{"id":230722,"date":"2020-10-23T17:46:14","date_gmt":"2020-10-23T15:46:14","guid":{"rendered":"https:\/\/www.scribbr.nl\/?p=230722"},"modified":"2022-07-06T15:33:47","modified_gmt":"2022-07-06T13:33:47","slug":"normal-distribution","status":"publish","type":"post","link":"https:\/\/www.scribbr.com\/statistics\/normal-distribution\/","title":{"rendered":"Normal Distribution | Examples, Formulas, &#038; Uses"},"content":{"rendered":"<p>In a normal distribution, data is symmetrically distributed with no <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//skewness///">skew<\/a>. When plotted on a graph, the data follows a bell shape, with most values clustering around a <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//central-tendency///">central region<\/a> and tapering off as they go further away from the center.<\/p>\n<p>Normal distributions are also called Gaussian distributions or bell curves because of their shape.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//uploads//2020//10//Normal-distribution-of-SAT-scores.svg/" alt=\"Bar chart showing a normal distribution of SAT scores\" width=\"600\" height=\"371\" class=\"aligncenter size-full wp-image-230734\" role=\"img\" \/><\/p>\n<p><!--more--><\/p>\n<h2>Why do normal distributions matter?<\/h2>\n<p>All kinds of variables in natural and social sciences are normally or approximately normally distributed. Height, birth weight, reading ability, job satisfaction, or SAT scores are just a few examples of such variables.<\/p>\n<p>Because normally distributed variables are so common, many<a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//statistical-tests///"> statistical tests<\/a> are designed for normally distributed populations.<\/p>\n<p>Understanding the properties of normal distributions means you can use <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//inferential-statistics///">inferential statistics<\/a> to compare different groups and make estimates about populations using samples.<\/p>\n<h2>What are the properties of normal distributions?<\/h2>\n<p>Normal distributions have key characteristics that are easy to spot in graphs:<\/p>\n<ul>\n<li>The <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//mean///">mean<\/a>, <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//median///">median<\/a> and <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//mode///">mode<\/a> are exactly the same.<\/li>\n<li>The distribution is symmetric about the mean\u2014half the values fall below the mean and half above the mean.<\/li>\n<li>The distribution can be described by two values: the mean and the <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//standard-deviation///">standard deviation<\/a>.<\/li>\n<\/ul>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//uploads//2020//10//normal-distribution.png/" alt=\"Mean, median, mode, and standard deviation in a normal distribution\" width=\"720\" height=\"405\" class=\"aligncenter wp-image-230747 size-full\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distribution.png 720w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distribution-300x169.png 300w\" sizes=\"(max-width: 720px) 100vw, 720px\" \/><\/p>\n<p>The mean is the location parameter while the standard deviation is the scale parameter.<\/p>\n<p>The mean determines where the peak of the curve is centered. Increasing the mean moves the curve right, while decreasing it moves the curve left.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.de//wp-content//uploads//2020//10//normal-distributions-with-different-means-1024x633.png/" alt=\"Normal distributions with different means\" width=\"600\" height=\"371\" class=\"aligncenter wp-image-230760\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-1024x633.png 1024w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-300x186.png 300w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-768x475.png 768w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means.png 1200w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>The standard deviation stretches or squeezes the curve. A small standard deviation results in a narrow curve, while a large standard deviation leads to a wide curve.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.de//wp-content//uploads//2020//10//normal-distributions-with-different-sds-1024x633.png/" alt=\"Normal distributions with different standard deviations\" width=\"600\" height=\"371\" class=\"aligncenter wp-image-230773\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-sds-1024x633.png 1024w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-sds-300x186.png 300w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-sds-768x475.png 768w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-sds.png 1200w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h2>Empirical rule<\/h2>\n<p>The <strong>empirical rule<\/strong>, or the 68-95-99.7 rule, tells you where most of your values lie in a normal distribution:<\/p>\n<ul>\n<li>Around 68% of values are within 1 standard deviation from the mean.<\/li>\n<li>Around 95% of values are within 2 standard deviations from the mean.<\/li>\n<li>Around 99.7% of values are within 3 standard deviations from the mean.<\/li>\n<\/ul>\n<figure class=\"kb-textbox border-left green\"><figcaption>Example: Using the empirical rule in a normal distribution<\/figcaption>You collect SAT scores from students in a new test preparation course. The data follows a normal distribution with a mean score (<em>M<\/em>) of 1150 and a standard deviation (<em>SD<\/em>) of 150.<\/p>\n<p>Following the empirical rule:<\/p>\n<ul>\n<li>Around 68% of scores are between 1,000 and 1,300, 1 standard deviation above and below the mean.<\/li>\n<li>Around 95% of scores are between 850 and 1,450, 2 standard deviations above and below the mean.<\/li>\n<li>Around 99.7% of scores are between 700 and 1,600, 3 standard deviations above and below the mean.<\/li>\n<\/ul>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//uploads//2020//10//empirical-rule1.png/" alt=\"The empirical rule in normal distributions\" width=\"600\" height=\"338\" class=\"aligncenter wp-image-230786\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/empirical-rule1.png 720w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/empirical-rule1-300x169.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure>\n<p>The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don\u2019t follow this pattern.<\/p>\n<p>If data from small samples do not closely follow this pattern, then other distributions like the <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//t-distribution///">t-distribution<\/a> may be more appropriate. Once you identify the distribution of your variable, you can apply appropriate statistical tests.<\/p>\n<h2>Central limit theorem<\/h2>\n<p>The <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//central-limit-theorem///">central limit theorem<\/a> is the basis for how normal distributions work in statistics.<\/p>\n<p>In research, to get a good idea of a<a href=https://www.scribbr.com/"https:////www.scribbr.com//methodology//population-vs-sample///"> population<\/a> mean, ideally you\u2019d collect data from multiple <a href=https://www.scribbr.com/"https:////www.scribbr.com//methodology//simple-random-sampling///">random samples<\/a> within the population. A <strong>sampling distribution of the mean<\/strong> is the distribution of the means of these different samples.<\/p>\n<p>The central limit theorem shows the following:<\/p>\n<ul>\n<li>Law of Large Numbers: As you increase sample size (or the number of samples), then the sample mean will approach the population mean.<\/li>\n<li>With multiple large samples, the sampling distribution of the mean is normally distributed, even if your original variable is not normally distributed.<\/li>\n<\/ul>\n<p><a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//statistical-tests//#parametric\">Parametric statistical tests<\/a> typically assume that samples come from normally distributed populations, but the central limit theorem means that this assumption isn\u2019t necessary to meet when you have a large enough sample.<\/p>\n<p>You can use parametric tests for large samples from populations with any kind of distribution as long as <a href=https://www.scribbr.com/"https:////www.scribbr.com//frequently-asked-questions//assumptions-of-statistical-tests///">other important assumptions<\/a> are met. A sample size of 30 or more is generally considered large.<\/p>\n<p>For small samples, the assumption of normality is important because the sampling distribution of the mean isn\u2019t known. For accurate results, you have to be sure that the population is normally distributed before you can use parametric tests with small samples.<\/p>\n<h2>Formula of the normal curve<\/h2>\n<p>Once you have the mean and standard deviation of a normal distribution, you can fit a normal curve to your data using a <strong>probability density function<\/strong>.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//uploads//2020//10//normal-curve-fitted.png/" alt=\"A normal curve fitted to a normal distribution of SAT scores\" width=\"600\" height=\"338\" class=\"aligncenter wp-image-230799\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-curve-fitted.png 720w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-curve-fitted-300x169.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>In a probability density function, the area under the curve tells you probability. The normal distribution is a <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//probability-distributions///"><strong>probability distribution<\/strong><\/a>, so the total area under the curve is always 1 or 100%.<\/p>\n<p>The formula for the normal probability density function looks fairly complicated. But to use it, you only need to know the population mean and standard deviation.<\/p>\n<p>For any value of <em>x<\/em>, you can plug in the mean and standard deviation into the formula to find the probability density of the variable taking on that value of <em>x<\/em>.<\/p>\n<table style=\";margin-left: auto; margin-right: auto;\" class=\"kb-table responsive-inline\">\n<thead>\n<tr>\n<th style=\"background-color: #f8f7f5;\">Normal probability density formula<\/th>\n<th style=\"background-color: #f8f7f5;\">Explanation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//ql-cache//quicklatex.com-16763e3b6e3f2bd556458282d64f2627_l3.png/" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#92;&#112;&#105;&#125;&#125;&#101;&#94;&#123;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#40;&#120;&#45;&#92;&#109;&#117;&#41;&#94;&#50;&#125;&#123;&#50;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"204\" style=\"vertical-align: -17px;\"\/><\/td>\n<td>\n<ul>\n<li><em>f<\/em>(<em>x<\/em>) = probability<\/li>\n<li><em>x<\/em> = value of the variable<\/li>\n<li>\u03bc = mean<\/li>\n<li>\u03c3 = standard deviation<\/li>\n<li>\u03c3<sup>2<\/sup> = variance<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure class=\"kb-textbox border-left blue\"><figcaption>Example: Using the probability density function<\/figcaption>You want to know the probability that SAT scores in your sample exceed 1380.<\/p>\n<p>On your graph of the probability density function, the probability is the shaded area under the curve that lies to the right of where your SAT scores equal 1380.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.de//wp-content//uploads//2020//10//pdf-of-sat-scores.png/" alt=\"The probability density function of SAT scores\" width=\"600\" height=\"338\" class=\"aligncenter wp-image-230825\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/pdf-of-sat-scores.png 720w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/pdf-of-sat-scores-300x169.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><br \/>\nYou can find the probability value of this score using the standard normal distribution.<\/figure>\n<h2>What is the standard normal distribution?<\/h2>\n<p>The <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//standard-normal-distribution///"><strong>standard normal distribution<\/strong><\/a>, also called the <strong><em>z<\/em>-distribution<\/strong>, is a special normal distribution where the mean is 0 and the standard deviation is 1.<\/p>\n<p>Every normal distribution is a version of the standard normal distribution that\u2019s been stretched or squeezed and moved horizontally right or left.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.de//wp-content//uploads//2020//10//normal-distributions-with-different-means-and-sds.png/" alt=\"Normal distributions with different means and SDs\" width=\"600\" height=\"371\" class=\"aligncenter wp-image-230838\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-and-sds.png 1200w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-and-sds-300x186.png 300w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-and-sds-1024x633.png 1024w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/normal-distributions-with-different-means-and-sds-768x475.png 768w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>While individual observations from normal distributions are referred to as<em> x<\/em>, they are referred to as <em>z<\/em> in the <em>z<\/em>-distribution. Every normal distribution can be converted to the standard normal distribution by turning the individual values into <em>z<\/em>-scores.<\/p>\n<p><em>Z<\/em>-scores tell you how many standard deviations away from the mean each value lies.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.de//wp-content//uploads//2020//10//standard-normal-distribution.png/" alt=\"The standard normal distribution has a mean of 0 and a standard deviation of 1.\" width=\"600\" height=\"371\" class=\"aligncenter wp-image-230851\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/standard-normal-distribution.png 1200w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/standard-normal-distribution-300x186.png 300w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/standard-normal-distribution-1024x633.png 1024w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/standard-normal-distribution-768x475.png 768w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>You only need to know the mean and standard deviation of your distribution to find the <em>z<\/em>-score of a value.<\/p>\n<table style=\";margin-left: auto; margin-right: auto;\" class=\"kb-table responsive-inline\">\n<thead>\n<tr>\n<th style=\"background-color: #f8f7f5;\"><em>Z<\/em>-score Formula<\/th>\n<th style=\"background-color: #f8f7f5;\">Explanation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//ql-cache//quicklatex.com-8c0d11455fb878c25f952eb08d64dc4c_l3.png/" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#92;&#109;&#117;&#125;&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"83\" style=\"vertical-align: -13px;\"\/><\/td>\n<td>\n<ul>\n<li><em>x<\/em> = individual value<\/li>\n<li>\u03bc = mean<\/li>\n<li>\u03c3 = standard deviation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We convert normal distributions into the standard normal distribution for several reasons:<\/p>\n<ul>\n<li>To find the probability of observations in a distribution falling above or below a given value.<\/li>\n<li>To find the probability that a sample mean significantly differs from a known population mean.<\/li>\n<li>To compare scores on different distributions with different means and standard deviations.<\/li>\n<\/ul>\n<h3>Finding probability using the <em>z<\/em>-distribution<\/h3>\n<p>Each <em>z<\/em>-score is associated with a probability, or <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//p-value///"><em>p<\/em>-value<\/a>, that tells you the likelihood of values below that <em>z<\/em>-score occurring. If you convert an individual value into a<em> z<\/em>-score, you can then find the probability of all values up to that value occurring in a normal distribution.<\/p>\n<figure class=\"kb-textbox border-left blue\"><figcaption>Example: Finding probability using the <em>z<\/em>-distribution<\/figcaption>To find the probability of SAT scores in your sample exceeding 1380, you first find the <em>z<\/em>-score.<\/p>\n<p>The mean of our distribution is 1150, and the standard deviation is 150. The <em>z<\/em>-score tells you how many standard deviations away 1380 is from the mean.<\/p>\n<table style=\";margin-left: auto; margin-right: auto;\" class=\"kb-table responsive-inline\">\n<thead>\n<tr>\n<th>Formula<\/th>\n<th>Calculation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//ql-cache//quicklatex.com-8c0d11455fb878c25f952eb08d64dc4c_l3.png/" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#92;&#109;&#117;&#125;&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"83\" style=\"vertical-align: -13px;\"\/><em><br \/>\n<\/em><\/td>\n<td><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//ql-cache//quicklatex.com-cadfa86b4c2e6cfc05075dcaf21ea73f_l3.png/" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#92;&#44;&#51;&#56;&#48;&#45;&#49;&#92;&#44;&#49;&#53;&#48;&#125;&#123;&#49;&#53;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"142\" style=\"vertical-align: -13px;\"\/><br \/>\n<img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//ql-cache//quicklatex.com-9a3ced517748dd0306e5b935a57962ec_l3.png/" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#49;&#46;&#53;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\"\/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For a <em>z<\/em>-score of 1.53, the <em>p<\/em>-value is 0.937.\u00a0 This is the probability of SAT scores being 1380 or less (93.7%), and it&#8217;s the area under the curve left of the shaded area.<\/p>\n<p><img src=https://www.scribbr.com/"https:////www.scribbr.de//wp-content//uploads//2020//10//standard-normal-shaded-area.png/" alt=\"Finding probability using the z-distribution\" width=\"600\" height=\"338\" class=\"aligncenter wp-image-230877\" srcset=\"https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/standard-normal-shaded-area.png 720w, https:\/\/www.scribbr.com\/wp-content\/uploads\/2020\/10\/standard-normal-shaded-area-300x169.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<p>To find the shaded area, you take away 0.937 from 1, which is the total area under the curve.<\/p>\n<p style=\"text-align: center;\">Probability of <em>x <\/em>&gt; 1380 = 1 \u2013 0.937 = <strong>0.063<\/strong><\/p>\n<p>That means it is likely that only 6.3% of SAT scores in your sample exceed 1380.<\/figure>\n    <h2>Frequently asked questions about normal distributions<\/h2>\n<dl class=\"faq-list faq-list--arrows w-100 my-5\" itemscope itemtype=\"https:\/\/schema.org\/FAQPage\">\n            <div itemscope itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n            <dt id=\"what-is-a-normal-distribution\" class=\"faq-list__title\">\n                <a href=https://www.scribbr.com/"https:////www.scribbr.com//frequently-asked-questions//what-is-a-normal-distribution///" class=\"faq-list__link qa-faq-anchor\">\n                    <span itemprop=\"name\">\n                        What is a normal distribution?\n                    <\/span>\n                <\/a>\n            <\/dt>\n            <dd id=\"what-is-a-normal-distribution-answer\" class=\"faq-list__answer qa-faq-answer kb-article-styles\" itemscope itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                <div itemprop=\"text\">\n                                        <p>In a <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//normal-distribution///">normal distribution<\/a>, data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.<\/p>\n<p>The <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//central-tendency///">measures of central tendency<\/a> (mean, mode, and median) are exactly the same in a normal distribution.<\/p>\n<p><a class=\"colorbox\" href=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//uploads//2022//05//normal-distribution.png/"><img src=https://www.scribbr.com/"https:////www.scribbr.com//wp-content//uploads//2022//05//normal-distribution-small.png/" alt=\"Normal distribution\" width=\"500px\" class=\"aligncenter mt-4 mb-3\" \/><\/a><\/p>\n\n                <\/div>\n            <\/dd>\n        <\/div>\n            <div itemscope itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n            <dt id=\"what-is-a-standard-normal-distribution\" class=\"faq-list__title\">\n                <a href=https://www.scribbr.com/"https:////www.scribbr.com//frequently-asked-questions//what-is-a-standard-normal-distribution///" class=\"faq-list__link qa-faq-anchor\">\n                    <span itemprop=\"name\">\n                        What is a standard normal distribution?\n                    <\/span>\n                <\/a>\n            <\/dt>\n            <dd id=\"what-is-a-standard-normal-distribution-answer\" class=\"faq-list__answer qa-faq-answer kb-article-styles\" itemscope itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                <div itemprop=\"text\">\n                                        <p>The <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//standard-normal-distribution///">standard normal distribution<\/a>, also called the <em>z<\/em>-distribution, is a special <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//normal-distribution///">normal distribution<\/a> where the <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//mean///">mean<\/a> is 0 and the<a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//standard-deviation///"> standard deviation<\/a> is 1.<\/p>\n<p>Any normal distribution can be converted into the standard normal distribution by turning the individual values into <em>z<\/em>-scores.\u00a0In a <em>z<\/em>-distribution, <em>z<\/em>-scores tell you how many standard deviations away from the mean each value lies.<\/p>\n\n                <\/div>\n            <\/dd>\n        <\/div>\n            <div itemscope itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n            <dt id=\"what-is-the-empirical-rule\" class=\"faq-list__title\">\n                <a href=https://www.scribbr.com/"https:////www.scribbr.com//frequently-asked-questions//what-is-the-empirical-rule///" class=\"faq-list__link qa-faq-anchor\">\n                    <span itemprop=\"name\">\n                        What is the empirical rule?\n                    <\/span>\n                <\/a>\n            <\/dt>\n            <dd id=\"what-is-the-empirical-rule-answer\" class=\"faq-list__answer qa-faq-answer kb-article-styles\" itemscope itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                <div itemprop=\"text\">\n                                        <p>The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//normal-distribution///">normal distribution<\/a>:<\/p>\n<ul>\n<li>Around 68% of values are within 1 <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//standard-deviation///">standard deviation<\/a> of the mean.<\/li>\n<li>Around 95% of values are within 2 standard deviations of the mean.<\/li>\n<li>Around 99.7% of values are within 3 standard deviations of the mean.<\/li>\n<\/ul>\n<p>The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don\u2019t follow this pattern.<\/p>\n\n                <\/div>\n            <\/dd>\n        <\/div>\n            <div itemscope itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n            <dt id=\"what-is-a-t-distribution\" class=\"faq-list__title\">\n                <a href=https://www.scribbr.com/"https:////www.scribbr.com//frequently-asked-questions//what-is-a-t-distribution///" class=\"faq-list__link qa-faq-anchor\">\n                    <span itemprop=\"name\">\n                        What is a t-distribution?\n                    <\/span>\n                <\/a>\n            <\/dt>\n            <dd id=\"what-is-a-t-distribution-answer\" class=\"faq-list__answer qa-faq-answer kb-article-styles\" itemscope itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                <div itemprop=\"text\">\n                                        <p>The<a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//t-distribution///"><em> t<\/em>-distribution<\/a> is a way of describing a set of observations where most observations fall close to the <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//mean///">mean<\/a>, and the rest of the observations make up the tails on either side. It is a type of <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//normal-distribution///">normal distribution<\/a> used for smaller sample sizes, where the <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//variance///">variance<\/a> in the data is unknown.<\/p>\n<p>The <em>t<\/em>-distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the <a href=https://www.scribbr.com/"https:////www.scribbr.com//statistics//standard-deviation///">standard deviation<\/a>.<\/p>\n\n                <\/div>\n            <\/dd>\n        <\/div>\n    <\/dl>\n","protected":false},"excerpt":{"rendered":"<p>In a normal distribution, data is symmetrically distributed with no skew. When plotted on a graph, the data follows a bell shape, with most values clustering around a central region and tapering off as they go further away from the center. Normal distributions are also called Gaussian distributions or bell curves because of their shape.<\/p>\n","protected":false},"author":115,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"","_relevanssi_noindex_reason":""},"categories":[39099],"tags":[],"acf":[],"yoast_head":"<title>Normal Distribution | Examples, Formulas, &amp; Uses<\/title>\n<!-- Added by HTTrack --><meta http-equiv="content-type" content="text/html;charset=UTF-8" /><!-- /Added by HTTrack -->
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