{"id":269,"date":"2017-08-23T04:36:17","date_gmt":"2017-08-23T04:36:17","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=269"},"modified":"2018-06-13T20:36:51","modified_gmt":"2018-06-13T20:36:51","slug":"summarizing-data","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/summarizing-data\/","title":{"rendered":"Summarizing Data"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/the-effects-on-measures-of-changes-in-dimensions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/further-reading-8\">Further Reading \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Summarizing Data<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson you will learn to represent data in multiple ways and compute ways to summarize that data. You will also explore potential sources of bias.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>We spent quite a bit of time reviewing volume, surface area, and areas of different three-dimensional figures. We&#8217;ve also refreshed your memory about some significant formulas and provided some extra practice using them.<\/li>\n<\/ul>\n<section>\n<h3>All About Data<\/h3>\n<p>In this section, we\u2019ll explore various aspects of data, from collection to representation and analysis. We won\u2019t do it in that order, though, because it\u2019s just more meaningful to first look at data and how it\u2019s represented. Then you will be able to look at data more critically. That is, you can ask yourself questions about how the data was collected and whether it truly represents what its author says it does.<\/p>\n<p>A <abbr title=\"A set of numbers and how often they occur within a statistical population.\">distribution<\/abbr> is an arrangement of values that gives information about the frequency with which they occur. A <em>histogram<\/em> is one representation of a distribution. A <em>frequency table<\/em> is another.<\/p>\n<p>The center of a distribution of values is simply the middle of that distribution. A measure of central tendency is a way of defining the center of a distribution. Perhaps this sounds more complicated than it really is. Just think of a measure of central tendency as one way to summarize a set of (data) values.<\/p>\n<p>In this section we&#8217;ll define and compute the three most common measures of central tendency: the <abbr title=\"The average value of a set of numbers.\">mean<\/abbr>, the m<abbr title=\"The middle value in a set of numbers.\">edian<\/abbr>, and the <abbr title=\"The number in a set that occurs most frequently.\">mode<\/abbr>.<\/p>\n<p>The <abbr title=\"the difference between the highest and lowest data values in the set.\">range<\/abbr> is another measure that gives information about a set of data, but it is not a measure of central tendency. That\u2019s because it doesn\u2019t define the center of the distribution of the data. It\u2019s actually called a measure of dispersion, because it describes the way that the data are dispersed.<\/p>\n<h3>Measures of Central Tendency<\/h3>\n<h4>The Mean<\/h4>\n<p>The mean of a set of data is the arithmetic average. That\u2019s the one you\u2019re used to computing. It\u2019s not called the average in statistics, because there are different kinds of averages.<\/p>\n<p>To find the mean, simply add up the values in a set of data and divide by the number of values in the set.<\/p>\n<div style=\"display: none;\">\n<div id=\"mean\" style=\"padding: 10px; background: #fff;\">\n<h4>The Mean<\/h4>\n<p>For example, the mean of the set {2, 4, 6, 8, 15} is equal to the sum of these numbers divided by the number of members of the set, 35 over 7.. Interestingly, The sets {7, 7, 7, 7, 7} and {0, 0, 0, 0, 35} also have a mean of seven.<\/p>\n<p>You can see that the mean gives some information about a data set, but not all of the information you might want.<\/p>\n<\/div>\n<\/div>\n<h4>The Median<\/h4>\n<p>The median of a set of data is the value in the middle of the set. To find the median, order the values in the set from least to greatest (or from greatest to least).<\/p>\n<div style=\"display: none;\">\n<div id=\"median\" style=\"padding: 10px; background: #fff;\">\n<h4>The Median<\/h4>\n<p>For example, the median of the set {2, 4, <abbr title=\"The set {1, 6, 4, 10, 7} is equivalent to the ordered set {1, 4, 6, 7, 10}. The middle number in this set is 6.\">6<\/abbr>, 8, 15} is six. And the median of the set {1, <abbr title=\"The set {1, 6, 4, 10, 7} is equivalent to the ordered set {1, 4, 6, 7, 10}. The middle number in this set is 6.\">6<\/abbr>, 4, 10, 7} is also six.<\/p>\n<p>What if there is an even number of values in the set? What strategy for finding the median of a set with an even number of values would you suggest? Perhaps it\u2019s intuitive.<\/p>\n<p>The people who agree on these matters have decided that the median of a set with an even number of values is the mean, or average, of the two middle values.<\/p>\n<p>For example, the median of the set {0, 2, 4, 6, 8, 10} is the mean of four and six, or five. The median of the set {1, 1, 1, 9, 9, 9} is the mean of one and nine, which is also five.<\/p>\n<\/div>\n<\/div>\n<h4>The Mode<\/h4>\n<p>The mode of a set of data is the value that occurs most often. The mode of the set {4, 5, 3, 5, 4, 2, 1, 2, 4} is four, because four occurs more often than any other number in the set.<\/p>\n<p>Unfortunately, it\u2019s not as simple as it sounds. Sometimes a set may have more than one mode or no mode at all.<\/p>\n<p>So how many modes are in the set {5, 4, 2, 4, 5, 2}? The answer is three. Every value in the set is also a mode of the set. Take some time to convince yourself that this is true.<\/p>\n<h4>A Measure of Dispersion: The Range<\/h4>\n<p>The range of a set of data is the difference between the greatest and least values in the set. That makes the range positive. To find the range of a set of data, subtract the least value in the set from the greatest value.<\/p>\n<div style=\"display: none;\">\n<div id=\"range\" style=\"padding: 10px; background: #fff;\">\n<h4>The Range<\/h4>\n<p>For example, the range of the set {1, 4, 6, 4, 2, 6, 3, 5, 9} is nine minus one, or eight. The range of the set {0, 0, 0, 0, 0, 0, 0, 0, 0, 8} is eight minus zero, which is also eight.<\/p>\n<\/div>\n<\/div>\n<p>You can see that the range gives a pretty poor summary of a set of data, because it relies on only two values in the set. <abbr title=\"Typically, the median of a set like this one is most meaningful, because the range of values is so great.\">What measure<\/abbr> would be more meaningful for a set of salaries that included yours and Bill Gates\u2019s?<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>For the set below, which measure is <em>least<\/em>?<\/p>\n<blockquote><p>{1, 8, 7, 3, 7, 6, 5, 6, 2}<\/p><\/blockquote>\n<ol>\n<li>Mean<\/li>\n<li>Median<\/li>\n<li>Mode<\/li>\n<li>Range<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice A is the correct answer. The mean of this set is equal to the sum of the values in the set, forty-five, divided by the number of values in the set, nine. So the mean is five. The median of the set is the middle number when the set is ordered. The ordered set is {1, 2, 3, 5, 6, 6, 7, 7, 8}, which has a median of six. The mode of the set is the value that occurs most often. The set has two modes: six and seven. The range of the set is the difference between the greatest and least values in the set. The range is eight minus one, or seven. The least of these measures is the mean, which is five.<\/p>\n<\/section>\n<h3>Review<\/h3>\n<ul>\n<li>The mean of a set of data is the arithmetic average.<\/li>\n<li>The median of a set of data is the value in the middle of the set.<\/li>\n<li>The mode of a set of data is the value that occurs most often.<\/li>\n<li>The range of a set of data is the difference between the greatest and least value in the set.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/the-effects-on-measures-of-changes-in-dimensions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/further-reading-8\">Further Reading \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Further Reading \u27a1 Summarizing Data Objective In this lesson you will learn to represent data in multiple ways and compute ways to summarize that data. You will also explore potential sources of bias. Previously Covered We spent quite a bit of time reviewing volume, surface area, and areas of different three-dimensional figures. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-269","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=269"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/269\/revisions"}],"predecessor-version":[{"id":1432,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/269\/revisions\/1432"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}