{"id":12504,"date":"2012-12-05T09:00:33","date_gmt":"2012-12-05T17:00:33","guid":{"rendered":"https:\/\/magoosh.com\/gre\/?p=12504"},"modified":"2017-01-24T09:31:25","modified_gmt":"2017-01-24T17:31:25","slug":"gre-math-histograms","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gre\/gre-math-histograms\/","title":{"rendered":"GRE Math: Histograms"},"content":{"rendered":"<p>First, a practice question about the following scenario.<\/p>\n<p>In a survey, 86 high school students were randomly selected and asked how many hours of television they had watched in the previous week.\u00a0 The histogram below displays their answers.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gre\/files\/2012\/12\/newest-histogram.jpg\"><img decoding=\"async\" class=\"alignnone wp-image-12984\" src=\"https:\/\/magoosh.com\/gre\/files\/2012\/12\/newest-histogram.jpg\" alt=\"newest histogram\" width=\"553\" height=\"290\" srcset=\"https:\/\/magoosh.com\/gre\/files\/2012\/12\/newest-histogram.jpg 790w, https:\/\/magoosh.com\/gre\/files\/2012\/12\/newest-histogram-300x157.jpg 300w\" sizes=\"(max-width: 553px) 100vw, 553px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>1)<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gre\/files\/2012\/11\/mvm_img2.png\"><img decoding=\"async\" class=\"alignnone wp-image-12507\" src=\"https:\/\/magoosh.com\/gre\/files\/2012\/11\/mvm_img2.png\" alt=\"\" width=\"613\" height=\"77\" srcset=\"https:\/\/magoosh.com\/gre\/files\/2012\/11\/mvm_img2.png 757w, https:\/\/magoosh.com\/gre\/files\/2012\/11\/mvm_img2-300x37.png 300w\" sizes=\"(max-width: 613px) 100vw, 613px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h2>Histogram<\/h2>\n<p>First, a reminder on histograms.\u00a0 Histograms are not simple <a href=\"https:\/\/magoosh.com\/gmat\/2012\/integrated-reasoning-question-type-bar-charts\/\" target=\"_blank\" rel=\"noopener noreferrer\">bar or column charts<\/a>.\u00a0\u00a0 A histogram, like a <a href=\"https:\/\/magoosh.com\/gre\/boxplots-on-the-gre\/\" target=\"_blank\" rel=\"noopener noreferrer\">boxplot<\/a>, shows the distribution of a single quantitative variable.\u00a0 Here, we ask each high school student, &#8220;How many hours of TV did you watch last week?&#8221;, and each high school student gives us a numerical answer.\u00a0 After interviewing 86 students, we have a list of 86 numbers.\u00a0 The histogram is a way to display visually the distribution of those 86 numbers.<\/p>\n<p>The histogram &#8220;chunks&#8221; the values into sections that occupy equal ranges of the variable, and it tells how many numbers on the list fall into that particular chunk.\u00a0 For example, the left-most column on this chart has a height of 13: this means, of the 86 students surveyed, 13 of them gave a numerical response somewhere from 1 hr to 5 hrs.\u00a0 Similarly, each bar tells us how many responses were in that particular range of hours of TV watched.<\/p>\n<p>&nbsp;<\/p>\n<h2>The median<\/h2>\n<p>The <a title=\"GRE Statistics\" href=\"https:\/\/magoosh.com\/gre\/gre-statistics\/\">median<\/a> is the middle of the list.\u00a0 Here, there is an even number of entries on the list, so the median would be the average of the two middle terms &#8212; the average of the 43rd and 44th numbers on the list.\u00a0\u00a0 We can tell that the first column accounts for the first 13 folks on the list, and that the first two columns account for the first 13 + 35 = 48 folks on the list, so by the time we got to the last person on the list in the second column, we would have already passed the 43rd and 44th entries, which\u00a0 means the median would be somewhere in that second column, somewhere between 6-10.<\/p>\n<p>&nbsp;<\/p>\n<h2>The mean<\/h2>\n<p>To calculate the <a title=\"GRE Statistics\" href=\"https:\/\/magoosh.com\/gre\/gre-statistics\/\">mean<\/a>, we would have to add up the exact values of all 86 entries on the list, and then divide that sum by 86.\u00a0\u00a0 In a histogram, we do not have access to exact values: we only know the ranges of numbers &#8212; for example, there are seventeen entries between 11 hrs and 15 hrs, but we don&#8217;t know exactly how many students said 11 hrs, how many said 12 hrs, etc.\u00a0\u00a0 Therefore, <strong><em>it is impossible to calculate the mean from a histogram<\/em><\/strong>.\u00a0 No one will ask you to do that.\u00a0 No one could reasonably expect you to do that, precisely because it is, in fact, impossible.<\/p>\n<p>&nbsp;<\/p>\n<h2>Median vs. Mean<\/h2>\n<p>If it&#8217;s impossible to calculate the mean, then how in tarnation can the GRE expect us to compare the mean to the median?\u00a0\u00a0 Well, here we need to know a slick little bit of statistical reasoning.\u00a0 Consider the following two lists:<\/p>\n<p>List A = {1, 2, 3, 4, 5}<\/p>\n<p>median = 3 and mean = 3<\/p>\n<p>List B = {1, 2, 3, 4, 100}<\/p>\n<p>median = 3 and mean = 22<\/p>\n<p>In changing from List A to List B, we took the last point and slid it out on the scale from x = 5 to x = 100.\u00a0 We made it an &#8220;<strong>outlier<\/strong>&#8220;, that is a point that is noticeably far from the other points.\u00a0\u00a0\u00a0 Notice that median didn&#8217;t change at all.\u00a0 The median doesn&#8217;t care about outliers.\u00a0 The median simply is not affected by outliers.\u00a0\u00a0 By, contrast, the mean changed substantially, because, unlike the median, <strong>the mean is sensitive to outliers<\/strong>.<\/p>\n<p>Now, consider a symmetrical distribution of numbers &#8212; it could be a perfect <a href=\"https:\/\/magoosh.com\/gre\/normal-distribution-on-the-gre\/\" target=\"_self\" rel=\"noopener noreferrer\">Bell Curve<\/a>, or it could be any other symmetrical distribution.\u00a0\u00a0 In any symmetrical distribution, the mean equals the median.\u00a0 Now, consider an asymmetrical distribution: if the outliers are yanked out to one side, then the median will stay put, but the mean will be yanked out in the same direction as the outliers.\u00a0\u00a0 <strong>Outliers pull the mean away from the median<\/strong>.\u00a0\u00a0 Therefore, if you simply notice on which side the outliers lie, then you know in which direction the mean was pulled away from the median.\u00a0 That makes it very easy to compare the two.\u00a0 The comparison is purely visual, and involves absolutely no calculations of any sort.\u00a0\u00a0 (Yes, sometimes you can &#8220;do math&#8221; simply <a href=\"https:\/\/magoosh.com\/gmat\/2012\/a-geometric-and-visual-approach-to-gmat-math\/\">by looking<\/a>!)<\/p>\n<p>Having read this, you may want to look at the QC above before reading the solution below.<\/p>\n<p>&nbsp;<\/p>\n<h2>Practice problem explanation<\/h2>\n<p>1) If you think you have to calculate both the median and the mean, then this question would be impossible, since it&#8217;s impossible to calculate the mean from a histogram.\u00a0\u00a0 If you know the trick discussed above, then all we have to notice is that the outliers, the points most distant from the central hump, are at the upper end.\u00a0 They are on the &#8220;high side&#8221; of the hours scale.\u00a0 The median probably just sits inside that central hump, but the mean has been pulled away from the median in the direction of the outliers, that is, in the direction of the high side of the scale.\u00a0\u00a0 That means, the mean is higher up on the hours scale than is the median.\u00a0 That means, the mean is greater than the median.\u00a0 Answer = <strong>A<\/strong><\/p>\n<p>Notice, this solution involves zero calculations.\u00a0 It is 100% visual.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>First, a practice question about the following scenario. In a survey, 86 high school students were randomly selected and asked how many hours of television they had watched in the previous week.\u00a0 The histogram below displays their answers. &nbsp; &nbsp; 1) &nbsp; Histogram First, a reminder on histograms.\u00a0 Histograms are not simple bar or column [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[39,25],"tags":[],"ppma_author":[12267],"class_list":["post-12504","post","type-post","status-publish","format-standard","hentry","category-data-analysis","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>GRE Math: Histograms - Magoosh Blog \u2014 GRE\u00ae Test<\/title>\n<meta name=\"description\" content=\"Learn more about histograms on the GRE by solving a practice problem in this post by Magoosh\u2019s Mike M\u1d9cGarry.\" \/>\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\/gre\/gre-math-histograms\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GRE Math: Histograms\" \/>\n<meta property=\"og:description\" content=\"Learn more about histograms on the GRE by solving a practice problem in this post by Magoosh\u2019s Mike M\u1d9cGarry.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gre\/gre-math-histograms\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GRE\u00ae Test\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Magoosh\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-12-05T17:00:33+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2017-01-24T17:31:25+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gre\/files\/2012\/12\/newest-histogram.jpg\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGRE\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGRE\" \/>\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=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-histograms\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-histograms\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gre\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"GRE Math: Histograms\",\"datePublished\":\"2012-12-05T17:00:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-histograms\/\"},\"wordCount\":848,\"commentCount\":27,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gre\/#organization\"},\"articleSection\":[\"GRE Data Analysis\",\"GRE Math\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gre\/gre-math-histograms\/\",\"url\":\"https:\/\/magoosh.com\/gre\/gre-math-histograms\/\",\"name\":\"GRE Math: Histograms - 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