{"id":37,"date":"2021-06-22T15:30:10","date_gmt":"2021-06-22T15:30:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-center-of-the-data\/"},"modified":"2023-12-05T08:56:11","modified_gmt":"2023-12-05T08:56:11","slug":"measures-of-the-center-of-the-data","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-center-of-the-data\/","title":{"raw":"Measures of Center: Means and Medians","rendered":"Measures of Center: Means and Medians"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Calculate means, medians, and modes for a set of data<\/li>\r\n \t<li>Determine if a mean or median is a better representation for the center of a set of data<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe \"center\" of a data set is also a way of describing location. The two most widely used measures of the \"center\" of the data are the <strong>mean <\/strong>(average) and the <strong>median<\/strong>. To calculate the mean weight of [latex]50[\/latex] people, add the [latex]50[\/latex] weights together and divide by [latex]50[\/latex]. To find the median weight of the [latex]50[\/latex] people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.\r\n<div class=\"textbox shaded\">\r\n<h3>Note<\/h3>\r\nThe words \"mean\" and \"average\" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is \"arithmetic mean\" and \"average\" is technically a center location. However, in practice among non-statisticians, \"average\" is commonly accepted for \"arithmetic mean.\"\r\n\r\n<\/div>\r\nWhen each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an [latex]x[\/latex] with a bar over it (read \"[latex]x[\/latex] bar\"): [latex]\\displaystyle\\overline{{x}}[\/latex].\r\n\r\nThe Greek letter [latex]\u03bc[\/latex] (pronounced \"mew\") represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.\r\n\r\nTo see that both ways of calculating the mean are the same, consider this sample:\r\n\r\n[latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]\r\n\r\n<center>[latex]\\displaystyle\\overline{{x}}=\\frac{{{1}+{1}+{1}+{2}+{2}+{3}+{4}+{4}+{4}+{4}+{4}}}{{11}}={2.7}[\/latex]<\/center><center>[latex]\\displaystyle\\overline{{x}}=\\frac{{{3}{({1})}+{2}{({2})}+{1}{({3})}+{5}{({4})}}}{{11}}={2.7}[\/latex]<\/center>In the second example, the frequencies are [latex]3[\/latex], [latex]2[\/latex], [latex]1[\/latex], and [latex]5[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Evaluating Algebraic Expression<\/h3>\r\nTo evaluate an algebraic expression, you replace the variable in the expression with a value.\r\n\r\nExample: Evaluate the following algebraic expression where [latex]n=99[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\frac{n+1}{2} = \\frac{99+1}{2} = \\frac{100}{2} = 50[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Summation Notation<\/h3>\r\nSummation notation is used when multiple numbers in a set need to be added together. The Greek letter capital sigma, [latex]\\sigma[\/latex], is used to represent the addition of the numbers in a set.\r\n\r\n<\/div>\r\nYou can quickly find the location of the median by using the expression [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}[\/latex].\r\n\r\nThe letter [latex]n[\/latex] is the total number of data values in the sample. If [latex]n[\/latex] is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If [latex]n[\/latex] is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is [latex]97[\/latex], then [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}=\\frac{{{97}+{1}}}{{2}}={49}[\/latex]. The median is the [latex]49[\/latex]th value in the ordered data. If the total number of data values is [latex]100[\/latex], then [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}=\\frac{{{100}+{1}}}{{2}}[\/latex] = [latex]50.5[\/latex]. The median occurs midway between the [latex]50[\/latex]th and [latex]51[\/latex]st values. The location of the median and the value of the median are <strong>not<\/strong> the same. The upper case letter [latex]M[\/latex] is often used to represent the median. The next example illustrates the location of the median and the value of the median.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):\r\n\r\n[latex]3[\/latex]; [latex]4[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]10[\/latex]; [latex]11[\/latex]; [latex]12[\/latex]; [latex]13[\/latex]; [latex]14[\/latex]; [latex]15[\/latex]; [latex]15[\/latex]; [latex]16[\/latex]; [latex]16[\/latex]; [latex]17[\/latex]; [latex]17[\/latex]; [latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]22[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]29[\/latex]; [latex]31[\/latex]; [latex]32[\/latex]; [latex]33[\/latex]; [latex]33[\/latex]; [latex]34[\/latex]; [latex]34[\/latex]; [latex]35[\/latex]; [latex]37[\/latex]; [latex]40[\/latex]; [latex]44[\/latex]; [latex]44[\/latex]; [latex]47[\/latex]\r\n\r\nCalculate the mean and the median.\r\n[reveal-answer q=\"124075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124075\"]\r\n\r\nThe calculation for the mean is [latex]\\displaystyle\\overline{{x}}=\\frac{{{[{3}+{4}+{({8})}{({2})}+{10}+{11}+{12}+{13}+{14}+{({15})}{({2})}+{({16})}{({2})}+\\ldots+{35}+{37}+{40}+{({44})}{({2})}+{47}]}}}{{40}}={23.6}[\/latex]\r\n\r\nTo find the median, [latex]M[\/latex], first use the formula for the location. The location is: [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}=\\frac{{{40}+{1}}}{{2}}={20.5}[\/latex]\r\n\r\nStarting at the smallest value, the median is located between the [latex]20[\/latex]th and [latex]21[\/latex]st values (the two [latex]24[\/latex]s):\r\n\r\n[latex]3[\/latex]; [latex]4[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]10[\/latex]; [latex]11[\/latex]; [latex]12[\/latex]; [latex]13[\/latex]; [latex]14[\/latex]; [latex]15[\/latex]; [latex]15[\/latex]; [latex]16[\/latex]; [latex]16[\/latex]; [latex]17[\/latex]; [latex]17[\/latex]; [latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]22[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]29[\/latex]; [latex]31[\/latex]; [latex]32[\/latex]; [latex]33[\/latex]; [latex]33[\/latex]; [latex]34[\/latex]; [latex]34[\/latex]; [latex]35[\/latex]; [latex]37[\/latex]; [latex]40[\/latex]; [latex]44[\/latex]; [latex]44[\/latex]; [latex]47[\/latex]\r\n\r\n[latex]\\displaystyle{M}=\\frac{{{24}+{24}}}{{2}}={24}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><strong>Finding the Mean and the Median<\/strong> <strong>Using the TI-83, 83+, 84, 84+ Calculator<\/strong><\/h2>\r\nClear list L1. Pres STAT 4:ClrList. Enter 2nd 1 for list L1. Press ENTER.\r\n\r\nEnter data into the list editor. Press STAT 1:EDIT.\r\n\r\nPut the data values into list L1.\r\n\r\nPress STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and then ENTER.\r\n\r\nPress the down and up arrow keys to scroll.\r\n\r\n[latex]\\displaystyle\\overline{{x}}[\/latex]= [latex]23.6[\/latex], [latex]M[\/latex] = [latex]24[\/latex]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median.\r\n\r\n[latex]3[\/latex]; [latex]4[\/latex]; [latex]5[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex];[latex]10[\/latex]; [latex]10[\/latex]; [latex]11[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]13[\/latex]; [latex]14[\/latex]; [latex]14[\/latex]; [latex]15[\/latex]; [latex]15[\/latex]; [latex]17[\/latex]; [latex]17[\/latex]; [latex]18[\/latex]; [latex]19[\/latex];[latex]19[\/latex]; [latex]19[\/latex]; [latex]21[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]22[\/latex]; [latex]23[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]\r\n\r\n[reveal-answer q=\"124076\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124076\"]\r\nMean:\r\n\r\n[latex]3 + 4 + 5 + 7 + 7 + 7 + 7 + 8 + 8 + 9 + 9 + 10 + 10 + 10 + 10 + 10 + 11 + 12 + 12 + 13 + 14 + 14 + 15 + 15 + [\/latex] [latex]17 + 17 + 18 + 19 + 19 + 19 + 21 + 21 + 22 + 22 + 23 + 24 + 24 + 24 = 544 [\/latex]\r\n\r\n[latex]\\displaystyle\\frac{{544}}{{39}}={13.95}[\/latex]\r\n\r\nMedian: Starting at the smallest value, the median is the [latex]20 [\/latex]th term, which is [latex]13[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSuppose that in a small town of [latex]50[\/latex] people, one person earns $[latex]5,000,000[\/latex] per year and the other [latex]49[\/latex] each earn $[latex]30,000 [\/latex]. Which is the better measure of the \"center\": the mean or the median?\r\n\r\n[reveal-answer q=\"124077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124077\"]\r\n\r\n[latex]\\displaystyle\\overline{{x}}=\\frac{{5,000,000}+{49}({30,000})}{{50}}={129400}[\/latex], [latex]\\\\{M}={30000}[\/latex]\r\n\r\n(There are [latex]49[\/latex] people who earn $[latex]30,000[\/latex] and one person who earns $[latex]5,000,000[\/latex].)\r\n\r\nThe median is a better measure of the \"center\" than the mean because [latex]49[\/latex] of the values are [latex]30,000[\/latex] and one is [latex]5,000,000[\/latex]. The [latex]5,000,000[\/latex] is an outlier. The [latex]30,000[\/latex] gives us a better sense of the middle of the data.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIn a sample of [latex]60[\/latex] households, one house is worth $[latex]2,500,000[\/latex]. Half of the rest are worth $[latex]280,000[\/latex], and all the others are worth $[latex]315,000[\/latex]. Which is the better measure of the \"center\": the mean or the median?\r\n[reveal-answer q=\"124078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124078\"]\r\nThe median is the better measure of the \"center\" because [latex]59[\/latex] of the values are $[latex]280,000[\/latex] and one is $[latex]2,500,000[\/latex]. The $[latex]2,500,000[\/latex] is an outlier. Either $[latex]280,000[\/latex] or $[latex]315,000[\/latex] gives us a better sense of the middle of the data.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother measure of the center is the mode. The <strong>mode<\/strong> is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nStatistics exam scores for [latex]20[\/latex] students are as follows:\r\n\r\n[latex]50[\/latex], [latex]53[\/latex], [latex]59[\/latex], [latex]59[\/latex], [latex]63[\/latex], [latex]63[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]76[\/latex], [latex]78[\/latex], [latex]81[\/latex], [latex]83[\/latex], [latex]84[\/latex], [latex]84[\/latex], [latex]84[\/latex], [latex]90[\/latex], [latex]93[\/latex]\r\n\r\nFind the mode.\r\n\r\n[reveal-answer q=\"124079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124079\"]\r\n\r\nThe most frequent score is [latex]72[\/latex], which occurs five times. Mode = [latex]72[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe number of books checked out from the library from [latex]25[\/latex] students are as follows:\r\n\r\n[latex]0[\/latex], [latex]0[\/latex], [latex]0[\/latex], [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex], [latex]3[\/latex], [latex]4[\/latex], [latex]4[\/latex], [latex]5[\/latex], [latex]5[\/latex], [latex]7[\/latex], [latex]7[\/latex], [latex]7[\/latex], [latex]7[\/latex], [latex]8[\/latex], [latex]8[\/latex], [latex]8[\/latex], [latex]9[\/latex], [latex]10[\/latex], [latex]10[\/latex], [latex]11[\/latex], [latex]11[\/latex], [latex]12[\/latex], [latex]12[\/latex]\r\n\r\nFind the mode.\r\n\r\n[reveal-answer q=\"124080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124080\"]\r\nThe most frequent number of books is [latex]7[\/latex], which occurs four times. Mode = [latex]7[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFive real estate exam scores are [latex]430[\/latex], [latex]430[\/latex], [latex]480[\/latex], [latex]480[\/latex], [latex]495[\/latex]. The data set is bimodal because the scores [latex]430[\/latex] and [latex]480[\/latex] each occur twice.\r\n\r\nWhen is the mode the best measure of the \"center\"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.\r\n<div class=\"textbox shaded\">\r\n<h3>Note<\/h3>\r\nThe mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red.\r\n\r\n<\/div>\r\nStatistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFive credit scores are [latex]680[\/latex], [latex]680[\/latex], [latex]700[\/latex], [latex]720[\/latex], [latex]720[\/latex]. The data set is bimodal because the scores [latex]680[\/latex] and [latex]720[\/latex] each occur twice. Consider the annual earnings of workers at a factory. The mode is [latex]$25,000[\/latex] and occurs [latex]150[\/latex] times out of [latex]301[\/latex]. The median is [latex]$50,000[\/latex] and the mean is [latex]$47,500[\/latex]. What would be the best measure of the \"center\"?\r\n\r\n[reveal-answer q=\"124081\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124081\"]\r\nBecause $[latex]25,000[\/latex] occurs nearly half the time, the mode would be the best measure of the center because the median and mean don't represent what most people make at the factory.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video from Khan Academy on finding the mean, median, and mode of a set of data.\r\n\r\nhttps:\/\/youtu.be\/k3aKKasOmIw","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Calculate means, medians, and modes for a set of data<\/li>\n<li>Determine if a mean or median is a better representation for the center of a set of data<\/li>\n<\/ul>\n<\/div>\n<p>The &#8220;center&#8221; of a data set is also a way of describing location. The two most widely used measures of the &#8220;center&#8221; of the data are the <strong>mean <\/strong>(average) and the <strong>median<\/strong>. To calculate the mean weight of [latex]50[\/latex] people, add the [latex]50[\/latex] weights together and divide by [latex]50[\/latex]. To find the median weight of the [latex]50[\/latex] people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.<\/p>\n<div class=\"textbox shaded\">\n<h3>Note<\/h3>\n<p>The words &#8220;mean&#8221; and &#8220;average&#8221; are often used interchangeably. The substitution of one word for the other is common practice. The technical term is &#8220;arithmetic mean&#8221; and &#8220;average&#8221; is technically a center location. However, in practice among non-statisticians, &#8220;average&#8221; is commonly accepted for &#8220;arithmetic mean.&#8221;<\/p>\n<\/div>\n<p>When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an [latex]x[\/latex] with a bar over it (read &#8220;[latex]x[\/latex] bar&#8221;): [latex]\\displaystyle\\overline{{x}}[\/latex].<\/p>\n<p>The Greek letter [latex]\u03bc[\/latex] (pronounced &#8220;mew&#8221;) represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.<\/p>\n<p>To see that both ways of calculating the mean are the same, consider this sample:<\/p>\n<p>[latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]3[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\overline{{x}}=\\frac{{{1}+{1}+{1}+{2}+{2}+{3}+{4}+{4}+{4}+{4}+{4}}}{{11}}={2.7}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\overline{{x}}=\\frac{{{3}{({1})}+{2}{({2})}+{1}{({3})}+{5}{({4})}}}{{11}}={2.7}[\/latex]<\/div>\n<p>In the second example, the frequencies are [latex]3[\/latex], [latex]2[\/latex], [latex]1[\/latex], and [latex]5[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Evaluating Algebraic Expression<\/h3>\n<p>To evaluate an algebraic expression, you replace the variable in the expression with a value.<\/p>\n<p>Example: Evaluate the following algebraic expression where [latex]n=99[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{n+1}{2} = \\frac{99+1}{2} = \\frac{100}{2} = 50[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: Summation Notation<\/h3>\n<p>Summation notation is used when multiple numbers in a set need to be added together. The Greek letter capital sigma, [latex]\\sigma[\/latex], is used to represent the addition of the numbers in a set.<\/p>\n<\/div>\n<p>You can quickly find the location of the median by using the expression [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}[\/latex].<\/p>\n<p>The letter [latex]n[\/latex] is the total number of data values in the sample. If [latex]n[\/latex] is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If [latex]n[\/latex] is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is [latex]97[\/latex], then [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}=\\frac{{{97}+{1}}}{{2}}={49}[\/latex]. The median is the [latex]49[\/latex]th value in the ordered data. If the total number of data values is [latex]100[\/latex], then [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}=\\frac{{{100}+{1}}}{{2}}[\/latex] = [latex]50.5[\/latex]. The median occurs midway between the [latex]50[\/latex]th and [latex]51[\/latex]st values. The location of the median and the value of the median are <strong>not<\/strong> the same. The upper case letter [latex]M[\/latex] is often used to represent the median. The next example illustrates the location of the median and the value of the median.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):<\/p>\n<p>[latex]3[\/latex]; [latex]4[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]10[\/latex]; [latex]11[\/latex]; [latex]12[\/latex]; [latex]13[\/latex]; [latex]14[\/latex]; [latex]15[\/latex]; [latex]15[\/latex]; [latex]16[\/latex]; [latex]16[\/latex]; [latex]17[\/latex]; [latex]17[\/latex]; [latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]22[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]29[\/latex]; [latex]31[\/latex]; [latex]32[\/latex]; [latex]33[\/latex]; [latex]33[\/latex]; [latex]34[\/latex]; [latex]34[\/latex]; [latex]35[\/latex]; [latex]37[\/latex]; [latex]40[\/latex]; [latex]44[\/latex]; [latex]44[\/latex]; [latex]47[\/latex]<\/p>\n<p>Calculate the mean and the median.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124075\">Show Solution<\/span><\/p>\n<div id=\"q124075\" class=\"hidden-answer\" style=\"display: none\">\n<p>The calculation for the mean is [latex]\\displaystyle\\overline{{x}}=\\frac{{{[{3}+{4}+{({8})}{({2})}+{10}+{11}+{12}+{13}+{14}+{({15})}{({2})}+{({16})}{({2})}+\\ldots+{35}+{37}+{40}+{({44})}{({2})}+{47}]}}}{{40}}={23.6}[\/latex]<\/p>\n<p>To find the median, [latex]M[\/latex], first use the formula for the location. The location is: [latex]\\displaystyle\\frac{{{n}+{1}}}{{2}}=\\frac{{{40}+{1}}}{{2}}={20.5}[\/latex]<\/p>\n<p>Starting at the smallest value, the median is located between the [latex]20[\/latex]th and [latex]21[\/latex]st values (the two [latex]24[\/latex]s):<\/p>\n<p>[latex]3[\/latex]; [latex]4[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]10[\/latex]; [latex]11[\/latex]; [latex]12[\/latex]; [latex]13[\/latex]; [latex]14[\/latex]; [latex]15[\/latex]; [latex]15[\/latex]; [latex]16[\/latex]; [latex]16[\/latex]; [latex]17[\/latex]; [latex]17[\/latex]; [latex]18[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]22[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]25[\/latex]; [latex]26[\/latex]; [latex]26[\/latex]; [latex]27[\/latex]; [latex]27[\/latex]; [latex]29[\/latex]; [latex]29[\/latex]; [latex]31[\/latex]; [latex]32[\/latex]; [latex]33[\/latex]; [latex]33[\/latex]; [latex]34[\/latex]; [latex]34[\/latex]; [latex]35[\/latex]; [latex]37[\/latex]; [latex]40[\/latex]; [latex]44[\/latex]; [latex]44[\/latex]; [latex]47[\/latex]<\/p>\n<p>[latex]\\displaystyle{M}=\\frac{{{24}+{24}}}{{2}}={24}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><strong>Finding the Mean and the Median<\/strong> <strong>Using the TI-83, 83+, 84, 84+ Calculator<\/strong><\/h2>\n<p>Clear list L1. Pres STAT 4:ClrList. Enter 2nd 1 for list L1. Press ENTER.<\/p>\n<p>Enter data into the list editor. Press STAT 1:EDIT.<\/p>\n<p>Put the data values into list L1.<\/p>\n<p>Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and then ENTER.<\/p>\n<p>Press the down and up arrow keys to scroll.<\/p>\n<p>[latex]\\displaystyle\\overline{{x}}[\/latex]= [latex]23.6[\/latex], [latex]M[\/latex] = [latex]24[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median.<\/p>\n<p>[latex]3[\/latex]; [latex]4[\/latex]; [latex]5[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]7[\/latex]; [latex]8[\/latex]; [latex]8[\/latex]; [latex]9[\/latex]; [latex]9[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex];[latex]10[\/latex]; [latex]10[\/latex]; [latex]11[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]13[\/latex]; [latex]14[\/latex]; [latex]14[\/latex]; [latex]15[\/latex]; [latex]15[\/latex]; [latex]17[\/latex]; [latex]17[\/latex]; [latex]18[\/latex]; [latex]19[\/latex];[latex]19[\/latex]; [latex]19[\/latex]; [latex]21[\/latex]; [latex]21[\/latex]; [latex]22[\/latex]; [latex]22[\/latex]; [latex]23[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]; [latex]24[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124076\">Show Solution<\/span><\/p>\n<div id=\"q124076\" class=\"hidden-answer\" style=\"display: none\">\nMean:<\/p>\n<p>[latex]3 + 4 + 5 + 7 + 7 + 7 + 7 + 8 + 8 + 9 + 9 + 10 + 10 + 10 + 10 + 10 + 11 + 12 + 12 + 13 + 14 + 14 + 15 + 15 +[\/latex] [latex]17 + 17 + 18 + 19 + 19 + 19 + 21 + 21 + 22 + 22 + 23 + 24 + 24 + 24 = 544[\/latex]<\/p>\n<p>[latex]\\displaystyle\\frac{{544}}{{39}}={13.95}[\/latex]<\/p>\n<p>Median: Starting at the smallest value, the median is the [latex]20[\/latex]th term, which is [latex]13[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Suppose that in a small town of [latex]50[\/latex] people, one person earns $[latex]5,000,000[\/latex] per year and the other [latex]49[\/latex] each earn $[latex]30,000[\/latex]. Which is the better measure of the &#8220;center&#8221;: the mean or the median?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124077\">Show Solution<\/span><\/p>\n<div id=\"q124077\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\displaystyle\\overline{{x}}=\\frac{{5,000,000}+{49}({30,000})}{{50}}={129400}[\/latex], [latex]\\\\{M}={30000}[\/latex]<\/p>\n<p>(There are [latex]49[\/latex] people who earn $[latex]30,000[\/latex] and one person who earns $[latex]5,000,000[\/latex].)<\/p>\n<p>The median is a better measure of the &#8220;center&#8221; than the mean because [latex]49[\/latex] of the values are [latex]30,000[\/latex] and one is [latex]5,000,000[\/latex]. The [latex]5,000,000[\/latex] is an outlier. The [latex]30,000[\/latex] gives us a better sense of the middle of the data.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>In a sample of [latex]60[\/latex] households, one house is worth $[latex]2,500,000[\/latex]. Half of the rest are worth $[latex]280,000[\/latex], and all the others are worth $[latex]315,000[\/latex]. Which is the better measure of the &#8220;center&#8221;: the mean or the median?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124078\">Show Solution<\/span><\/p>\n<div id=\"q124078\" class=\"hidden-answer\" style=\"display: none\">\nThe median is the better measure of the &#8220;center&#8221; because [latex]59[\/latex] of the values are $[latex]280,000[\/latex] and one is $[latex]2,500,000[\/latex]. The $[latex]2,500,000[\/latex] is an outlier. Either $[latex]280,000[\/latex] or $[latex]315,000[\/latex] gives us a better sense of the middle of the data.\n<\/div>\n<\/div>\n<\/div>\n<p>Another measure of the center is the mode. The <strong>mode<\/strong> is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Statistics exam scores for [latex]20[\/latex] students are as follows:<\/p>\n<p>[latex]50[\/latex], [latex]53[\/latex], [latex]59[\/latex], [latex]59[\/latex], [latex]63[\/latex], [latex]63[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]72[\/latex], [latex]76[\/latex], [latex]78[\/latex], [latex]81[\/latex], [latex]83[\/latex], [latex]84[\/latex], [latex]84[\/latex], [latex]84[\/latex], [latex]90[\/latex], [latex]93[\/latex]<\/p>\n<p>Find the mode.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124079\">Show Solution<\/span><\/p>\n<div id=\"q124079\" class=\"hidden-answer\" style=\"display: none\">\n<p>The most frequent score is [latex]72[\/latex], which occurs five times. Mode = [latex]72[\/latex].\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The number of books checked out from the library from [latex]25[\/latex] students are as follows:<\/p>\n<p>[latex]0[\/latex], [latex]0[\/latex], [latex]0[\/latex], [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex], [latex]3[\/latex], [latex]4[\/latex], [latex]4[\/latex], [latex]5[\/latex], [latex]5[\/latex], [latex]7[\/latex], [latex]7[\/latex], [latex]7[\/latex], [latex]7[\/latex], [latex]8[\/latex], [latex]8[\/latex], [latex]8[\/latex], [latex]9[\/latex], [latex]10[\/latex], [latex]10[\/latex], [latex]11[\/latex], [latex]11[\/latex], [latex]12[\/latex], [latex]12[\/latex]<\/p>\n<p>Find the mode.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124080\">Show Solution<\/span><\/p>\n<div id=\"q124080\" class=\"hidden-answer\" style=\"display: none\">\nThe most frequent number of books is [latex]7[\/latex], which occurs four times. Mode = [latex]7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Five real estate exam scores are [latex]430[\/latex], [latex]430[\/latex], [latex]480[\/latex], [latex]480[\/latex], [latex]495[\/latex]. The data set is bimodal because the scores [latex]430[\/latex] and [latex]480[\/latex] each occur twice.<\/p>\n<p>When is the mode the best measure of the &#8220;center&#8221;? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.<\/p>\n<div class=\"textbox shaded\">\n<h3>Note<\/h3>\n<p>The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red.<\/p>\n<\/div>\n<p>Statistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Five credit scores are [latex]680[\/latex], [latex]680[\/latex], [latex]700[\/latex], [latex]720[\/latex], [latex]720[\/latex]. The data set is bimodal because the scores [latex]680[\/latex] and [latex]720[\/latex] each occur twice. Consider the annual earnings of workers at a factory. The mode is [latex]$25,000[\/latex] and occurs [latex]150[\/latex] times out of [latex]301[\/latex]. The median is [latex]$50,000[\/latex] and the mean is [latex]$47,500[\/latex]. What would be the best measure of the &#8220;center&#8221;?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124081\">Show Solution<\/span><\/p>\n<div id=\"q124081\" class=\"hidden-answer\" style=\"display: none\">\nBecause $[latex]25,000[\/latex] occurs nearly half the time, the mode would be the best measure of the center because the median and mean don&#8217;t represent what most people make at the factory.\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video from Khan Academy on finding the mean, median, and mode of a set of data.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/k3aKKasOmIw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-37\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax, Statistics, Measures of the Center of the Data. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-5-measures-of-the-center-of-the-data\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-5-measures-of-the-center-of-the-data<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy. <strong>Authored by<\/strong>: Khan Academy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/k3aKKasOmIw\">https:\/\/youtu.be\/k3aKKasOmIw<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Measures of the Center of the Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-5-measures-of-the-center-of-the-data\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"copyrighted_video\",\"description\":\"Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy\",\"author\":\"Khan 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