{"id":2691,"date":"2012-09-10T09:00:54","date_gmt":"2012-09-10T16:00:54","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2691"},"modified":"2019-04-20T01:33:31","modified_gmt":"2019-04-20T08:33:31","slug":"standard-deviation-on-the-gmat","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/","title":{"rendered":"Standard Deviation on the GMAT"},"content":{"rendered":"<p>First, here are some challenging practice questions:<\/p>\n<p>1) Set S has a mean of 10 and a standard deviation of 1.5.\u00a0 We are going to add two additional numbers to Set S.\u00a0 Which pair of numbers would decrease the standard deviation the most?<\/p>\n<ul>\n\t(A) {2, 10}<br \/>\n\t(B) {10, 18}<br \/>\n\t(C) {7, 13}<br \/>\n\t(D) {9, 11}<br \/>\n\t(E) {16, 16}\n<\/ul>\n<p>2) Set Q consists of the following five numbers: Q = {5, 8, 13, 21, 34}.\u00a0 Which of the following sets has the same standard deviation as Set Q?<\/p>\n<p>I. {35, 38, 43, 51, 64}<br \/>\nII. {10, 16, 26, 42, 68}<br \/>\nIII. {46, 59, 67, 72, 75}<\/p>\n<ul>\n\t(A) I only<br \/>\n\t(B) I &amp; II<br \/>\n\t(C) I &amp; III<br \/>\n\t(D) II &amp; III<br \/>\n\t(E) I, II, &amp; III\n<\/ul>\n<p>3) Consider the following sets:<br \/>\nL = {3, 4, 5, 5, 6, 7}<br \/>\nM = {2, 2, 2, 8, 8, 8}<br \/>\nN = {15, 15, 15, 15, 15, 15}<\/p>\n<p>Rank those three sets from least standard deviation to greatest standard deviation.<\/p>\n<ul>\n\t(A) L, M, N<br \/>\n\t(B) M, L, N<br \/>\n\t(C) M, N, L<br \/>\n\t(D) N, L, M<br \/>\n\t(E) N, M, L\n<\/ul>\n<p>Do these three questions make your head spin?\u00a0 You have found a good blog article to help you!\u00a0 Explanations to these will appear at the end.<\/p>\n<h2>Spread<\/h2>\n<p>When we are summarizing a list of numbers, typically we want to know the <strong>center<\/strong> and the <strong>spread<\/strong>.\u00a0 (If we are doing an advanced analysis, we would also want to know the shape of the distribution: that can come into play in <a href=\"https:\/\/magoosh.com\/gmat\/complete-guide-to-gmat-integrated-reasoning\/\">IR questions<\/a>.)<\/p>\n<p>The two most typical measures of center are <a href=\"https:\/\/magoosh.com\/gmat\/common-gmat-topic-descriptive-statistics\/\">mean and median<\/a>.\u00a0 Center gives us an idea of where the middle of the distribution of numbers falls.<\/p>\n<p>Measures of spread give us an idea of the spacing of the numbers, how much they are &#8220;spread&#8221; out from each other.\u00a0 A relatively crude measure of spread is the <a href=\"https:\/\/magoosh.com\/gmat\/common-gmat-topic-descriptive-statistics\/\">range<\/a>, which really only tells us about the extreme high and the extreme low, not all the data points in the middle.\u00a0 A more sophisticated measure of spread is the standard deviation.<\/p>\n<h2>Standard Deviation<\/h2>\n<p>Every list of numbers has a mean.\u00a0 Therefore, every number on the list has a <strong>deviation from the mean<\/strong>: that is how far that number is from the mean.<\/p>\n<p>deviation from the mean = (value) \u2013 (mean)<\/p>\n<p>Technically, numbers below the mean have a negative deviation from the mean, and numbers above the mean have a positive deviation from the mean.\u00a0 \u00a0In the list {2, 4, 6, 8, 10}, the mean = 6, so 8 has a deviation from the mean of +2, and 2 has a deviation from the mean of -4.\u00a0 So, parallel to this first list is a second list, the list of deviations from the mean.\u00a0 (It&#8217;s a good exercise to convince yourself why this second list always has a mean of zero.)<\/p>\n<p><em>Here is the technical procedure for calculating the standard deviation<\/em>.\u00a0 We already have List #1, original data set, and List #2, deviations from the mean for each value in List #1.\u00a0 Now, List #3 will be the List #2 squared &#8212; the squared deviations from the mean.\u00a0 This is the list we average: that average is something called the &#8220;variance.&#8221;\u00a0 Then, to undo the effects of squaring, we take a square root, and that final answer is the standard deviation.\u00a0 The OG explains this procedure in the Math Review.\u00a0 If you understand and remember this, great, but chances are good that you don&#8217;t need to know it in all its gory detail if you know the rough and ready facts below.<\/p>\n<p>&nbsp;<\/p>\n<h2>Rough and ready facts about standard deviation<\/h2>\n<p>1) The standard deviation gives us an estimate of the size of a typical deviation from the mean.\u00a0 It&#8217;s a way of &#8220;averaging&#8221; the deviations from the mean, though it is not strictly the mean of that list.<\/p>\n<p>2) If every element in the data set is equal, they all equal the mean, each deviation from the mean is zero, and the standard deviation is<strong> zero<\/strong>.\u00a0 This is the <strong>lowest possible standard deviation<\/strong> for any set to have. (That&#8217;s an excellent GMAT shortcut to know!)<\/p>\n<p>3) If you add the same number to every number on the list, or if you subtract the same number from every number on a list, or if you subtract each number on the list from the same number, all of the new lists produced would have exactly the same standard deviation as the original.\u00a0 Addition and subtraction slides values up and down the number line, but does not change any of the spacing between the numbers.<\/p>\n<p>4) If you multiply the numbers on a list by any values (other than \u00b11), or if you raise the numbers on a list to a power, that <em>always<\/em> changes the standard deviation.\u00a0 Multiplying changes the spacing on the list.\u00a0 In particular, if you multiply each number by k, then you multiply the standard deviation by |k|.<\/p>\n<p>5) If all the numbers on the list are <em>the same distance from the mean<\/em>, that distance <em>is<\/em> the standard deviation.\u00a0 For example, in the set {17, 17, 17, 23, 23, 23}, the mean = 20, and each number is exactly 3 units from the mean, so the standard deviation is 3.<\/p>\n<p>6) If you do anything that &#8220;bunches the numbers together&#8221;, that decrease the standard deviation.\u00a0 If you do anything that &#8220;pulls the numbers further apart&#8221;, that increase the standard deviation.<\/p>\n<p>7) If you include <span style=\"text-decoration: underline\">new<\/span> numbers in the set &#8212; that is tricky, because adding in most numbers will <em>change the mean of the entire set<\/em>, which will change the deviation from the mean for each number on the list, which changes the standard deviation.\u00a0 If you include an additional number or a few additional numbers that are far away from the other numbers, this inclusion will wildly increase the standard deviation.<\/p>\n<p>8) If you include two new numbers that are <span style=\"text-decoration: underline\">symmetrical around the mean<\/span>, then that will not change the mean.\u00a0 If the distance of these two numbers from the mean is greater than the standard deviation, adding them will increase the standard deviation (there&#8217;s a larger &#8220;average&#8221; distance from the mean).\u00a0 If the distance of these two numbers from the mean is less than the standard deviation, adding them will decrease the standard deviation (there&#8217;s a smaller &#8220;average&#8221; distance from the mean).<\/p>\n<p>9) This is an extreme instance of the last case discussed in the previous point.\u00a0 If you include two new numbers <span style=\"text-decoration: underline\">equal to the mean<\/span> (and therefore, with a deviation from the mean of zero), of course that decreases the standard deviation, but we can say more than that.\u00a0 <span style=\"text-decoration: underline\">Of all possible new numbers<\/span> you could include in a set, the new numbers that will <strong><em>most decrease<\/em><\/strong> the overall standard deviation of the set are new entries equal to the mean.\u00a0 That is the single most efficient way to decrease the standard deviation of a set by including new entries to the list.<\/p>\n<p>I realize that&#8217;s a great deal of information.\u00a0 The more you understand how standard deviation works, the more you will understand the interconnection of these &#8220;rough and ready&#8221; facts, which will make the entire list easier to remember.<\/p>\n<p>At this point, you may want to go back to the three practice questions at the beginning of this post, and see if you have any insights.<\/p>\n<p>&nbsp;<\/p>\n<h2>Practice problem solutions<\/h2>\n<p>1) This is a very tricky problem.\u00a0 Starting list has mean = 10 and standard deviation of 1.5.<\/p>\n<p>(A) {2, 10} &#8212; these two don&#8217;t have a mean of 10, so adding them will change the mean; further, one number is &#8220;far away&#8221;, which will wildly decrease the mean, increasing the deviations from the mean of almost every number on the list, and therefore increasing the standard deviation.\u00a0 <strong>WRONG<\/strong><\/p>\n<p>B. {10, 18} &#8212; these two don&#8217;t have a mean of 10, so adding them will change the mean; further, one number is &#8220;far away&#8221;, which will wildly increase the mean, increasing the deviations from the mean of almost every number on the list, and therefore increasing the standard deviation.\u00a0 BTW, (A) &amp; (B) are essentially the same change &#8212; add the mean and add one number eight units from the mean.\u00a0 <strong>WRONG<\/strong><\/p>\n<p>C. {7, 13} &#8212; centered on 10, so this will not change the mean.\u00a0 Both of these are a distance of 3 units from the mean, and this is larger than the standard deviation, so it increases the size of the typical deviation from the mean.\u00a0 <strong>WRONG<\/strong><\/p>\n<p>D. {9, 11} &#8212; centered on 10, so this will not change the mean.\u00a0 Both of these are a distance of 1 units from the mean, and this is less than the standard deviation, so it decreases the size of the typical deviation from the mean.\u00a0 <strong>RIGHT<\/strong><\/p>\n<p>E. {16, 16} &#8212; these are two values far away from everything else, so this will wildly increase the standard deviation.\u00a0 <strong>WRONG<\/strong><\/p>\n<p>Answer = <strong>D<\/strong><\/p>\n<p>2) Original set: Q = {5, 8, 13, 21, 34}.<\/p>\n<p>Notice that Set I is just every number in Q plus 30.\u00a0 When you add the same number to every number in a set, you simply shift it up without changing the spacing, so this doesn&#8217;t change the standard deviation at all.\u00a0 Set I has the same standard deviation as Q.<\/p>\n<p>Notice that Set II is just every number in Q multiplied by 2.\u00a0 Multiplying by a number <em>does<\/em> change the spacing, so this <em>does<\/em> change the standard deviation.\u00a0 Set II does <strong>not<\/strong> have the same standard deviation as Q.<\/p>\n<p>This one is very tricky, and probably is at the outer limit of what the GMAT could ever expect you to see.\u00a0 The spacing between the numbers in Set III, from right to left, is the same as the spacing between the numbers in Q from left to right.\u00a0 Another way to say that is: every number in Set III is a number in Q subtracted from 80.\u00a0 Again, would be very hard to &#8220;notice&#8221;, but once you see that, of course adding and subtraction the same number doesn&#8217;t change the standard deviation.\u00a0 Set III has the same standard deviation.<\/p>\n<p>The correct combination is I and III, so the answer is <strong>C<\/strong>.<\/p>\n<p>3) OK, well first of all, set N has six numbers that are all the same.\u00a0 When all the members of a set are identical, the standard deviation is zero, which is the smallest possible standard deviation.\u00a0 So, automatically, N, must have the lowest.\u00a0 Right away, we can eliminate (A) &amp; (B) &amp; (C).\u00a0 In fact, even if we could do nothing else in this problem, we could guess randomly from the remaining two answers, and the odds would be in our favor.\u00a0 See <a href=\"https:\/\/magoosh.com\/gmat\/guessing-strategies-for-the-gmat\/\">this post<\/a> for more on that strategy.<\/p>\n<p>Now we have to compare the standard deviations of Set L and Set M.\u00a0 In Set L, the mean is clearly 5: two of the entries equal 5, so they have a deviation from the mean of zero, and no entry is more than two units from the mean.\u00a0 By contrast, in Set M, the mean is also 5, and here, every number is 3 units away from the mean, so the standard deviation of M is 3.\u00a0 No number in Set L is as much as 3 units away from the mean, so whatever the standard deviation of L is, it absolutely must be less than 3.\u00a0 That means, Set L has the second largest standard deviation, and Set M has the largest of the three.\u00a0 N, L, M in increasing order.\u00a0 Answer = <strong>D<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>First, here are some challenging practice questions: 1) Set S has a mean of 10 and a standard deviation of 1.5.\u00a0 We are going to add two additional numbers to Set S.\u00a0 Which pair of numbers would decrease the standard deviation the most? (A) {2, 10} (B) {10, 18} (C) {7, 13} (D) {9, 11} [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[112],"tags":[],"ppma_author":[13209],"class_list":["post-2691","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Standard Deviation on the GMAT - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"description\" content=\"Prepare for standard deviation on the GMAT with these practice problems and tips from our very own Magoosh GMAT expert.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Standard Deviation on the GMAT\" \/>\n<meta property=\"og:description\" content=\"Prepare for standard deviation on the GMAT with these practice problems and tips from our very own Magoosh GMAT expert.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-09-10T16:00:54+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-04-20T08:33:31+00:00\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"9 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"Standard Deviation on the GMAT\",\"datePublished\":\"2012-09-10T16:00:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/\"},\"wordCount\":1813,\"commentCount\":49,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Math\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/standard-deviation-on-the-gmat\/\",\"name\":\"Standard Deviation on the GMAT - 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2691","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=2691"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2691\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=2691"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=2691"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=2691"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=2691"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}