{"id":985,"date":"2021-10-10T00:29:30","date_gmt":"2021-10-10T00:29:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=985"},"modified":"2021-11-30T22:55:21","modified_gmt":"2021-11-30T22:55:21","slug":"5-1-3-mean-and-weighted-mean","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/5-1-3-mean-and-weighted-mean\/","title":{"raw":"5.1.3: Descriptive Statistics: Mean","rendered":"5.1.3: Descriptive Statistics: Mean"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the mean of a set of numbers&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6529,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Calculate the mean of a set of numbers<\/span><\/li>\r\n \t<li>Calculate the mean from a frequency table or a grouped frequency table.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>KEY words<\/h3>\r\n<ul>\r\n \t<li><strong>Mean<\/strong>: the arithmetic average<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h2>Mean<\/h2>\r\nSo far we have studied two averages; the median and the mode. A third average is the <strong><em>mean<\/em><\/strong>, which is often called the <strong><em>arithmetic average<\/em><\/strong>. It is computed by dividing the sum of the data values by the number of data values. Generally speaking, when people talk about average, it is usually the mean they are referring to. Technically, the mean is the arithmetic average, and average is a central location, but mean and average are used interchangeably in common practice. \u00a0The mean is a central value of a <em>finite<\/em> set of numbers: specifically, the sum of the values divided by the number of values.\r\n<div class=\"textbox shaded\">\r\n<h3>Mean<\/h3>\r\nThe mean of a set of [latex]n[\/latex] numerical data points is the arithmetic average of the numbers.\r\n\r\n[latex]\\text{mean}={\\frac{\\text{sum of values in data set}}{n}}[\/latex]\r\n\r\nwhere [latex]n[\\latex] is the number of data points.\r\n\r\n<\/div>\r\nSuppose Eva\u2019s first three test scores were [latex]85,88,\\text{and }94[\/latex]. To find the mean score, we would add them and divide by [latex]3[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ {\\frac{85+88+94}{3}} &amp;=&amp;{\\frac{267}{3}}\\\\ &amp;=&amp;89\\end{array}[\/latex]<\/p>\r\nThe mean test score is [latex]89[\/latex] points.\r\n\r\nThe mean of a set of data is denoted by either the Greek letter [latex]\u03bc[\/latex] (mu) when representing the<em> <strong>population mean<\/strong><\/em>, or\u00a0[latex]\\displaystyle\\overline{{x}}[\/latex] (read\u00a0\"[latex]x[\/latex] bar\") when representing a <em><strong>sample mean<\/strong><\/em>.\u00a0One 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<div class=\"textbox shaded\">\r\n<h3>Calculate the mean of a set of numbers.<\/h3>\r\n<ol id=\"eip-id1168468272241\" class=\"stepwise\">\r\n \t<li>Write the formula for the mean\r\n[latex]\\text{mean}={\\frac{\\text{sum of values in the data set}}{n}}[\/latex]<\/li>\r\n \t<li>Find the sum of all the values in the set. Write the sum in the numerator.<\/li>\r\n \t<li>Count the number, [latex]n[\/latex], of values in the set. Write this number in the denominator.<\/li>\r\n \t<li>Divide the numerator by the denominator.<\/li>\r\n \t<li>Check to see that the mean is <em>reasonable<\/em>. The mean is an average which is a measure of the center of the data set, so it should be greater than the least number and less than the greatest number in the set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the mean of the numbers [latex]8,12,15, 9,\\text{ and }6[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168467435661\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 501.9px; height: 30px;\">Write the formula for the mean:<\/td>\r\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\frac{\\text{sum of all the numbers}}{n}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 501.9px; height: 30px;\">Write the sum of the numbers in the numerator.<\/td>\r\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\frac{8+12+15+9+6}{n}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"width: 501.9px; height: 30px;\">Count how many numbers are in the set. There are [latex]5[\/latex] numbers in the set, so [latex]n=5[\/latex] .<\/td>\r\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\frac{8+12+15+9+6}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 501.9px; height: 15px;\">Add the numbers in the numerator.<\/td>\r\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}={\\frac{50}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 501.9px; height: 15px;\">Then divide.<\/td>\r\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}=10[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30.8281px;\">\r\n<td style=\"width: 501.9px; height: 30.8281px;\">Check to see that the mean is 'typical': [latex]10[\/latex] is neither less than [latex]6[\/latex] nor greater than [latex]15[\/latex].<\/td>\r\n<td style=\"width: 333.1px; height: 30.8281px;\">The mean is [latex]10[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146388[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146389[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;\">A table that shows the frequency of each data value in a sample is called a\u00a0<\/span><strong style=\"font-size: 1em;\"><em>frequency table<\/em><\/strong><span style=\"font-size: 1em;\">. It is used when<\/span>\u00a0each value in the data set is not unique so that the frequency of the value can be shown. 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.\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:\r\n<table style=\"border-collapse: collapse; width: 25.8232%; height: 348px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 50%; height: 12px;\">Number of Months<\/th>\r\n<th style=\"width: 50%; height: 12px;\">Frequency<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]11[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]13[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]14[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]15[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]17[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]21[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]22[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]25[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]26[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]27[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]29[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]31[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]32[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]33[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]34[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]35[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]37[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]40[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]44[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">[latex]47[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCalculate the mean.\r\n<h4>Solution<\/h4>\r\nThe calculation for the mean is:\r\n\r\n[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}\\text{months}[\/latex]\r\n\r\n<\/div>\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, median, and mode.\r\n<table style=\"border-collapse: collapse; width: 2.5021234329204742%; height: 470px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">Number of Months<\/td>\r\n<td style=\"width: 50%;\">Frequency<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]11[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]13[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]14[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]15[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]17[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]19[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]21[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]22[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]23[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm076\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"hjm076\"]\r\nMean:\r\n\r\nSum of values = [latex]3 + 4 + 5 + {({7})}{({4})} + {({8)}}{({2})} + {({9})}{({2})} + {({10})}{({5})} + 11 + {({12})}{({2})} + 13 + {({14})}{({2})} + {({15})}{({2})} + {({17})}{({2})} + 18 + {({19})}{({3})} + {({21})}{({2})} + {({22})}{({2})} + 23 + {({24})}{({4})} = 544 [\/latex] months\r\n\r\n[latex]\\displaystyle\\overline{{x}} = \\displaystyle\\frac{{544}}{{39}}={13.95}[\/latex] months\r\n\r\nMedian: Since there are 39 values the median will be in the 20th place. Starting at the smallest value, the median is the [latex]20 [\/latex]th term, which is [latex]13[\/latex] months.\r\n\r\nMode: The highest frequency is [latex]5[\/latex] that occurs at [latex]10[\/latex] months, so the mode is 10 months\r\n\r\n[\/hidden-answer]\r\n\r\nNotice that mean = [latex]13.95[\/latex] months; median = [latex]13[\/latex] months; mode =\u00a0[latex]10[\/latex] months. Although they are all different, they are each a measure of average.\r\n\r\n<\/div>\r\n<h3>Mean and Median. What's the Difference?<\/h3>\r\nSince mean and median are both measures of the center of the data set, how do we know which one is more representative of the data set? The next example, considers this question:\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\nMean: [latex]\\displaystyle\\overline{{x}}=\\frac{{5,000,000}+{49}({30,000})}{{50}}={129,400}[\/latex]\r\n\r\nMedian: [latex]\\\\{M}={30,000}[\/latex]\r\n\r\nThere are [latex]49[\/latex] people who earn $[latex]30,000[\/latex] (the mode is $30,000) and only one person who earns $[latex]5,000,000[\/latex], so median\u00a0is a better measure of the \"center\" than the mean. The $[latex]30,000[\/latex] gives us a better sense of the middle of the data and is more representative of the data.\r\n\r\nThe $[latex]5,000,000[\/latex] value is an <strong><em>outlier<\/em><\/strong>. Outliers affect the mean much more than they affect the median.\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIn a sample of [latex]61[\/latex] households, one house is worth $[latex]2,500,000[\/latex]. Half of the rest (30) are worth $[latex]280,000[\/latex], and all the others (30) 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\" than the mean because [latex]60[\/latex] of the values are either $[latex]280,000[\/latex] or $[latex]315,000[\/latex] and only 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\nWhen the data set consists of values that are basically symmetric about the center, the mean and the median get closer together. When the data set consists of values that are <em>skewed<\/em> to the left or right of center, or if the data set contains <em>outliers<\/em>, the median is the better choice.\r\n\r\nIn addition, the<em> Law of Large Numbers<\/em> says that if we take samples of larger and larger size from any population, then the mean, [latex]\\displaystyle\\overline{{x}}[\/latex], of the sample is very likely to get closer and closer to the population mean, [latex]\u00b5[\/latex].\r\n<h3>Calculating the Mean of Grouped Frequency Tables<\/h3>\r\nWhen only grouped data is available, we do not know the individual data values (we only know classes and class frequencies); therefore, we cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a <em><strong>frequency table<\/strong><\/em>. Remember that a frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean:\r\n[latex]\\displaystyle\\text{mean}=\\frac{{\\text{data sum}}}{{\\text{number of data values}}}[\/latex]. We simply need to modify the definition to fit within the restrictions of a frequency table.\r\n\r\nSince we do not know the individual data values we can instead find the midpoint of each class. The midpoint is the mean of the lower and upper interval constraints: [latex]\\displaystyle\\frac{{\\text{lower boundary } + \\text{upper boundary}}}{{2}}[\/latex]. Then, the best estimate of the mean is:\r\n<div class=\"textbox shaded\">\r\n<h3>GROUPED FREQUENCY Mean<\/h3>\r\nThe best estimate of a grouped frequency mean is:\r\n\r\n[latex]\\text{mean}={\\frac{\\text{sum of the product of each class frequency and midpoint }}{\\text{sum of each frequency}}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\" style=\"text-align: left;\">\r\n<h3>example<\/h3>\r\nA frequency table displaying professor Payne's last math test is shown. Find the best estimate of the class mean.\r\n<table style=\"height: 261px; width: 458px;\">\r\n<tbody>\r\n<tr>\r\n<td>Grade Interval<\/td>\r\n<td>Number of Students<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]50\u201356.5[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nStart by finding the midpoints for all the classes:\r\n<table style=\"height: 261px; width: 459px;\">\r\n<tbody>\r\n<tr>\r\n<td>Grade Class<\/td>\r\n<td>\u00a0Midpoint<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]50\u201356.5[\/latex]<\/td>\r\n<td>[latex]53.25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\r\n<td>[latex]65.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\r\n<td>[latex]71.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\r\n<td>[latex]77.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\r\n<td>[latex]83.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\r\n<td>[latex]95.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThen calculate the sum of the product of each interval frequency and midpoint:\r\n\r\n<center>[latex]53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.25[\/latex]<\/center>&nbsp;\r\n\r\nThen divide by the total frequencies = 19\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\mu=\\frac{{1460.25}}{{19}}={76.86}[\/latex]<\/p>\r\nNOTE: the test scores represent a population (professor Payne's class) so the mean is denoted by [latex]\u03bc[\/latex].\r\n\r\nWhen calculating this use a calculator to determine the numerator then immediately divide by the denominator. Do not round the numerator before dividing as this will introduce rounding error.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMaris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:\r\n<table style=\"height: 174px; width: 456px;\">\r\n<tbody>\r\n<tr>\r\n<th>Hours Teenagers Spend on Video Games<\/th>\r\n<th>Number of Teenagers<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0\u20133.5[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3.5\u20137.5[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7.5\u201311.5[\/latex]<\/td>\r\n<td>\u00a0[latex]12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11.5\u201315.5[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]15.5\u201319.5[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat is the best estimate for the mean number of hours spent playing video games?\r\n\r\n[reveal-answer q=\"124083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124083\"]\r\n\r\nFind the midpoint of each class, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers.\r\n\r\nThe midpoints are [latex]1.75[\/latex], [latex]5.5[\/latex], [latex]9.5[\/latex], [latex]13.5[\/latex], [latex]17.5[\/latex].\r\n\r\n[latex]Mean = \\displaystyle\\overline{{x}} = \\frac{(1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9)}{38} = 10.78\\text{ hours}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the mean of a set of numbers&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6529,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Calculate the mean of a set of numbers<\/span><\/li>\n<li>Calculate the mean from a frequency table or a grouped frequency table.<\/li>\n<\/ul>\n<\/div>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>KEY words<\/h3>\n<ul>\n<li><strong>Mean<\/strong>: the arithmetic average<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>Mean<\/h2>\n<p>So far we have studied two averages; the median and the mode. A third average is the <strong><em>mean<\/em><\/strong>, which is often called the <strong><em>arithmetic average<\/em><\/strong>. It is computed by dividing the sum of the data values by the number of data values. Generally speaking, when people talk about average, it is usually the mean they are referring to. Technically, the mean is the arithmetic average, and average is a central location, but mean and average are used interchangeably in common practice. \u00a0The mean is a central value of a <em>finite<\/em> set of numbers: specifically, the sum of the values divided by the number of values.<\/p>\n<div class=\"textbox shaded\">\n<h3>Mean<\/h3>\n<p>The mean of a set of [latex]n[\/latex] numerical data points is the arithmetic average of the numbers.<\/p>\n<p>[latex]\\text{mean}={\\frac{\\text{sum of values in data set}}{n}}[\/latex]<\/p>\n<p>where [latex]n[\\latex] is the number of data points.    <\/p><\/div>\n<p>  Suppose Eva\u2019s first three test scores were [latex]85,88,\\text{and }94[\/latex]. To find the mean score, we would add them and divide by [latex]3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ {\\frac{85+88+94}{3}} &=&{\\frac{267}{3}}\\\\ &=&89\\end{array}[\/latex]<\/p>\n<p>The mean test score is [latex]89[\/latex] points.<\/p>\n<p>The mean of a set of data is denoted by either the Greek letter [latex]\u03bc[\/latex] (mu) when representing the<em> <strong>population mean<\/strong><\/em>, or\u00a0[latex]\\displaystyle\\overline{{x}}[\/latex] (read\u00a0\"[latex]x[\/latex] bar\") when representing a <em><strong>sample mean<\/strong><\/em>.\u00a0One 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<div class=\"textbox shaded\">\n<h3>Calculate the mean of a set of numbers.<\/h3>\n<ol id=\"eip-id1168468272241\" class=\"stepwise\">\n<li>Write the formula for the mean<br \/>\n[latex]\\text{mean}={\\frac{\\text{sum of values in the data set}}{n}}[\/latex]<\/li>\n<li>Find the sum of all the values in the set. Write the sum in the numerator.<\/li>\n<li>Count the number, [latex]n[\/latex], of values in the set. Write this number in the denominator.<\/li>\n<li>Divide the numerator by the denominator.<\/li>\n<li>Check to see that the mean is <em>reasonable<\/em>. The mean is an average which is a measure of the center of the data set, so it should be greater than the least number and less than the greatest number in the set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the mean of the numbers [latex]8,12,15, 9,\\text{ and }6[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467435661\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"width: 501.9px; height: 30px;\">Write the formula for the mean:<\/td>\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\frac{\\text{sum of all the numbers}}{n}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 501.9px; height: 30px;\">Write the sum of the numbers in the numerator.<\/td>\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\frac{8+12+15+9+6}{n}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"width: 501.9px; height: 30px;\">Count how many numbers are in the set. There are [latex]5[\/latex] numbers in the set, so [latex]n=5[\/latex] .<\/td>\n<td style=\"width: 333.1px; height: 30px;\">[latex]\\text{mean}={\\frac{8+12+15+9+6}{5}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 501.9px; height: 15px;\">Add the numbers in the numerator.<\/td>\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}={\\frac{50}{5}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 501.9px; height: 15px;\">Then divide.<\/td>\n<td style=\"width: 333.1px; height: 15px;\">[latex]\\text{mean}=10[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30.8281px;\">\n<td style=\"width: 501.9px; height: 30.8281px;\">Check to see that the mean is 'typical': [latex]10[\/latex] is neither less than [latex]6[\/latex] nor greater than [latex]15[\/latex].<\/td>\n<td style=\"width: 333.1px; height: 30.8281px;\">The mean is [latex]10[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146388\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146388&theme=oea&iframe_resize_id=ohm146388&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146389\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146389&theme=oea&iframe_resize_id=ohm146389&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;\">A table that shows the frequency of each data value in a sample is called a\u00a0<\/span><strong style=\"font-size: 1em;\"><em>frequency table<\/em><\/strong><span style=\"font-size: 1em;\">. It is used when<\/span>\u00a0each value in the data set is not unique so that the frequency of the value can be shown. 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.<\/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:<\/p>\n<table style=\"border-collapse: collapse; width: 25.8232%; height: 348px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 50%; height: 12px;\">Number of Months<\/th>\n<th style=\"width: 50%; height: 12px;\">Frequency<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]11[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]13[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]14[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]15[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]17[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]18[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]21[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]22[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]24[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]25[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]26[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]27[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]29[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]31[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]32[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]33[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]34[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]35[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]37[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]40[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]44[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">[latex]47[\/latex]<\/td>\n<td style=\"width: 50%; height: 12px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Calculate the mean.<\/p>\n<h4>Solution<\/h4>\n<p>The calculation for the mean is:<\/p>\n<p>[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}\\text{months}[\/latex]<\/p>\n<\/div>\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, median, and mode.<\/p>\n<table style=\"border-collapse: collapse; width: 2.5021234329204742%; height: 470px;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">Number of Months<\/td>\n<td style=\"width: 50%;\">Frequency<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]7[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]11[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]13[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]14[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]15[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]17[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]18[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]19[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]21[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]22[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]23[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]24[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm076\">Show Solution<\/span><\/p>\n<div id=\"qhjm076\" class=\"hidden-answer\" style=\"display: none\">\nMean:<\/p>\n<p>Sum of values = [latex]3 + 4 + 5 + {({7})}{({4})} + {({8)}}{({2})} + {({9})}{({2})} + {({10})}{({5})} + 11 + {({12})}{({2})} + 13 + {({14})}{({2})} + {({15})}{({2})} + {({17})}{({2})} + 18 + {({19})}{({3})} + {({21})}{({2})} + {({22})}{({2})} + 23 + {({24})}{({4})} = 544[\/latex] months<\/p>\n<p>[latex]\\displaystyle\\overline{{x}} = \\displaystyle\\frac{{544}}{{39}}={13.95}[\/latex] months<\/p>\n<p>Median: Since there are 39 values the median will be in the 20th place. Starting at the smallest value, the median is the [latex]20[\/latex]th term, which is [latex]13[\/latex] months.<\/p>\n<p>Mode: The highest frequency is [latex]5[\/latex] that occurs at [latex]10[\/latex] months, so the mode is 10 months<\/p>\n<\/div>\n<\/div>\n<p>Notice that mean = [latex]13.95[\/latex] months; median = [latex]13[\/latex] months; mode =\u00a0[latex]10[\/latex] months. Although they are all different, they are each a measure of average.<\/p>\n<\/div>\n<h3>Mean and Median. What's the Difference?<\/h3>\n<p>Since mean and median are both measures of the center of the data set, how do we know which one is more representative of the data set? The next example, considers this question:<\/p>\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 \"center\": 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>Mean: [latex]\\displaystyle\\overline{{x}}=\\frac{{5,000,000}+{49}({30,000})}{{50}}={129,400}[\/latex]<\/p>\n<p>Median: [latex]\\\\{M}={30,000}[\/latex]<\/p>\n<p>There are [latex]49[\/latex] people who earn $[latex]30,000[\/latex] (the mode is $30,000) and only one person who earns $[latex]5,000,000[\/latex], so median\u00a0is a better measure of the \"center\" than the mean. The $[latex]30,000[\/latex] gives us a better sense of the middle of the data and is more representative of the data.<\/p>\n<p>The $[latex]5,000,000[\/latex] value is an <strong><em>outlier<\/em><\/strong>. Outliers affect the mean much more than they affect the median.\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>In a sample of [latex]61[\/latex] households, one house is worth $[latex]2,500,000[\/latex]. Half of the rest (30) are worth $[latex]280,000[\/latex], and all the others (30) are worth $[latex]315,000[\/latex]. Which is the better measure of the \"center\": 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 \"center\" than the mean because [latex]60[\/latex] of the values are either $[latex]280,000[\/latex] or $[latex]315,000[\/latex] and only 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>When the data set consists of values that are basically symmetric about the center, the mean and the median get closer together. When the data set consists of values that are <em>skewed<\/em> to the left or right of center, or if the data set contains <em>outliers<\/em>, the median is the better choice.<\/p>\n<p>In addition, the<em> Law of Large Numbers<\/em> says that if we take samples of larger and larger size from any population, then the mean, [latex]\\displaystyle\\overline{{x}}[\/latex], of the sample is very likely to get closer and closer to the population mean, [latex]\u00b5[\/latex].<\/p>\n<h3>Calculating the Mean of Grouped Frequency Tables<\/h3>\n<p>When only grouped data is available, we do not know the individual data values (we only know classes and class frequencies); therefore, we cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a <em><strong>frequency table<\/strong><\/em>. Remember that a frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean:<br \/>\n[latex]\\displaystyle\\text{mean}=\\frac{{\\text{data sum}}}{{\\text{number of data values}}}[\/latex]. We simply need to modify the definition to fit within the restrictions of a frequency table.<\/p>\n<p>Since we do not know the individual data values we can instead find the midpoint of each class. The midpoint is the mean of the lower and upper interval constraints: [latex]\\displaystyle\\frac{{\\text{lower boundary } + \\text{upper boundary}}}{{2}}[\/latex]. Then, the best estimate of the mean is:<\/p>\n<div class=\"textbox shaded\">\n<h3>GROUPED FREQUENCY Mean<\/h3>\n<p>The best estimate of a grouped frequency mean is:<\/p>\n<p>[latex]\\text{mean}={\\frac{\\text{sum of the product of each class frequency and midpoint }}{\\text{sum of each frequency}}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\" style=\"text-align: left;\">\n<h3>example<\/h3>\n<p>A frequency table displaying professor Payne's last math test is shown. Find the best estimate of the class mean.<\/p>\n<table style=\"height: 261px; width: 458px;\">\n<tbody>\n<tr>\n<td>Grade Interval<\/td>\n<td>Number of Students<\/td>\n<\/tr>\n<tr>\n<td>[latex]50\u201356.5[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>Start by finding the midpoints for all the classes:<\/p>\n<table style=\"height: 261px; width: 459px;\">\n<tbody>\n<tr>\n<td>Grade Class<\/td>\n<td>\u00a0Midpoint<\/td>\n<\/tr>\n<tr>\n<td>[latex]50\u201356.5[\/latex]<\/td>\n<td>[latex]53.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\n<td>[latex]59.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\n<td>[latex]65.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\n<td>[latex]71.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\n<td>[latex]77.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\n<td>[latex]83.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\n<td>[latex]89.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\n<td>[latex]95.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Then calculate the sum of the product of each interval frequency and midpoint:<\/p>\n<div style=\"text-align: center;\">[latex]53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.25[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Then divide by the total frequencies = 19<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\mu=\\frac{{1460.25}}{{19}}={76.86}[\/latex]<\/p>\n<p>NOTE: the test scores represent a population (professor Payne's class) so the mean is denoted by [latex]\u03bc[\/latex].<\/p>\n<p>When calculating this use a calculator to determine the numerator then immediately divide by the denominator. Do not round the numerator before dividing as this will introduce rounding error.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:<\/p>\n<table style=\"height: 174px; width: 456px;\">\n<tbody>\n<tr>\n<th>Hours Teenagers Spend on Video Games<\/th>\n<th>Number of Teenagers<\/th>\n<\/tr>\n<tr>\n<td>[latex]0\u20133.5[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3.5\u20137.5[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]7.5\u201311.5[\/latex]<\/td>\n<td>\u00a0[latex]12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]11.5\u201315.5[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]15.5\u201319.5[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What is the best estimate for the mean number of hours spent playing video games?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124083\">Show Solution<\/span><\/p>\n<div id=\"q124083\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the midpoint of each class, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers.<\/p>\n<p>The midpoints are [latex]1.75[\/latex], [latex]5.5[\/latex], [latex]9.5[\/latex], [latex]13.5[\/latex], [latex]17.5[\/latex].<\/p>\n<p>[latex]Mean = \\displaystyle\\overline{{x}} = \\frac{(1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9)}{38} = 10.78\\text{ hours}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<p>&nbsp;<\/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-985\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Mean: Rewritten and organized Lumen content. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 146390, 146394, 146389, 146388. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Find the Mean of a Data Set. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/0io9U8Jcjeo\">https:\/\/youtu.be\/0io9U8Jcjeo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>OpenStax, Statistics, Measures of the Center of the Data. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowski, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/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>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":370291,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Mean: Rewritten and organized Lumen content\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Ex: Find the Mean of a Data Set\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/0io9U8Jcjeo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146390, 146394, 146389, 146388\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Measures of the Center of the Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics \",\"author\":\"Barbara Illowski, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-985","chapter","type-chapter","status-publish","hentry"],"part":657,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/985","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":34,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/985\/revisions"}],"predecessor-version":[{"id":1916,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/985\/revisions\/1916"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/657"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/985\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=985"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=985"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=985"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}