-ZEaMqE h4,\n    .lazyblock-content-ZEaMqE h5 {\n        color: ;\n    }\n\n    #content-ZEaMqE {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZEaMqE {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZEaMqE {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZEaMqE-container  px-0 \"\n     id=\"content-ZEaMqE\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZEaMqE\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">There are two standard deviation equations, one for populations and one for samples. However, depending on whether you expand and simplify that formula, each equation can be written out in two ways.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-1K6tj5 .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-1K6tj5 {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1K6tj5 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-1K6tj5  px-0\" id=\"heading-1K6tj5\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-1K6tj5\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"population-standard-deviation-formula-%cf%83\"><\/a>\n            <h3>Population standard deviation formula (\u03c3)<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZGEXit p,\n    .lazyblock-content-ZGEXit p a,\n    .lazyblock-content-ZGEXit h1,\n    .lazyblock-content-ZGEXit h2,\n    .lazyblock-content-ZGEXit h3,\n    .lazyblock-content-ZGEXit h4,\n    .lazyblock-content-ZGEXit h5 {\n        color: ;\n    }\n\n    #content-ZGEXit {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZGEXit {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZGEXit {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZGEXit-container  px-0 \"\n     id=\"content-ZGEXit\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZGEXit\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">The most straightforward way to calculate standard deviation is with its mathematical definition as the square root of the population variance:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-iMvuh {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-iMvuh \" id=\"image-iMvuh\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Population-standard-deviation-formula.jpg\" alt=\"Population standard deviation formula\" title=\"Population standard deviation formula\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-1kQ75S p,\n    .lazyblock-content-1kQ75S p a,\n    .lazyblock-content-1kQ75S h1,\n    .lazyblock-content-1kQ75S h2,\n    .lazyblock-content-1kQ75S h3,\n    .lazyblock-content-1kQ75S h4,\n    .lazyblock-content-1kQ75S h5 {\n        color: ;\n    }\n\n    #content-1kQ75S {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-1kQ75S {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-1kQ75S {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-1kQ75S-container  px-0 \"\n     id=\"content-1kQ75S\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-1kQ75S\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><em><strong><span style=\"font-size: 14px;\">Equation 1: Population standard deviation formula (Image source: made by author)<\/span><\/strong><\/em><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-12qKiC p,\n    .lazyblock-content-12qKiC p a,\n    .lazyblock-content-12qKiC h1,\n    .lazyblock-content-12qKiC h2,\n    .lazyblock-content-12qKiC h3,\n    .lazyblock-content-12qKiC h4,\n    .lazyblock-content-12qKiC h5 {\n        color: ;\n    }\n\n    #content-12qKiC {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-12qKiC {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-12qKiC {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-12qKiC-container  px-0 \"\n     id=\"content-12qKiC\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-12qKiC\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Where:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u03c3 is the population standard deviation<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u03a3 denotes the sum of&#8230;<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">i<\/span><span style=\"font-weight: 400;\"> is each data point<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u03bc is the population mean<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">N is the total number of data points<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-ZQHdFj .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-ZQHdFj {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-ZQHdFj {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-ZQHdFj  px-0\" id=\"heading-ZQHdFj\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-ZQHdFj\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"expanded-population-sd-formula\"><\/a>\n            <h3>Expanded population SD formula<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z21VCJS p,\n    .lazyblock-content-Z21VCJS p a,\n    .lazyblock-content-Z21VCJS h1,\n    .lazyblock-content-Z21VCJS h2,\n    .lazyblock-content-Z21VCJS h3,\n    .lazyblock-content-Z21VCJS h4,\n    .lazyblock-content-Z21VCJS h5 {\n        color: ;\n    }\n\n    #content-Z21VCJS {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z21VCJS {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z21VCJS {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z21VCJS-container  px-0 \"\n     id=\"content-Z21VCJS\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z21VCJS\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">We can expand and simplify the above formula to get a second, more computationally efficient way to calculate \u03c3:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-Zfdz8l {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-Zfdz8l \" id=\"image-Zfdz8l\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Population-standard-deviation-expanded-formula.jpg\" alt=\"Population standard deviation expanded formula\" title=\"Population standard deviation expanded formula\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z1c8WOd p,\n    .lazyblock-content-Z1c8WOd p a,\n    .lazyblock-content-Z1c8WOd h1,\n    .lazyblock-content-Z1c8WOd h2,\n    .lazyblock-content-Z1c8WOd h3,\n    .lazyblock-content-Z1c8WOd h4,\n    .lazyblock-content-Z1c8WOd h5 {\n        color: ;\n    }\n\n    #content-Z1c8WOd {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z1c8WOd {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1c8WOd {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1c8WOd-container  px-0 \"\n     id=\"content-Z1c8WOd\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1c8WOd\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><span style=\"font-size: 14px;\"><strong><em>Equation 2: Population standard deviation expanded formula (Image source: made by author)<\/em><\/strong><\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-293wUI p,\n    .lazyblock-content-293wUI p a,\n    .lazyblock-content-293wUI h1,\n    .lazyblock-content-293wUI h2,\n    .lazyblock-content-293wUI h3,\n    .lazyblock-content-293wUI h4,\n    .lazyblock-content-293wUI h5 {\n        color: ;\n    }\n\n    #content-293wUI {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-293wUI {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-293wUI {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-293wUI-container  px-0 \"\n     id=\"content-293wUI\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-293wUI\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">The variables are the same as above. The most notable difference is that we don\u2019t need the population\u2019s mean value (\u03bc).<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-R46dF .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-R46dF {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-R46dF {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-R46dF  px-0\" id=\"heading-R46dF\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-R46dF\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"sample-standard-deviation-s\"><\/a>\n            <h3>Sample standard deviation (S)<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZVskCe p,\n    .lazyblock-content-ZVskCe p a,\n    .lazyblock-content-ZVskCe h1,\n    .lazyblock-content-ZVskCe h2,\n    .lazyblock-content-ZVskCe h3,\n    .lazyblock-content-ZVskCe h4,\n    .lazyblock-content-ZVskCe h5 {\n        color: ;\n    }\n\n    #content-ZVskCe {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZVskCe {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZVskCe {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZVskCe-container  px-0 \"\n     id=\"content-ZVskCe\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZVskCe\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">For the sample, the SD formula is very similar, with the difference that we subtract 1 from the denominator inside the square root:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-ZHmRL0 {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-ZHmRL0 \" id=\"image-ZHmRL0\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Sample-standard-deviation.jpg\" alt=\"Sample standard deviation\" title=\"Sample standard deviation\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-wD1xx p,\n    .lazyblock-content-wD1xx p a,\n    .lazyblock-content-wD1xx h1,\n    .lazyblock-content-wD1xx h2,\n    .lazyblock-content-wD1xx h3,\n    .lazyblock-content-wD1xx h4,\n    .lazyblock-content-wD1xx h5 {\n        color: ;\n    }\n\n    #content-wD1xx {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-wD1xx {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-wD1xx {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-wD1xx-container  px-0 \"\n     id=\"content-wD1xx\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-wD1xx\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><em><strong><span style=\"font-size: 14px;\">Equation 3: Sample standard deviation formula (Image source: made by author)<\/span><\/strong><\/em><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z19Qfwq p,\n    .lazyblock-content-Z19Qfwq p a,\n    .lazyblock-content-Z19Qfwq h1,\n    .lazyblock-content-Z19Qfwq h2,\n    .lazyblock-content-Z19Qfwq h3,\n    .lazyblock-content-Z19Qfwq h4,\n    .lazyblock-content-Z19Qfwq h5 {\n        color: ;\n    }\n\n    #content-Z19Qfwq {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z19Qfwq {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z19Qfwq {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z19Qfwq-container  px-0 \"\n     id=\"content-Z19Qfwq\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z19Qfwq\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Where:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">S is the sample standard deviation.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">x\u0304 is the sample mean.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">n is the sample size or number of data points<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The other symbols are the same as above.<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-ZBJgBg .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-ZBJgBg {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-ZBJgBg {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-ZBJgBg  px-0\" id=\"heading-ZBJgBg\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-ZBJgBg\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"expanded-sample-sd-formula\"><\/a>\n            <h3>Expanded sample SD formula<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-irH6T p,\n    .lazyblock-content-irH6T p a,\n    .lazyblock-content-irH6T h1,\n    .lazyblock-content-irH6T h2,\n    .lazyblock-content-irH6T h3,\n    .lazyblock-content-irH6T h4,\n    .lazyblock-content-irH6T h5 {\n        color: ;\n    }\n\n    #content-irH6T {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-irH6T {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-irH6T {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-irH6T-container  px-0 \"\n     id=\"content-irH6T\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-irH6T\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">As before, there\u2019s an expanded version of the sample SD formula:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-Z1rcofo {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-Z1rcofo \" id=\"image-Z1rcofo\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Sample-standard-deviation-expanded-formula.jpg\" alt=\"Sample standard deviation expanded formula\" title=\"Sample standard deviation expanded formula\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZOcfNn p,\n    .lazyblock-content-ZOcfNn p a,\n    .lazyblock-content-ZOcfNn h1,\n    .lazyblock-content-ZOcfNn h2,\n    .lazyblock-content-ZOcfNn h3,\n    .lazyblock-content-ZOcfNn h4,\n    .lazyblock-content-ZOcfNn h5 {\n        color: ;\n    }\n\n    #content-ZOcfNn {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZOcfNn {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZOcfNn {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZOcfNn-container  px-0 \"\n     id=\"content-ZOcfNn\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZOcfNn\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><strong><em><span style=\"font-size: 14px;\">Equation 4: Sample standard deviation expanded formula (Image source: made by author)<\/span><\/em><\/strong><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-2c2nlh p,\n    .lazyblock-content-2c2nlh p a,\n    .lazyblock-content-2c2nlh h1,\n    .lazyblock-content-2c2nlh h2,\n    .lazyblock-content-2c2nlh h3,\n    .lazyblock-content-2c2nlh h4,\n    .lazyblock-content-2c2nlh h5 {\n        color: ;\n    }\n\n    #content-2c2nlh {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-2c2nlh {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-2c2nlh {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-2c2nlh-container  px-0 \"\n     id=\"content-2c2nlh\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-2c2nlh\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">The variables are the same as before.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-Z2hedbi .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-Z2hedbi {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z2hedbi {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-Z2hedbi  px-0\" id=\"heading-Z2hedbi\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-Z2hedbi\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"coefficient-of-variation-cv\"><\/a>\n            <h3>Coefficient of variation (CV)<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z3fvRS p,\n    .lazyblock-content-Z3fvRS p a,\n    .lazyblock-content-Z3fvRS h1,\n    .lazyblock-content-Z3fvRS h2,\n    .lazyblock-content-Z3fvRS h3,\n    .lazyblock-content-Z3fvRS h4,\n    .lazyblock-content-Z3fvRS h5 {\n        color: ;\n    }\n\n    #content-Z3fvRS {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z3fvRS {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z3fvRS {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z3fvRS-container  px-0 \"\n     id=\"content-Z3fvRS\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z3fvRS\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Sometimes, expressing standard deviation as a percentage of the mean is helpful. Doing so gives us the relative standard deviation, a.k.a, the coefficient of variation (CV):<\/span><\/p>\n<p><b><i>CV = (\u03c3 \/ \u03bc) * 100%\u00a0 \u00a0 \u00a0 \u00a0 <\/i><\/b><b><i>or\u00a0 \u00a0 \u00a0 \u00a0 CV = (S \/ x\u0304) * 100%<\/i><\/b><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-1ad9DC .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-1ad9DC {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1ad9DC {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-1ad9DC  px-0\" id=\"heading-1ad9DC\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-1ad9DC\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"interpreting-sd-results\"><\/a>\n            <h2>Interpreting SD results<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-1OLiK1 p,\n    .lazyblock-content-1OLiK1 p a,\n    .lazyblock-content-1OLiK1 h1,\n    .lazyblock-content-1OLiK1 h2,\n    .lazyblock-content-1OLiK1 h3,\n    .lazyblock-content-1OLiK1 h4,\n    .lazyblock-content-1OLiK1 h5 {\n        color: ;\n    }\n\n    #content-1OLiK1 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-1OLiK1 {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-1OLiK1 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-1OLiK1-container  px-0 \"\n     id=\"content-1OLiK1\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-1OLiK1\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">The standard deviation isn\u2019t just a number\u200c \u2014 \u200cit\u2019s a powerful tool for drawing meaningful conclusions from your data. Here\u2019s how you can interpret standard deviation in different contexts:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Comparing datasets<\/b><span style=\"font-weight: 400;\">: If two datasets have the same mean but different standard deviations, the one with the higher standard deviation has more variability.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Identifying outliers<\/b><span style=\"font-weight: 400;\">: Data points that fall more than two or three standard deviations from the mean are often considered outliers and may warrant further investigation.<br \/><br \/><em><span>Let\u2019s say Haley, the HR consultant, analyzes the results of a training needs survey. After reviewing the data, she notices the average rating for <\/span><b>communication skills training<\/b><span> is around 4 out of 5, with a <\/span><b>low standard deviation<\/b><span> indicating most employees are in agreement about its importance.<br \/><br \/><\/span><\/em><\/span><em><span>However, when she looks more closely, she spots a couple of responses far outside the general trend. One employee rated communication training as a 1 (very low importance), while another gave it a perfect 5 (extremely important), even though the majority of responses are clustered around the 4.<br \/><br \/><\/span><\/em><em><span>This is where <\/span><b>standard deviation<\/b><span> helps. These unusually low and high ratings are considered <\/span><b>outliers<\/b><span>, as they fall far away from the average score. With the standard deviation, Haley can easily identify these extreme data points that don&#8217;t align with the overall pattern.<br \/><br \/><\/span><\/em><em><span>By flagging these outliers, Haley can dive deeper into understanding why these individuals have differing views. It could highlight specific needs or concerns that may not be immediately obvious to the rest of the group. Maybe the employee who rated communication training as a 1 has a different job function where communication isn\u2019t as critical, or the one who gave it a 5 might be in a customer-facing role where communication skills are vital. <br \/><br \/><\/span><\/em><em><span>Using standard deviation to identify these outliers allows Haley to address individual concerns or explore specific areas where training programs might need further refinement.<br \/><br \/><\/span><\/em><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Process control<\/b><span style=\"font-weight: 400;\">: In <\/span><a href=\"https:\/\/www.growth-hackers.net\/best-manufacturing-marketing-agencies\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">manufacturing<\/span><\/a><span style=\"font-weight: 400;\"> or quality control, standard deviation helps track process variability and identify deviations from desired standards.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Financial analysis<\/b><span>: Standard deviation measures the volatility and deviation of stock returns, bonds, and other financial instruments.<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-2hQu4N {\n        padding-top: 2rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-2hQu4N {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-2hQu4N-container  #ffffff custom-padding \" id=\"section-2hQu4N\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-6 px-0 custom-padding row\">\n                    \n\n<style>\n    #heading-1241dU .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-1241dU {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1241dU {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-1241dU  px-0\" id=\"heading-1241dU\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-1241dU\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"strengths-and-weaknesses-of-standard-deviation-as-a-measure-of-dispersion\"><\/a>\n            <h2>Strengths and weaknesses of standard deviation as a measure of dispersion<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-1GbOJm p,\n    .lazyblock-content-1GbOJm p a,\n    .lazyblock-content-1GbOJm h1,\n    .lazyblock-content-1GbOJm h2,\n    .lazyblock-content-1GbOJm h3,\n    .lazyblock-content-1GbOJm h4,\n    .lazyblock-content-1GbOJm h5 {\n        color: ;\n    }\n\n    #content-1GbOJm {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-1GbOJm {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-1GbOJm {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-1GbOJm-container  px-0 \"\n     id=\"content-1GbOJm\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-1GbOJm\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Like any statistical tool, standard deviation has its strengths and weaknesses.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-Z2ho0YW .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-Z2ho0YW {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z2ho0YW {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-Z2ho0YW  px-0\" id=\"heading-Z2ho0YW\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-Z2ho0YW\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"strengths-of-sd-for-data-analysis\"><\/a>\n            <h3>Strengths of SD for data analysis<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z12ifNf p,\n    .lazyblock-content-Z12ifNf p a,\n    .lazyblock-content-Z12ifNf h1,\n    .lazyblock-content-Z12ifNf h2,\n    .lazyblock-content-Z12ifNf h3,\n    .lazyblock-content-Z12ifNf h4,\n    .lazyblock-content-Z12ifNf h5 {\n        color: ;\n    }\n\n    #content-Z12ifNf {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z12ifNf {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z12ifNf {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z12ifNf-container  px-0 \"\n     id=\"content-Z12ifNf\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z12ifNf\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Widely used and understood<\/b><span style=\"font-weight: 400;\">: Standard deviation is a well-established measure used across numerous fields, making it easy to communicate your findings.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>It helps identify outliers<\/b><span style=\"font-weight: 400;\">: It reflects the impact of extreme values, providing a more complete picture of the data\u2019s variability.<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-1WAp4E .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-1WAp4E {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1WAp4E {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-1WAp4E  px-0\" id=\"heading-1WAp4E\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-1WAp4E\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"weaknesses-of-standard-deviation-for-data-analysis\"><\/a>\n            <h3>Weaknesses of standard deviation for data analysis<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-nFeeU p,\n    .lazyblock-content-nFeeU p a,\n    .lazyblock-content-nFeeU h1,\n    .lazyblock-content-nFeeU h2,\n    .lazyblock-content-nFeeU h3,\n    .lazyblock-content-nFeeU h4,\n    .lazyblock-content-nFeeU h5 {\n        color: ;\n    }\n\n    #content-nFeeU {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-nFeeU {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-nFeeU {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-nFeeU-container  px-0 \"\n     id=\"content-nFeeU\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-nFeeU\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <ul>\n<li><b>Assumption of normality<\/b><span style=\"font-weight: 400;\">: It\u2019s most effective when the data follows a normal distribution. Other measures may be more appropriate for skewed or non-normal probability distributions<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-ZAmlUf {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-ZAmlUf \" id=\"image-ZAmlUf\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Normal-distribution.jpg\" alt=\"Normal distribution\" title=\"Normal distribution\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-2jwBWK p,\n    .lazyblock-content-2jwBWK p a,\n    .lazyblock-content-2jwBWK h1,\n    .lazyblock-content-2jwBWK h2,\n    .lazyblock-content-2jwBWK h3,\n    .lazyblock-content-2jwBWK h4,\n    .lazyblock-content-2jwBWK h5 {\n        color: ;\n    }\n\n    #content-2jwBWK {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-2jwBWK {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-2jwBWK {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-2jwBWK-container  px-0 \"\n     id=\"content-2jwBWK\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-2jwBWK\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\">Source: https:\/\/integratedmlai.com\/normal-distribution-an-introductory-guide-to-pdf-and-cdf\/<\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-2p2EDf p,\n    .lazyblock-content-2p2EDf p a,\n    .lazyblock-content-2p2EDf h1,\n    .lazyblock-content-2p2EDf h2,\n    .lazyblock-content-2p2EDf h3,\n    .lazyblock-content-2p2EDf h4,\n    .lazyblock-content-2p2EDf h5 {\n        color: ;\n    }\n\n    #content-2p2EDf {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-2p2EDf {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-2p2EDf {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-2p2EDf-container  px-0 \"\n     id=\"content-2p2EDf\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-2p2EDf\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <ul>\n<li><b>Sensitivity to outliers<\/b><span style=\"font-weight: 400;\">: While this is a strength, it can also be a weakness if extreme values are due to errors that skew the data.<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-ZzQSTc {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-ZzQSTc \" id=\"image-ZzQSTc\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Score.jpg\" alt=\"Score\" title=\"Score\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z1ePGxv p,\n    .lazyblock-content-Z1ePGxv p a,\n    .lazyblock-content-Z1ePGxv h1,\n    .lazyblock-content-Z1ePGxv h2,\n    .lazyblock-content-Z1ePGxv h3,\n    .lazyblock-content-Z1ePGxv h4,\n    .lazyblock-content-Z1ePGxv h5 {\n        color: ;\n    }\n\n    #content-Z1ePGxv {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z1ePGxv {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1ePGxv {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1ePGxv-container  px-0 \"\n     id=\"content-Z1ePGxv\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1ePGxv\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><a href=\"https:\/\/mathspace.co\/textbooks\/syllabuses\/Syllabus-1159\/topics\/Topic-21945\/subtopics\/Subtopic-280027\/\" target=\"_blank\" rel=\"noopener\">Source<\/a><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-ZScnw3 .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-ZScnw3 {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-ZScnw3 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-ZScnw3  px-0\" id=\"heading-ZScnw3\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-ZScnw3\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"calculating-the-standard-deviation-an-example\"><\/a>\n            <h2>Calculating the standard deviation &#8211; An example<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-1tUeFE p,\n    .lazyblock-content-1tUeFE p a,\n    .lazyblock-content-1tUeFE h1,\n    .lazyblock-content-1tUeFE h2,\n    .lazyblock-content-1tUeFE h3,\n    .lazyblock-content-1tUeFE h4,\n    .lazyblock-content-1tUeFE h5 {\n        color: ;\n    }\n\n    #content-1tUeFE {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-1tUeFE {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-1tUeFE {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-1tUeFE-container  px-0 \"\n     id=\"content-1tUeFE\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-1tUeFE\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">To illustrate the practical application of statistical analysis using standard deviation, let\u2019s do an example calculation of SD in data center management.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Standard deviation is a powerful tool that could be applied for data center management and optimization. This measure serves software like Nlyte to analyze:\u00a0<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Identify servers consuming excessive or insufficient power<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Analyze power usage and consumption across servers<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Monitor for temperature fluctuations<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Understand workload distribution<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Let\u2019s consider a scenario where we\u2019re monitoring the real-time power consumption (in watts) of 20 servers in a data center:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #section-1D4WK8 {\n        padding-top: 1rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-1D4WK8 {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-1D4WK8-container  #ffffff custom-padding \" id=\"section-1D4WK8\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-12 px-0 custom-padding row\">\n                    \n\n<style>\n    .lazyblock-content-ZOfBjH p,\n    .lazyblock-content-ZOfBjH p a,\n    .lazyblock-content-ZOfBjH h1,\n    .lazyblock-content-ZOfBjH h2,\n    .lazyblock-content-ZOfBjH h3,\n    .lazyblock-content-ZOfBjH h4,\n    .lazyblock-content-ZOfBjH h5 {\n        color: ;\n    }\n\n    #content-ZOfBjH {\n        margin-top: 0rem;\n        margin-bottom: -6rem;\n    }\n\n    #content-ZOfBjH {\n        margin-left: 0rem;\n        margin-right: 0rem;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZOfBjH {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZOfBjH-container   \"\n     id=\"content-ZOfBjH\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZOfBjH\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <table class=\" aligncenter alignleft\" style=\"border-collapse: collapse; width: 100%; height: auto;\">\n<thead>\n<tr>\n<td>\n<p><strong>Server\u00a0<\/strong><\/p>\n<\/td>\n<td scope=\"rowgroup\">\n<p><strong>Power<br \/>Consumption<br \/>(Watts)<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Server\u00a0<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Power <\/strong><br \/><strong>Consumption <\/strong><br \/><strong>(Watts)<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">1<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">350<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">11<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">350<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">2<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">365<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">12<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">385<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">3<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">340<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">13<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">340<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">4<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">380<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">14<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">395<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">5<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">355<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">15<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">365<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">6<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">370<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">16<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">370<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">7<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">345<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">17<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">345<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">8<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">390<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">18<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">390<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">9<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">360<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">19<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">355<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 5px; vertical-align: middle; height: 5px;\"><span class=\"OYPEnA font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">10<\/span><\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">375<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">20<\/span><\/p>\n<\/td>\n<td style=\"width: 5px; height: 5px;\">\n<p><span style=\"font-weight: 400;\">380<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    .lazyblock-content-24diic p,\n    .lazyblock-content-24diic p a,\n    .lazyblock-content-24diic h1,\n    .lazyblock-content-24diic h2,\n    .lazyblock-content-24diic h3,\n    .lazyblock-content-24diic h4,\n    .lazyblock-content-24diic h5 {\n        color: ;\n    }\n\n    #content-24diic {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-24diic {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-24diic {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-24diic-container  px-0 \"\n     id=\"content-24diic\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-24diic\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p>Now, let\u2019s calculate the SD using both versions of the equations outlined above. As is normally the case, we\u2019ll use equations 3 and 4 for the sample standard deviation.<\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-Z2hnmeI .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-Z2hnmeI {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z2hnmeI {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-Z2hnmeI  px-0\" id=\"heading-Z2hnmeI\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-Z2hnmeI\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"how-to-calculate-standard-deviation-the-usual-way\"><\/a>\n            <h2>How to calculate standard deviation the usual way\u200c<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z1k4TlJ p,\n    .lazyblock-content-Z1k4TlJ p a,\n    .lazyblock-content-Z1k4TlJ h1,\n    .lazyblock-content-Z1k4TlJ h2,\n    .lazyblock-content-Z1k4TlJ h3,\n    .lazyblock-content-Z1k4TlJ h4,\n    .lazyblock-content-Z1k4TlJ h5 {\n        color: ;\n    }\n\n    #content-Z1k4TlJ {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z1k4TlJ {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1k4TlJ {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1k4TlJ-container  px-0 \"\n     id=\"content-Z1k4TlJ\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1k4TlJ\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">If you want to practice using the SD equation for a dataset like the one shown above, here\u2019s what you must do:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-Z1mwcWo .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-Z1mwcWo {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z1mwcWo {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-Z1mwcWo  px-0\" id=\"heading-Z1mwcWo\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-Z1mwcWo\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-1-calculate-the-mean-x\"><\/a>\n            <h3>Step #1: Calculate the mean (x\u0304)<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z2726jE p,\n    .lazyblock-content-Z2726jE p a,\n    .lazyblock-content-Z2726jE h1,\n    .lazyblock-content-Z2726jE h2,\n    .lazyblock-content-Z2726jE h3,\n    .lazyblock-content-Z2726jE h4,\n    .lazyblock-content-Z2726jE h5 {\n        color: ;\n    }\n\n    #content-Z2726jE {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z2726jE {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z2726jE {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z2726jE-container  px-0 \"\n     id=\"content-Z2726jE\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z2726jE\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Add all power consumption values and divide by the total number of servers (20). Since this is a sample, x\u0304 will denote this mean:<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">x\u0304 = (350 + 365 + &#8230; + 380) \/ 20 = 365.3<\/span><\/i><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-ZahYvl .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-ZahYvl {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-ZahYvl {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-ZahYvl  px-0\" id=\"heading-ZahYvl\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-ZahYvl\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-2-calculate-the-deviations-from-the-mean-xi-x\"><\/a>\n            <h3>Step #2: Calculate the deviations from the mean (xi-x\u0304)<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Zv0oPj p,\n    .lazyblock-content-Zv0oPj p a,\n    .lazyblock-content-Zv0oPj h1,\n    .lazyblock-content-Zv0oPj h2,\n    .lazyblock-content-Zv0oPj h3,\n    .lazyblock-content-Zv0oPj h4,\n    .lazyblock-content-Zv0oPj h5 {\n        color: ;\n    }\n\n    #content-Zv0oPj {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Zv0oPj {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Zv0oPj {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Zv0oPj-container  px-0 \"\n     id=\"content-Zv0oPj\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Zv0oPj\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Subtract the mean from each power consumption value:<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">x<\/span><\/i><i><span style=\"font-weight: 400;\">1<\/span><\/i><i><span style=\"font-weight: 400;\"> &#8211; x\u0304 = 350 \u2013 365.3 = -15.3<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">x<\/span><\/i><i><span style=\"font-weight: 400;\">2<\/span><\/i><i><span style=\"font-weight: 400;\"> &#8211; x\u0304 = 365 \u2013 365.3 = -0.3<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">&#8230;<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">x<\/span><\/i><i><span style=\"font-weight: 400;\">20<\/span><\/i><i><span style=\"font-weight: 400;\"> &#8211; x\u0304 = 380 \u2013 365.3 = 9.8<\/span><\/i><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-lXQcM .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-lXQcM {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-lXQcM {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-lXQcM  px-0\" id=\"heading-lXQcM\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-lXQcM\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-3-calculate-the-squared-deviations\"><\/a>\n            <h3>Step #3: Calculate the squared deviations<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Zt3x7g p,\n    .lazyblock-content-Zt3x7g p a,\n    .lazyblock-content-Zt3x7g h1,\n    .lazyblock-content-Zt3x7g h2,\n    .lazyblock-content-Zt3x7g h3,\n    .lazyblock-content-Zt3x7g h4,\n    .lazyblock-content-Zt3x7g h5 {\n        color: ;\n    }\n\n    #content-Zt3x7g {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Zt3x7g {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Zt3x7g {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Zt3x7g-container  px-0 \"\n     id=\"content-Zt3x7g\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Zt3x7g\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Square each of the differences you just calculated to get the squared deviations. Here\u2019s what we \u200chave so far:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #section-1m8Ftm {\n        padding-top: 1rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-1m8Ftm {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-1m8Ftm-container  #ffffff custom-padding \" id=\"section-1m8Ftm\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-12 px-0 custom-padding row\">\n                    \n\n<style>\n    .lazyblock-content-Z1O8iuH p,\n    .lazyblock-content-Z1O8iuH p a,\n    .lazyblock-content-Z1O8iuH h1,\n    .lazyblock-content-Z1O8iuH h2,\n    .lazyblock-content-Z1O8iuH h3,\n    .lazyblock-content-Z1O8iuH h4,\n    .lazyblock-content-Z1O8iuH h5 {\n        color: ;\n    }\n\n    #content-Z1O8iuH {\n        margin-top: 0rem;\n        margin-bottom: -6rem;\n    }\n\n    #content-Z1O8iuH {\n        margin-left: 0rem;\n        margin-right: 0rem;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1O8iuH {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1O8iuH-container   \"\n     id=\"content-Z1O8iuH\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1O8iuH\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <table class=\" aligncenter\" style=\"border-collapse: collapse; width: 100%; height: auto;\">\n<thead>\n<tr>\n<td style=\"width: 81px;\">\n<p><b>Server (i)<\/b><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><b>xi<\/b><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><b>xi &#8211; x\u0304<\/b><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><b>(xi &#8211; x\u0304)\u00b2<\/b><\/p>\n<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">1<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">350<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-15.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">232.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">2<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">365<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-0.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">0.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">3<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">340<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-25.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">637.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">4<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">380<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">14.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">217.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">5<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">355<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-10.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">105.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">6<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">370<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">4.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">22.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">7<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">345<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-20.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">410.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">8<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">390<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">24.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">612.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">9<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">360<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-5.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">27.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">10<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">375<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">9.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">95.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">11<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">350<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-15.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">232.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">12<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">385<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">19.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">390.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">13<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">340<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-25.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">637.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">14<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">395<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">29.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">885.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">15<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">365<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-0.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">0.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">16<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">370<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">4.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">22.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">17<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">345<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-20.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">410.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">18<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">390<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">24.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">612.6<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">19<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">355<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">-10.3<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">105.1<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">\n<p><span style=\"font-weight: 400;\">20<\/span><\/p>\n<\/td>\n<td style=\"width: 31px;\">\n<p><span style=\"font-weight: 400;\">380<\/span><\/p>\n<\/td>\n<td style=\"width: 41px;\">\n<p><span style=\"font-weight: 400;\">14.8<\/span><\/p>\n<\/td>\n<td style=\"width: 63px;\">\n<p><span style=\"font-weight: 400;\">217.6<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #heading-1Jqelw .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-1Jqelw {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1Jqelw {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-1Jqelw  px-0\" id=\"heading-1Jqelw\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-1Jqelw\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-4-find-the-average-of-the-squared-differences\"><\/a>\n            <h3>Step #4: Find the average of the squared differences<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZUcGC6 p,\n    .lazyblock-content-ZUcGC6 p a,\n    .lazyblock-content-ZUcGC6 h1,\n    .lazyblock-content-ZUcGC6 h2,\n    .lazyblock-content-ZUcGC6 h3,\n    .lazyblock-content-ZUcGC6 h4,\n    .lazyblock-content-ZUcGC6 h5 {\n        color: ;\n    }\n\n    #content-ZUcGC6 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZUcGC6 {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZUcGC6 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZUcGC6-container  px-0 \"\n     id=\"content-ZUcGC6\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZUcGC6\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Add up all the squared differences (in the last column), and divide by the total number of servers minus 1 (if you\u2019re calculating \u03c3, you don\u2019t need to subtract 1). The result is the sample variance, S\u00b2:<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">(-15.3)\u00b2 + (-0.3)\u00b2 + &#8230; + (14.8)\u00b2 = 5873.8<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">S\u00b2 = 5873.8 \/ (20 \u2013 1) = 309.1 W\u00b2<\/span><\/i><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-272Sxw .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-272Sxw {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-272Sxw {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-272Sxw  px-0\" id=\"heading-272Sxw\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-272Sxw\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-5-take-the-square-root-of-the-variance\"><\/a>\n            <h3>Step #5: Take the square root of the variance<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-l4h68 p,\n    .lazyblock-content-l4h68 p a,\n    .lazyblock-content-l4h68 h1,\n    .lazyblock-content-l4h68 h2,\n    .lazyblock-content-l4h68 h3,\n    .lazyblock-content-l4h68 h4,\n    .lazyblock-content-l4h68 h5 {\n        color: ;\n    }\n\n    #content-l4h68 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-l4h68 {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-l4h68 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-l4h68-container  px-0 \"\n     id=\"content-l4h68\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-l4h68\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><i><span style=\"font-weight: 400;\">S = \u221a(S\u00b2) = \u221a309.1 = 17.6 W<\/span><\/i><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-ZSqo8D .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-ZSqo8D {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-ZSqo8D {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-ZSqo8D  px-0\" id=\"heading-ZSqo8D\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-ZSqo8D\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"how-to-calculate-sample-standard-deviation-the-easy-way-a-step-by-step-guide\"><\/a>\n            <h2>How to calculate sample standard deviation the easy way: a step-by-step guide<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z16Uzqk p,\n    .lazyblock-content-Z16Uzqk p a,\n    .lazyblock-content-Z16Uzqk h1,\n    .lazyblock-content-Z16Uzqk h2,\n    .lazyblock-content-Z16Uzqk h3,\n    .lazyblock-content-Z16Uzqk h4,\n    .lazyblock-content-Z16Uzqk h5 {\n        color: ;\n    }\n\n    #content-Z16Uzqk {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z16Uzqk {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z16Uzqk {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z16Uzqk-container  px-0 \"\n     id=\"content-Z16Uzqk\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z16Uzqk\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Now, let\u2019s do the same calculation but use the simplified expanded formula to see how it makes calculation easier.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-1PM9kn .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-1PM9kn {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1PM9kn {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-1PM9kn  px-0\" id=\"heading-1PM9kn\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-1PM9kn\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-1-square-each-value\"><\/a>\n            <h3>Step #1: Square each value<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-2fsuyp p,\n    .lazyblock-content-2fsuyp p a,\n    .lazyblock-content-2fsuyp h1,\n    .lazyblock-content-2fsuyp h2,\n    .lazyblock-content-2fsuyp h3,\n    .lazyblock-content-2fsuyp h4,\n    .lazyblock-content-2fsuyp h5 {\n        color: ;\n    }\n\n    #content-2fsuyp {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-2fsuyp {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-2fsuyp {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-2fsuyp-container  px-0 \"\n     id=\"content-2fsuyp\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-2fsuyp\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><i><span style=\"font-weight: 400;\">x<\/span><\/i><i><span style=\"font-weight: 400;\">1<\/span><\/i><i><span style=\"font-weight: 400;\">\u00b2\u00a0 = 350\u00b2 = 122,500<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">x<\/span><\/i><i><span style=\"font-weight: 400;\">2<\/span><\/i><i><span style=\"font-weight: 400;\">\u00b2\u00a0 = 365\u00b2 = 133,225<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">&#8230;<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">x<\/span><\/i><i><span style=\"font-weight: 400;\">20<\/span><\/i><i><span style=\"font-weight: 400;\">\u00b2\u00a0 = 380\u00b2 = 144,400<\/span><\/i><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-2ij6Ef .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-2ij6Ef {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-2ij6Ef {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-2ij6Ef  px-0\" id=\"heading-2ij6Ef\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-2ij6Ef\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-2-find-the-sum-of-the-squares\"><\/a>\n            <h3>Step #2: Find the sum of the squares<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z13RHmT p,\n    .lazyblock-content-Z13RHmT p a,\n    .lazyblock-content-Z13RHmT h1,\n    .lazyblock-content-Z13RHmT h2,\n    .lazyblock-content-Z13RHmT h3,\n    .lazyblock-content-Z13RHmT h4,\n    .lazyblock-content-Z13RHmT h5 {\n        color: ;\n    }\n\n    #content-Z13RHmT {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z13RHmT {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z13RHmT {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z13RHmT-container  px-0 \"\n     id=\"content-Z13RHmT\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z13RHmT\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Add up all the squared values:<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">\u03a3xi\u00b2 = 122,500 + 133,225 + &#8230; + 144,400 = 2,674,025 W\u00b2<\/span><\/i><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-UO6St .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-UO6St {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-UO6St {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-UO6St  px-0\" id=\"heading-UO6St\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-UO6St\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-3-find-the-sum-of-the-original-values\"><\/a>\n            <h3>Step #3: Find the sum of the original values<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z228qI3 p,\n    .lazyblock-content-Z228qI3 p a,\n    .lazyblock-content-Z228qI3 h1,\n    .lazyblock-content-Z228qI3 h2,\n    .lazyblock-content-Z228qI3 h3,\n    .lazyblock-content-Z228qI3 h4,\n    .lazyblock-content-Z228qI3 h5 {\n        color: ;\n    }\n\n    #content-Z228qI3 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z228qI3 {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z228qI3 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z228qI3-container  px-0 \"\n     id=\"content-Z228qI3\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z228qI3\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><i><span style=\"font-weight: 400;\">\u03a3xi = 350 + 365 + &#8230; + 380 = 7,305 W<\/span><\/i><\/p>\n<p><span style=\"font-weight: 400;\">This is what we would have so far:<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #section-21PtQq {\n        padding-top: 1rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-21PtQq {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-21PtQq-container  #ffffff custom-padding \" id=\"section-21PtQq\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-12 px-0 custom-padding row\">\n                    \n\n<style>\n    .lazyblock-content-ZolC5M p,\n    .lazyblock-content-ZolC5M p a,\n    .lazyblock-content-ZolC5M h1,\n    .lazyblock-content-ZolC5M h2,\n    .lazyblock-content-ZolC5M h3,\n    .lazyblock-content-ZolC5M h4,\n    .lazyblock-content-ZolC5M h5 {\n        color: ;\n    }\n\n    #content-ZolC5M {\n        margin-top: 0rem;\n        margin-bottom: -6rem;\n    }\n\n    #content-ZolC5M {\n        margin-left: 0rem;\n        margin-right: 0rem;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZolC5M {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZolC5M-container   \"\n     id=\"content-ZolC5M\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZolC5M\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <table class=\" aligncenter\" style=\"border-collapse: collapse; width: 100%; height: auto;\">\n<thead>\n<tr>\n<td style=\"width: 30%;\">\n<p><b>Server (i)<\/b><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><b>xi<\/b><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><b>xi\u00b2<\/b><\/p>\n<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">1<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">350<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">122,500<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">2<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">365<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">133,225<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">3<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">340<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">115,600<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">4<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">380<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">144,400<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">5<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">355<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">126,025<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">6<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">370<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">136,900<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">7<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">345<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">119,025<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">8<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">390<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">152,100<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">9<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">360<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">129,600<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">10<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">375<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">140,625<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">11<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">350<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">122,500<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">12<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">385<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">148,225<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">13<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">340<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">115,600<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">14<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">395<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">156,025<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">15<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">365<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">133,225<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">16<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">370<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">136,900<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">17<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">345<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">119,025<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">18<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">390<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">152,100<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">19<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">355<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">126,025<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><span style=\"font-weight: 400;\">20<\/span><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><span style=\"font-weight: 400;\">380<\/span><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><span style=\"font-weight: 400;\">144,400<\/span><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30%;\">\n<p><b>Sum<\/b><\/p>\n<\/td>\n<td style=\"width: 25.2632%;\">\n<p><b>7,305<\/b><\/p>\n<\/td>\n<td style=\"width: 44.2105%;\">\n<p><b>2,674,025<\/b><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\">\u00a0<\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #heading-Z1wFg6q .heading h3 {\n        color: #421F3C;\n        font-size: 1.75rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-Z1wFg6q {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z1wFg6q {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n<div class=\"container-md lazyblock-heading-3-container lazyblock-heading-3-Z1wFg6q  px-0\" id=\"heading-Z1wFg6q\">\n    <div class=\"row heading lazyblock-heading-3 justify-content-start  lazyblock-heading-3-Z1wFg6q\">\n        <div class=\"col-12 col-lg-10 text-start\">\n            <a class=\"heading-3-anchor heading-anchor\" id=\"step-4-apply-the-expanded-formula\"><\/a>\n            <h3>Step #4: Apply the expanded formula<\/h3>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z25Q1Tv p,\n    .lazyblock-content-Z25Q1Tv p a,\n    .lazyblock-content-Z25Q1Tv h1,\n    .lazyblock-content-Z25Q1Tv h2,\n    .lazyblock-content-Z25Q1Tv h3,\n    .lazyblock-content-Z25Q1Tv h4,\n    .lazyblock-content-Z25Q1Tv h5 {\n        color: ;\n    }\n\n    #content-Z25Q1Tv {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z25Q1Tv {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z25Q1Tv {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z25Q1Tv-container  px-0 \"\n     id=\"content-Z25Q1Tv\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z25Q1Tv\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Here again, we\u2019ll use <\/span><b><i>(n \u2013 1)<\/i><\/b><span style=\"font-weight: 400;\"> as the denominator inside the square root since we\u2019re using a sample:<\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">S = \u221a {[2,674,025 &#8211; (7,305)\u00b2\/20)]\/(20 \u2013 1)} = 17.6 W<\/span><\/i><\/p>\n<p><span style=\"font-weight: 400;\">Both methods yield the same result, but the latter requires roughly half the calculations.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-Z1BF4v8 {\n        padding-top: 4rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-Z1BF4v8 {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #421f3c;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-Z1BF4v8-container  #421f3c custom-padding \" id=\"section-Z1BF4v8\"\n    style=\"opacity: 100%; background-color: #421f3c;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-12 px-0 custom-padding row\">\n                    \n<div class=\"container-md\">\n    <div class=\"row lazyblock-content-content align-items-start\">\n\n        <div class=\"col-12 col-lg-6 order-1\">\n            <h2><span style=\"color: #ff7418; font-size: 32px;\">Wow customers with automated, scored reports\u200b<\/span><\/h2>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #ffffff;\">Here&#8217;s a quick introduction on how Pointerpro works, brought to you by one of our product experts, Chris.<\/span><\/p>\n        <\/div>\n\n        <div class=\"col-12 col-lg-6 order-2\">\n            <p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4yDn1mhRrOg?si=wkhSscEvw_-gFezd\" width=\"100%\" height=\"500\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-Z1NcTK2 .heading h2 {\n        color: #ffffff;\n        font-size: 2.5rem;\n        line-height: 2.5rem !important;\n    }\n\n    #heading-Z1NcTK2 {\n        margin-top: 2rem;\n        margin-bottom: 3rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z1NcTK2 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-Z1NcTK2  px-0\" id=\"heading-Z1NcTK2\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-center  lazyblock-heading-2-Z1NcTK2\">\n        <div class=\"col-12 col-lg-10 text-center\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"this-is-what-clients-say-about-us\"><\/a>\n            <h2>This is what clients say about us:\u200b<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #lazyblock-carousel-1yeeEl {\n        padding-top: 0rem;\n        padding-bottom: 2.8rem;\n    }\n<\/style>\n\n<script>\n    jQuery(document).ready(function ($) {\n        \/\/ Get the specific carousel instance using unique ID\n        var $carousel = $('#lazyblock-carousel-1yeeEl .pointerpro-testimonial-carousel-inner-wrapper');\n\n        if (!$carousel.hasClass('slick-initialized')) {\n            $carousel.slick({\n                infinite: true,\n                slidesToShow: 1,\n                slidesToScroll: 1,\n                autoplay: true,\n                autoplaySpeed: 8000,\n                adaptiveHeight: true,\n                dots: true,\n                responsive: [\n                    {\n                        breakpoint: 768,\n                        settings: {\n                            slidesToShow: 1,\n                            slidesToScroll: 1\n                        }\n                    }\n                ],\n                arrows: false,\n                prevArrow: '<i class=\"fas fa-arrow-left prev\"><\/i>',\n                nextArrow: '<i class=\"fas fa-arrow-right next\"><\/i>',\n            });\n        }\n    });\n<\/script>\n\n<div class=\"container-md lazyblock-testimonial-carousel-container lazyblock-testimonial-carousel-1yeeEl-container \"\n     id=\"lazyblock-carousel-1yeeEl\">\n    <div class=\"row mx-0 lazyblock-testimonial-carousel justify-content-center carousel-wrapper lazyblock-testimonial-carousel-1yeeEl\">\n        <div class=\"col-12 col-lg-10 px-0 pointerpro-testimonial-carousel-inner-wrapper\">\n                            <div>\n                    <p>&#8220;We use Pointerpro for all types of surveys and assessments across our global business, and employees love its ease of use and flexible reporting.&#8221;<\/p>                                            <img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Jim-McLean-Alere-1.jpg\" alt=\"Jim McLean Alere 1\">\n                                        <p class=\"testimonial-author\">Jim McLean<\/p>\n                    <p class=\"testimonial-function\">Director at Alere<\/p>\n                <\/div>\n                            <div>\n                    <p>&#8220;I give the new report builder 5 stars for its easy of use. Anyone without coding experience can start creating automated personalized reports quickly.&#8221;<\/p>                                            <img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Nicolas-gounin.jpg\" alt=\"Nicolas gounin\">\n                                        <p class=\"testimonial-author\">Nicolas Gounin<\/p>\n                    <p class=\"testimonial-function\">CFO &#038; COO at Egg Science<\/p>\n                <\/div>\n                            <div>\n                    <p>&#8220;You guys have done a great job making this as easy to use as possible and still robust in functionality.&#8221;<\/p>                                            <img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Dicksinson.jpg\" alt=\"Dicksinson\">\n                                        <p class=\"testimonial-author\">Ryan Dicksinson<\/p>\n                    <p class=\"testimonial-function\">Account Director at Reed Talent Solutions<\/p>\n                <\/div>\n                            <div>\n                    <p>\u201cIt\u2019s a great advantage to have formulas and the possibility for a really thorough analysis. There are hundreds of formulas, but the customer only sees the easy-to-read report. If you\u2019re looking for something like that, it\u2019s really nice to work with Pointerpro.\u201d<\/p>                                            <img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/sabine-e1589813236553-2.png\" alt=\"sabine e1589813236553 2\">\n                                        <p class=\"testimonial-author\">Sabine Wanmaker<\/p>\n                    <p class=\"testimonial-function\">Country Manager Netherlands at Better Minds at Work<\/p>\n                <\/div>\n                    <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZRfRzK p,\n    .lazyblock-content-ZRfRzK p a,\n    .lazyblock-content-ZRfRzK h1,\n    .lazyblock-content-ZRfRzK h2,\n    .lazyblock-content-ZRfRzK h3,\n    .lazyblock-content-ZRfRzK h4,\n    .lazyblock-content-ZRfRzK h5 {\n        color: ;\n    }\n\n    #content-ZRfRzK {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZRfRzK {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZRfRzK {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZRfRzK-container   \"\n     id=\"content-ZRfRzK\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZRfRzK\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><a href=\"https:\/\/s.pointerpro.com\/sugbaccz\" class=\"btn btn-primary btn-shortcode\" target=\"_self\">Book a demo<\/a><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-Z1jHlhS {\n        padding-top: 2.8rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-Z1jHlhS {\n        ;\n        ;\n        }\n    }\n\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; background-color: #ffffff;\">\n    <div class=\"container-xl lazyblock-section lazyblock-section-Z1jHlhS-container  #ffffff custom-padding \" id=\"section-Z1jHlhS\"\n    style=\"opacity: 100%; background-color: #ffffff;\">\n                            <div class=\"section-bg-color\"\n                 style=\"opacity: 100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-6 px-0 custom-padding row\">\n                    \n\n<style>\n    #heading-1lEtWX .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-1lEtWX {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-1lEtWX {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-1lEtWX  px-0\" id=\"heading-1lEtWX\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-1lEtWX\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"sample-vs-population-standard-deviation-in-data-analysis-s-vs-%cf%83\"><\/a>\n            <h2>Sample vs. population standard deviation in data analysis (S vs. \u03c3)<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-ZtAr0B p,\n    .lazyblock-content-ZtAr0B p a,\n    .lazyblock-content-ZtAr0B h1,\n    .lazyblock-content-ZtAr0B h2,\n    .lazyblock-content-ZtAr0B h3,\n    .lazyblock-content-ZtAr0B h4,\n    .lazyblock-content-ZtAr0B h5 {\n        color: ;\n    }\n\n    #content-ZtAr0B {\n        margin-top: 1rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-ZtAr0B {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-ZtAr0B {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-ZtAr0B-container  px-0 \"\n     id=\"content-ZtAr0B\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-ZtAr0B\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Standard deviation is a measure of variability or dispersion within a <\/span><a href=\"https:\/\/pointerpro.com\/blog\/quantitative-vs-qualitative-digital-marketing\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">quantitative dataset<\/span><\/a><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">While sample and population variances also measure this spread, the standard deviation is expressed in the same units as the original data in the statistical population, making it far easier to interpret.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-2Rc7l .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-2Rc7l {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-2Rc7l {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-2Rc7l  px-0\" id=\"heading-2Rc7l\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-2Rc7l\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"case-study-hr-drives-store-performance\"><\/a>\n            <h2>Case study: HR drives store performance<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z1QsXxv p,\n    .lazyblock-content-Z1QsXxv p a,\n    .lazyblock-content-Z1QsXxv h1,\n    .lazyblock-content-Z1QsXxv h2,\n    .lazyblock-content-Z1QsXxv h3,\n    .lazyblock-content-Z1QsXxv h4,\n    .lazyblock-content-Z1QsXxv h5 {\n        color: ;\n    }\n\n    #content-Z1QsXxv {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z1QsXxv {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1QsXxv {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1QsXxv-container  px-0 \"\n     id=\"content-Z1QsXxv\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1QsXxv\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><a href=\"https:\/\/www.aihr.com\/blog\/people-analytics-case-study-how-hr-made-customers-happy\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">A large restaurant chain<\/span><\/a><span style=\"font-weight: 400;\"> in the midst of failure asked a team of consultants for help to determine why performance was down and how they could improve. Data collection was nonexistent, so the consultants created a survey which focused on three key outcomes:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Customer satisfaction<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Employee retention<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Customer count<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">The business distributed an engagement survey that:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Linked employee outcomes to their real business outcomes<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Prioritized the factors that had the largest impact on business outcomes<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Showed the business impact of improvements in these factors<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Focused front-line managers on the factors that showed the largest impact<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #image-kosgH {\n        padding-top: 0rem;\n        padding-bottom: 0rem;\n    }\n            @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n<\/style>\n\n<div class=\"lazyblock-image-container \">\n    <div class=\"row lazyblock-image justify-content-center lazyblock-image-kosgH \" id=\"image-kosgH\">\n        <div class=\"col-12 col-lg-8 image\">\n                            <img decoding=\"async\" class=\"\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/Use-case-of-standard-deviation-in-HR.jpg\" alt=\"Use case of standard deviation in HR\" title=\"Use case of standard deviation in HR\"\/>\n                        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z15W7e0 p,\n    .lazyblock-content-Z15W7e0 p a,\n    .lazyblock-content-Z15W7e0 h1,\n    .lazyblock-content-Z15W7e0 h2,\n    .lazyblock-content-Z15W7e0 h3,\n    .lazyblock-content-Z15W7e0 h4,\n    .lazyblock-content-Z15W7e0 h5 {\n        color: ;\n    }\n\n    #content-Z15W7e0 {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z15W7e0 {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z15W7e0 {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z15W7e0-container  px-0 \"\n     id=\"content-Z15W7e0\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z15W7e0\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p style=\"text-align: center;\"><a href=\"https:\/\/www.aihr.com\/blog\/hr-analytics-case-studies\/\" target=\"_blank\" rel=\"noopener\">Source<\/a><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-1j8s5M p,\n    .lazyblock-content-1j8s5M p a,\n    .lazyblock-content-1j8s5M h1,\n    .lazyblock-content-1j8s5M h2,\n    .lazyblock-content-1j8s5M h3,\n    .lazyblock-content-1j8s5M h4,\n    .lazyblock-content-1j8s5M h5 {\n        color: ;\n    }\n\n    #content-1j8s5M {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-1j8s5M {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-1j8s5M {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-1j8s5M-container  px-0 \"\n     id=\"content-1j8s5M\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-1j8s5M\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">They found six factors contributed the most to business improvement and success:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Ethics<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Teamwork<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Job fit<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Senior leaders<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Communication<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Management<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">If the restaurant owners focused on promoting employees who scored a four or higher in these six characteristics, they could expect the following improvements in the three key business outcomes:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">16% increase in customer satisfaction<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">18,000 more customers per year<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">10% less staff turnover<\/span><\/li>\n<\/ul>        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    #heading-Z2iUeFE .heading h2 {\n        color: #421F3C;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-Z2iUeFE {\n        margin-top: 1rem;\n        margin-bottom: 1rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-Z2iUeFE {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-Z2iUeFE  px-0\" id=\"heading-Z2iUeFE\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-Z2iUeFE\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"standard-deviation-is-one-of-many-tools-in-your-data-analysis-belt\"><\/a>\n            <h2>Standard deviation is one of many tools in your data analysis belt<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n\n<style>\n    .lazyblock-content-Z1rhKeM p,\n    .lazyblock-content-Z1rhKeM p a,\n    .lazyblock-content-Z1rhKeM h1,\n    .lazyblock-content-Z1rhKeM h2,\n    .lazyblock-content-Z1rhKeM h3,\n    .lazyblock-content-Z1rhKeM h4,\n    .lazyblock-content-Z1rhKeM h5 {\n        color: ;\n    }\n\n    #content-Z1rhKeM {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-Z1rhKeM {\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n        #content-Z1rhKeM {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-content-container lazyblock-content-Z1rhKeM-container  px-0 \"\n     id=\"content-Z1rhKeM\">\n    <div class=\"row lazyblock-content justify-content-start  lazyblock-content-Z1rhKeM\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <p><span style=\"font-weight: 400;\">Because of its significance for the normal distribution function that models many real-world datasets, \u200cstandard deviation is the most widely used measure of dispersion in statistical analysis. However, it\u2019s important to remember that it\u2019s just one tool among many.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Other measures of dispersion include:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Variance<\/b><span style=\"font-weight: 400;\"> (a measure of average squared variances)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Range of values <\/b><span style=\"font-weight: 400;\">(the difference between the greatest and lowest value. A wider range means more dispersion)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>Interquartile range<\/b><span style=\"font-weight: 400;\"> (often abbreviated as IQR, which is the spread of the middle 50% of the data)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>The mean absolute deviation <\/b><span style=\"font-weight: 400;\">(MAD), and others.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">The most effective data analysts have a comprehensive understanding of various statistical measures and know when to apply each one to gain the most profound insights.<\/span><\/p>        <\/div>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-nhPW5 {\n        padding-top: 2.8rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-nhPW5 {\n        ;\n        ;\n        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100%;\"><\/div>\n                <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-6 px-0 custom-padding row\">\n                    \n\n<style>\n    #content-image-ZdAY1r {\n        padding-top: 3rem;\n        padding-bottom: 3rem;\n    }\n\n        \n    @media (max-width: 767.98px) {\n        .p-0 {\n            padding: 1rem !important;\n        }\n    }\n\n    #content-image-ZdAY1r {\n        margin-top: 0rem;\n        margin-bottom: 0rem;\n    }\n\n    #content-image-ZdAY1r .content-wrapper {\n        padding-left: 0rem;\n        padding-right: 0rem;\n    }\n\n    @media (max-width: 768px) {\n        #content-image-ZdAY1r .content-wrapper {\n            padding-left: 0 !important;\n        }\n    }\n\n    @media (max-width: 1440px) {\n        #section-ZdAY1r {\n            \/\/padding-left: -2.8rem;\n        }\n    }\n\n    \n \/*   .lazyblock-content-image-container {\n        position: relative;\n    }*\/\n\n<\/style>\n\n<div class=\"lazyblock-content-image-container lazyblock-content-image-ZdAY1r-container -container px-0 mx-0\">\n    <div class=\"row lazyblock-content-image \"\n         id=\"content-image-ZdAY1r\">\n        <div class=\"content-wrapper col-lg-6 col-12 pe-4 mb-lg-0 mb-4 order-1\">\n            <div class=\"d-flex flex-centered h-100\">\n                                                <div class=\"d-flex flex-wrap\">\n                    <div class=\"container-md px-0\">\n                        <div class=\"row lazyblock-content justify-content-start\">\n                            <div class=\"col-12 col-lg-12 text-start ps-4\">\n                    \n\n<style>\n    #heading-LLxkK .heading h2 {\n        color: #ffffff;\n        font-size: 2rem;\n        line-height: 3rem !important;\n    }\n\n    #heading-LLxkK {\n        margin-top: 0rem;\n        margin-bottom: 2rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-LLxkK {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-LLxkK  px-0\" id=\"heading-LLxkK\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-LLxkK\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"create-your-own-assessment-for-free\"><\/a>\n            <h2>Create your own assessment <br>for free!<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n                            <\/div>\n                        <\/div>\n                    <\/div>\n                <\/div>\n            <\/div>\n        <\/div>\n        <div class=\"image-wrapper col-lg-6 order-2\">\n            <div class=\"d-flex flex-centered h-100\">\n                                                                    <img decoding=\"async\" class=\"on-the-fly-ZdAY1r\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/PP-BANNER-GetStartedToday.png\"\n                         alt=\"\" title=\"\"\/>\n                            <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n\n<script>\n    document.addEventListener(\"DOMContentLoaded\", function() {\n        var containers = document.querySelectorAll('.lazyblock-content-image-container');\n\n        function checkViewport() {\n            var screenWidth = window.innerWidth;\n            if (screenWidth > 320 && screenWidth <= 768) {\n                containers.forEach(function(container) {\n                    container.classList.remove('px-0', 'mx-0');\n                });\n            } else {\n                containers.forEach(function(container) {\n                    container.classList.add('px-0', 'mx-0');\n                });\n            }\n        }\n\n        checkViewport();\n\n        window.addEventListener('resize', checkViewport);\n    });\n\n<\/script>\n\n<div class=\"row lazyblock-cta w-100\">\n    <div class=\"col-lg-12 text-center\">\n        <a class=\"btn btn-primary  \" href=\"https:\/\/app.pointerpro.com\/register?trial=true&amp;plan=professional\" target=\"_self\">Get started today<\/a>\n    <\/div>\n<\/div>\n\n                <\/div>\n        <\/div>\n    <\/div>\n<\/div>\n<!-- end -->\n\n<!-- if container is MD -->\n<!-- end -->\n\n\n\n\n<style>\n    #section-snaNG {\n        padding-top: 3rem;\n        padding-bottom: 2.8rem;\n    }\n\n    @media (min-width:1024px ) {\n        #section-snaNG {\n        ;\n        ;\n        }\n    }\n\n        .custom-padding {\n        padding-right: 0 !important;\n        padding-left: 0 !important;\n    }\n    @media (max-width: 1024px) {\n        .custom-padding {\n            padding-right: var(--bs-gutter-x, .75rem) !important;\n            padding-left: var(--bs-gutter-x, .75rem) !important;\n        }\n    }\n    <\/style>\n\n<!-- if container is fluid -->\n<div class=\"full-width\" style=\"position: relative; opacity: 100%; \">\n    <div class=\"container-xl lazyblock-section lazyblock-section-snaNG-container recommended-reading-single-post-container  custom-padding \" id=\"section-snaNG\"\n    style=\"opacity: 100%; \">\n                        <div class=\"row justify-content-center px-0 mx-0\">\n                <div class=\"col-lg-12 px-0 custom-padding row\">\n                    \n\n<style>\n    #heading-j8ssN .heading h2 {\n        color: #ff7418;\n        font-size: 1.5rem;\n        line-height: 2rem !important;\n    }\n\n    #heading-j8ssN {\n        margin-top: 0rem;\n        margin-bottom: 2rem;\n        margin-left: auto;\n        margin-right: auto;\n    }\n\n    @media (max-width: 768px) {\n        #heading-j8ssN {\n            margin-left: 0;\n            margin-right: 0;\n        }\n    }\n<\/style>\n\n<div class=\"container-md lazyblock-heading-2-container lazyblock-heading-2-j8ssN  px-0\" id=\"heading-j8ssN\">\n    <div class=\"row heading lazyblock-heading-2 justify-content-start  lazyblock-heading-2-j8ssN\">\n        <div class=\"col-12 col-lg-12 text-start\">\n            <a class=\"heading-2-anchor heading-anchor\" id=\"recommended-reading\"><\/a>\n            <h2>Recommended Reading<\/h2>\n        <\/div>\n    <\/div>\n<\/div>\n\n\n<div class=\"container-md lazyblock-teasers \">\n\n    \n    \n            <div class=\"row-wrapper child row row-cols-1 row-cols-lg-2 row-cols-xl-3 grid-gutter\" id=\"content_HOsUn\">\n                                                <div class=\"col pb-5\">\n                        <div class=\"card h-100 px-0 recommended-posts\">\n    <div class=\"recommended-item-wrapper\">\n        <div class=\"recommended-item-card\">\n                            <div class=\"recommended-item-img-wrapper\">\n                    <img decoding=\"async\" src=\"https:\/\/pointerpro.com\/wp-content\/uploads\/SURVEYxANYPLACExZAVENTEMANTWERP-124-1.jpg\" class=\"recommended-item-img\" alt=\"Are you popular? Brand awareness surveys reveal the truth.\">\n                <\/div>\n                        <div class=\"recommended-item-body flex-column h-100\">\n                <div class=\"recommended-item-title\">\n                    <h4><a href=\"https:\/\/pointerpro.com\/blog\/popular-brand-awareness-surveys\/\">Are you popular? Brand awareness surveys reveal the truth.<\/a><\/h4>\n                <\/div>\n                <div class=\"recommended-item-excerpt\">\n                    <p>By using a brand awareness survey you give your customers a voice. 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In this article we explain how.<\/p>\n","protected":false},"author":117,"featured_media":19589,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[198],"topic":[416],"class_list":["post-19531","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-general-en","tag-data-collection","topic-questionnaire-report-tips"],"acf":[],"_links":{"self":[{"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/posts\/19531","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/users\/117"}],"replies":[{"embeddable":true,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/comments?post=19531"}],"version-history":[{"count":40,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/posts\/19531\/revisions"}],"predecessor-version":[{"id":21929,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/posts\/19531\/revisions\/21929"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/media\/19589"}],"wp:attachment":[{"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/media?parent=19531"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/categories?post=19531"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/tags?post=19531"},{"taxonomy":"topic","embeddable":true,"href":"https:\/\/pointerpro.com\/wp-json\/wp\/v2\/topic?post=19531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}