{"id":57034,"date":"2019-11-26T20:12:42","date_gmt":"2019-11-26T20:12:42","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=57034"},"modified":"2026-03-25T11:42:51","modified_gmt":"2026-03-25T16:42:51","slug":"standard-deviation","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/standard-deviation\/","title":{"rendered":"Standard Deviation"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_DOIsplk2q6s\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_DOIsplk2q6s\" data-source-videoID=\"DOIsplk2q6s\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Standard Deviation Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Standard Deviation\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_DOIsplk2q6s:hover {cursor:pointer;} img#videoThumbnailImage_DOIsplk2q6s {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/331-standard-deviation-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_DOIsplk2q6s\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_DOIsplk2q6s\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Standard Deviation\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_DOIsplk2q6s\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_DOIsplk2q6s\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_DOIsplk2q6s\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction KfF_Function() {\n  var x = document.getElementById(\"KfF\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"KfF_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"KfF\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Analyzing_Data\" class=\"smooth-scroll\">Analyzing Data<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Normal_Distribution\" class=\"smooth-scroll\">Normal Distribution<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_to_Calculate_Standard_Deviation\" class=\"smooth-scroll\">How to Calculate Standard Deviation<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Standard_Deviation_Practice_Questions\" class=\"smooth-scroll\">Standard Deviation Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hey everyone! Today we\u2019re going to take a look at standard deviation when applied to a population.<\/p>\n<h2><span id=\"Analyzing_Data\" class=\"m-toc-anchor\"><\/span>Analyzing Data<\/h2>\n<p>\nLet\u2019s start by looking at what standard deviation can tell us about a set of data. <\/p>\n<p>Let\u2019s say we have a classroom of 100 middle school students, and we measure the height of each student. We now have our data set. <\/p>\n<p>We could find the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/mean-median-and-mode\/\">mean<\/a>, or simple average, of the data by adding all of the numbers together and dividing by how many students we measured, which in this case is 100. Let\u2019s say the mean height of this group of students is 58 inches tall.<\/p>\n<p>This number alone is quite limited. Without access to the original data, we don\u2019t know if every single student was that height, which would give us that average, or if half of them are 53 inches tall and the other half is 63 inches tall, which also gives us that average. <\/p>\n<h2><span id=\"Normal_Distribution\" class=\"m-toc-anchor\"><\/span>Normal Distribution<\/h2>\n<p>\nIn reality, our intuition would expect there to be a mix of students with most of the students around the average height and fewer students who are a bit shorter or taller and even fewer who are a lot shorter and taller than the average. And our intuition is right. This is what we call a normal distribution, and it looks like this: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65423\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution.png\" alt=\"normal distribution\" width=\"777\" height=\"421\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution.png 979w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution-300x162.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution-768x416.png 768w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>If we start in the center and move to the right until we reach the next vertical line, we\u2019ve moved one standard deviation above the mean, which is the middle. In this example, the standard deviation is 3 inches. <\/p>\n<p>We can see in the area under the graph that this accounts for just over 34% of all the students. If we go back to the middle and move left until we reach a vertical line, we\u2019ve moved one standard deviation below the mean, which accounts for another 34% or so of the students. So we can say that over 68% of the students are within 3 inches (our standard deviation) of the mean height for all the students. <\/p>\n<p>This is always true for a normal distribution. The only thing that changes is the value of the standard deviation itself. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65422\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution-2.png\" alt=\"3 inch normal distribution\" width=\"777\" height=\"504\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution-2.png 1003w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution-2-300x194.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/normal-distribution-2-768x498.png 768w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>For instance, if we had calculated the standard deviation and it came out to 2 inches instead of 3, then the same <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/percentages\/\">percentage<\/a> of students would be within one standard deviation of the mean, but when we look at the normal distribution it\u2019s taller and we see that our 34% fits within 2 inches of the center rather than 3 like in our previous distribution:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65419\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-inches.png\" alt=\"2 inch normal distribution\" width=\"777\" height=\"614\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-inches.png 974w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-inches-300x237.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-inches-768x607.png 768w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>That shows us what standard deviation is all about. In this case, it\u2019s the number of inches of height away from the average that will make up 34% of the population. In the first case it was three inches to account for that percentage of people. In our second example, it only took two inches to account for that percentage. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-65416\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-examples.png\" alt=\"standard deviation examples\" width=\"775\" height=\"1042\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-examples.png 728w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/2-examples-223x300.png 223w\" sizes=\"auto, (max-width: 775px) 100vw, 775px\" \/><\/p>\n<p>So when we calculate standard deviation from a sample set, that\u2019s what we\u2019re finding. And the number tells us something about all of the data. In our revised distribution we know that most of the students are within two inches, or one standard deviation, of the mean of 58 inches. In our first set, it took a standard deviation of 3 inches to get that many. So you could say it\u2019s a measure of how spread out our data is. <\/p>\n<h2><span id=\"How_to_Calculate_Standard_Deviation\" class=\"m-toc-anchor\"><\/span>How to Calculate Standard Deviation<\/h2>\n<p>\nSo at this point you might be asking: How do we calculate the standard deviation, anyway?<\/p>\n<p>Well, the easiest way is on a spreadsheet, where standard deviation (STDDEV) is a common function that can be used on a range of cells. But we can calculate it manually. Let\u2019s keep our data set small to make our lives a little bit easier. Say we have 10 students and we measure their heights and get values of 52, 55, 56, 56, 57, 58, 59, 61, 62, and 64. <\/p>\n<ol>\n<li>Step one is to find the mean.<\/li>\n<\/ol>\n<div class=\"examplesentence\">\\(\\frac{52+55+56+56+57+58+59+61+62+64}{10}\\)\\(=\\frac{580}{10}\\)\\(=58\\)<\/div>\n<p>\n&nbsp;<\/p>\n<ol start=\"2\">\n<li>Step two is to take the difference between each number and the mean and then square it.<\/li>\n<\/ol>\n<div class=\"examplesentence\">\\((52 \u2013 58)^2=36\\)<br \/>\n\\((55 \u2013 58)^2=9\\)<br \/>\n\\((56 \u2013 58)^2=4\\)<br \/>\n\\((56 \u2013 58)^2=4\\)<br \/>\n\\((57 \u2013 58)^2=1\\)<br \/>\n\\((58 \u2013 58)^2=0\\)<br \/>\n\\((59 \u2013 58)^2=1\\)<br \/>\n\\((61 \u2013 58)^2=9\\)<br \/>\n\\((62-58)^2=16\\)<br \/>\n\\((64-58)^2=36\\)<\/div>\n<p>\n&nbsp;<\/p>\n<ol start=\"23\">\n<li>Step three is take the average of those squared differences. This is called the variance.<\/li>\n<\/ol>\n<div class=\"examplesentence\">\\(36+9+4+4+1=54\\)<br \/>\n\\(0+1+9+16+36=62\\)<br \/>\n\\(54+62=116\\)<br \/>\n\\(\\frac{116}{10}=11.6\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo when I add them all up and divide by 10, I get 11.6.<\/p>\n<ol start=\"4\">\n<li>To find the standard deviation, I simply take the square root of the variance.<\/li>\n<\/ol>\n<p>The square root of 11.6 is approximately 3.4. <\/p>\n<div class=\"examplesentence\">\\(\\sqrt{11.6}\\approx 3.4\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo the standard deviation, in this case, is equal to 3.4 inches. Remember, the standard deviation has units, so that\u2019s inches in this case. <\/p>\n<p>In reality, you probably wouldn\u2019t calculate a standard deviation on such a small population, but this just gives you an idea. The process would still be the same four steps even if there were 1,000 students in your population, though that sure would take a lot longer to calculate manually. <\/p>\n<hr>\n<p>\nI hope this video was helpful for understanding standard deviation. Thanks for watching. See you next time!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Standard_Deviation_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Standard Deviation Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFor a given data set with a normal distribution, which of the following values for standard deviation will give us the narrowest normal bell curve?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">100 units<\/div><div class=\"PQ\"  id=\"PQ-1-2\">5 units<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">1.5 units<\/div><div class=\"PQ\"  id=\"PQ-1-4\">25 units<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The smallest of these values is 1.5 units, so that is the one that will give us the narrowest bell curve for the normal distribution. Remember, the standard deviation is a measure of how \u201cspread out\u201d the curve will be, with 68% of the population being within one standard deviation from the mean (34% above and 34% below). When those 68% are within just 1.5 units in either direction, the curve is very tall and narrow, compared to larger standard deviations like 25 or 100!<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFrom a given data set, it is determined that the variance is equal to 2,025. What is the standard deviation of the data set?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">41<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">45<\/div><div class=\"PQ\"  id=\"PQ-2-3\">55<\/div><div class=\"PQ\"  id=\"PQ-2-4\">62<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The relationship between variance and standard deviation is that standard deviation is equal to the square root of the variance.<\/p>\n<p style=\"text-align: center\">\\(\\sqrt{2{,}025}=45\\)<\/p>\n<p>In fact, variance is usually written as \\(\\sigma^2\\), because the lowercase Greek letter sigma \\((\\sigma)\\) represents standard deviation.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nOver the past four weeks, Deb\u2019s Florist has sold 29, 40, 33, and 36 bouquets each week, respectively. Calculate the standard deviation for this data set. (Hint: start by computing the average, then squaring the difference between each data point and the average. Find the variance by computing the mean of these values, and then take the square root to get the standard deviation.)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(\\sigma\\approx3.1667\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\(\\sigma=3.5\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(\\sigma=3.8\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">\\(\\sigma\\approx4.031\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, we compute the average of the four data points.<\/p>\n<p style=\"text-align:center; line-height: 50px\">\n\\(29+40+33+36=138\\), and \\(\\dfrac{138}{4}=34.5\\)<\/p>\n<p>The average of the data set is 34.5 bouquets per week. Now, how far is each data point from that average, and what are the squares of those values?<\/p>\n<p style=\"text-align:center; line-height: 35px\">\n\\((29-34.5)^2=(-5.5)^2=30.25\\)<br \/>\n\\((40-34.5)^2=(5.5)^2=30.25\\)<br \/>\n\\((33-34.5)^2=(-1.5)^2=2.25\\)<br \/>\n\\((36-34.5)^2=(1.5)^2=2.25\\)<\/p>\n<p>Now that we have these values, we will compute their mean to find the variance.<\/p>\n<p style=\"text-align:center; line-height: 60px\">\n\\(\\sigma^2=\\)\\(\\:\\dfrac{(30.25+30.25+2.25+2.25)}{4}\\)\\(\\:=\\dfrac{65}{4}=16.25\\)<\/p>\n<p>Now that we know the variance is 16.25, we just need to take the square root to find the standard deviation.<\/p>\n<p style=\"text-align:center;\">\n\\(\\sigma=\\sqrt{16.25}\\approx 4.031\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nCarmen is doing research for her taxi company by documenting gas prices in the area over the past 100 days. The average price of gas over the 100-day period was $2.86, and the standard deviation was $0.11. Approximately how many days was the price of gas within $0.11 of the average price?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">34 days<\/div><div class=\"PQ\"  id=\"PQ-4-2\">36 days<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">68 days<\/div><div class=\"PQ\"  id=\"PQ-4-4\">72 days<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Remember, we know that approximately 34% of the data points in a set should be between the mean and the value of the mean plus one standard deviation. Similarly, another 34% of the points will be between the mean and the value of the mean <em>minus<\/em> one standard deviation.<\/p>\n<p>Adding those together, 68% of the data points will be within one standard deviation (plus or minus) from the mean.<\/p>\n<p>Because our data set has one hundred points, we see that 68 points, or 68 days, should be within one standard deviation, or $0.11, of the mean price.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>Carlos\u2019s Art Gallery sold four paintings last weekend at auction for the following prices:<\/p>\n<ul>\n<li>$40,700<\/li>\n<li>$12,900<\/li>\n<li>$24,500<\/li>\n<li>$26,100<\/li>\n<\/ul>\n<p>Calculate the standard deviation of these prices.<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">$9,873.58<\/div><div class=\"PQ\"  id=\"PQ-5-2\">$10,143.77<\/div><div class=\"PQ\"  id=\"PQ-5-3\">$13,837.91<\/div><div class=\"PQ\"  id=\"PQ-5-4\">$14,950.12<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, we calculate the average of the four prices as follows:<\/p>\n<p style=\"text-align:center; line-height: 50px\">\n\\(40{,}700+12{,}900+24{,}500\\)\\(\\: + \\: 26{,}100=104{,}200\\), and \\(\\dfrac{104{,}200}{4}=26{,}050\\)<\/p>\n<p>Then, square the difference between each price and the average.<\/p>\n<p style=\"text-align:center; line-height: 35px\">\n<span style=\"font-size: 90%\">\\((40{,}700-26{,}050)^2=214{,}622{,}500\\)<\/span><br \/>\n<span style=\"font-size: 90%\">\\((12{,}900-26{,}050)^2=172{,}922{,}500\\)<\/span><br \/>\n<span style=\"font-size: 90%\">\\((24{,}500-26{,}050)^2=2{,}402{,}500\\)<\/span><br \/>\n<span style=\"font-size: 90%\">\\((26{,}100-26{,}050)^2=2{,}500\\)<\/span><\/p>\n<p>Now, we take the average of these values to determine the variance.<\/p>\n<p style=\"text-align:center; line-height: 50px\">\\(\\sigma^2=\\)<span style=\"font-size: 110%\">\\(\\:\\frac{(214{,}622{,}500+172{,}922{,}500+2{,}402{,}500+2{,}500)}{4}\\)<\/span>\\(\\:=97{,}487{,}500\\)<\/p>\n<p>Finally, take the square root to find the standard deviation.<\/p>\n<p style=\"text-align:center;\">\n\\(\\sigma=\\sqrt{97{,}487{,}500}\\approx9{,}873.58\\)<\/p>\n<p>The standard deviation of the prices of the four paintings is therefore $9,873.58.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/statistics\/\">Return to Statistics Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Statistics Videos<\/p>\n","protected":false},"author":1,"featured_media":91498,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-57034","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-statistics-videos","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/57034","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=57034"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/57034\/revisions"}],"predecessor-version":[{"id":279247,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/57034\/revisions\/279247"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/91498"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=57034"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}