{"id":138,"date":"2023-12-05T16:38:56","date_gmt":"2023-12-05T21:38:56","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=138"},"modified":"2024-06-11T15:57:08","modified_gmt":"2024-06-11T19:57:08","slug":"frequency-distributions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/frequency-distributions\/","title":{"raw":"Frequency Distributions","rendered":"Frequency Distributions"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">\u00a0Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nConstruct and understand frequency distributions\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>A frequency distribution is:<\/h2>\r\n<ul>\r\n \t<li>An organized tabulation\/graphical representation of the number of data in each category.<\/li>\r\n \t<li>It allows us to have a glance at the entire data conveniently[footnote]https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3117575\/#CIT1[\/footnote]<\/li>\r\n \t<li>See the sample frequency table below (<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-0-1_Student_Survey_Data.xlsx\">created from student survey data from 2019<\/a>):<\/li>\r\n<\/ul>\r\n<table class=\"grid\" style=\"border-collapse: collapse;width: 42.6234%;height: 187px;margin-left: 40px\" border=\"0\">\r\n<tbody style=\"padding-left: 40px\">\r\n<tr style=\"padding-left: 40px\">\r\n<th style=\"width: 26.7399%;text-align: center;height: 18px\">Class<\/th>\r\n<th style=\"width: 1.8315%;text-align: center;height: 18px\">Frequency<\/th>\r\n<th style=\"width: 16.4835%;text-align: center;height: 18px\">% Frequency<\/th>\r\n<\/tr>\r\n<tr style=\"padding-left: 40px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 143 \u2013 149<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">1<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">0.57%<\/td>\r\n<\/tr>\r\n<tr style=\"padding-left: 40px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 21px\">\u00a0 150 \u2013 156<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 21px\">13<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 21px\">7.43%<\/td>\r\n<\/tr>\r\n<tr style=\"padding-left: 40px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 157 \u2013 163<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">32<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">18.29%<\/td>\r\n<\/tr>\r\n<tr style=\"padding-left: 40px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 22px\">\u00a0 164 \u2013 170<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 22px\">47<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 22px\">26.86%<\/td>\r\n<\/tr>\r\n<tr style=\"padding-left: 40px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 171 \u2013 177<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">26<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">14.86%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 178 \u2013 184<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">44<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">25.14%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 185 \u2013 191<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">9<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">5.14%<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 192 \u2013 198<\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">3<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">1.71%<\/td>\r\n<\/tr>\r\n<tr style=\"padding-left: 40px\">\r\n<td style=\"width: 26.7399%;text-align: center;height: 18px\"><strong>Totals<\/strong><\/td>\r\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">175<\/td>\r\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">100%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Steps to Construct A Frequency Distribution<\/h1>\r\nTo construct a frequency table, follow the following steps:\r\n<ol>\r\n \t<li>Calculate the range (MIN and MAX values)<\/li>\r\n \t<li>Choose the number of classes\/intervals to divide the range into (more on this below)<\/li>\r\n \t<li>Calculate the width of each class (round to the closest 'convenient' number): \\[ \\text{Width (}W\\text{)} = \\frac{MAX - MIN}{\\text{#Classes (}C\\text{)}} \\]<\/li>\r\n \t<li>Pick an ARBITRARY starting point (less than or equal to the MIN) =\u00a0 first lower limit (LL). Ie: \\[ \\text{First Lower Limit (}LL_1\\text{)} = \\text{Starting Point}\\]<\/li>\r\n \t<li>Calculate lower limits. \\[ \\text{Lower Limit (}LL\\text{)} = \\text{Previous Lower Limit} + \\text{Width (}W\\text{)} \\]<\/li>\r\n \t<li>Determine the precision level in data. Ex: precision = 2 decimal places, or 0.01, for money.<\/li>\r\n \t<li>Subtract precision level from lower limits to calculate upper limits. \\[ \\text{Upper Limit (}LL\\text{)} = \\text{Next Lower Limit } - \\text{ Precision Level (ex: 0.01)} \\]<\/li>\r\n \t<li>Count the number of data in each class. In Excel, use a <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/create-a-pivottable-to-analyze-worksheet-data-a9a84538-bfe9-40a9-a8e9-f99134456576\">Pivot Table<\/a> or use =<a href=\"https:\/\/support.microsoft.com\/en-us\/office\/countif-function-e0de10c6-f885-4e71-abb4-1f464816df34\">COUNTIF<\/a>().<\/li>\r\n \t<li>Calculate the percent of data in each class by dividing by the total number of data: \\[\\text{% frequency}= \\frac{\\text{frequency (}f\\text{)}}{\\text{sample size (} n \\text{)}} \\]<\/li>\r\n<\/ol>\r\n<h1>Determining number of classes<\/h1>\r\nIf there are too many class intervals:\r\n<ul>\r\n \t<li>There is no reduction in the \"bulkiness\" of data<\/li>\r\n \t<li>And minor deviations also become noticeable[footnote]https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3117575\/#CIT2[\/footnote]<\/li>\r\n<\/ul>\r\nIf there are too few class intervals:\r\n<ul>\r\n \t<li>The shape of the distribution itself cannot be determined<\/li>\r\n \t<li>Generally, 6\u201314 intervals are adequate[footnote]Dawson B, Trapp RG. 4th ed. New York: McGraw Hill; 2004. Basic and clinical biostatistics. [Google Scholar][\/footnote].<\/li>\r\n<\/ul>\r\nSee the table (using the <a href=\"https:\/\/www.sixsigmadaily.com\/cell-intervals-rules-of-thumb\/#:~:text=As%20a%20rule%20of%20thumb,%2B%201%20intervals)%20as%20well.\">2<sup>k<\/sup> rule<\/a>) below for suggested numbers of classes:\r\n<table class=\"grid aligncenter\" style=\"height: 399px;margin-left: 40px;width: 400px\" width=\"333\">\r\n<tbody>\r\n<tr style=\"height: 18px\">\r\n<th style=\"width: 264.229px;height: 18px;vertical-align: middle;text-align: center\"><b>Sample Size<\/b><\/th>\r\n<th style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\"><strong>Number of Classes<\/strong><\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">9\u201316<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">17\u201332<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">33\u201364<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">65\u2013128<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">129\u2013256<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">257\u2013512<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;padding-left: 40px\">\r\n<td style=\"width: 264.229px;height: 18px;text-align: center\">513\u20131,024<\/td>\r\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 264.229px;text-align: center\">1,025\u20132,048<\/td>\r\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 264.229px;text-align: center\">2,049\u20134,096<\/td>\r\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 264.229px;text-align: center\">4,097\u20138,192<\/td>\r\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">13<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 264.229px;text-align: center\">8,193\u201316,384<\/td>\r\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">14<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Creating Frequency Tables 'Manually'<\/h1>\r\n<h2>Example 9.1.1<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup: <\/strong><\/span>Suppose we have taken a SAMPLE of 30 BCIT students (<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-1-1.xlsx\">click here to download as Excel sheet<\/a>), and asked them of their heights, in centimeters (below):[footnote]We used to survey the students and ask 25 questions. See the last survey results from 2019 collected here: https:\/\/docs.google.com\/spreadsheets\/d\/14hmeeeEvaI-3uwUDAwSot0l6UqYlurgwMKCMHb1CSy8\/edit?usp=sharing[\/footnote] (click here for the Excel file with these values):\r\n<table style=\"height: 54px;margin-left: 40px\">\r\n<tbody style=\"padding-left: 40px\">\r\n<tr style=\"height: 18px;padding-left: 40px\">\r\n<td style=\"height: 18px;width: 41.6875px;padding-left: 40px\">173<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">153<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">172<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">191<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">173<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">167<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">156<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">169<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">175<\/td>\r\n<td style=\"height: 18px;width: 41.9792px;padding-left: 40px\">169<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;padding-left: 40px\">\r\n<td style=\"height: 18px;width: 41.6875px;padding-left: 40px\">159<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">163<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">177<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">155<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">152<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">178<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">172<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">188<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">152<\/td>\r\n<td style=\"height: 18px;width: 41.9792px;padding-left: 40px\">171<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;padding-left: 40px\">\r\n<td style=\"height: 18px;width: 41.6875px;padding-left: 40px\">174<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">183<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">192<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">151<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">159<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">184<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">170<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">186<\/td>\r\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">155<\/td>\r\n<td style=\"height: 18px;width: 41.9792px;padding-left: 40px\">156<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"color: #003366\"><strong>Question: <\/strong><\/span>What is the frequency table for these heights?\r\n\r\n<span style=\"color: #003366\"><strong>Steps: <\/strong><\/span>We will follow the steps outlined in the first section:\r\n<ol>\r\n \t<li>Range: [latex] MAX = 192[\/latex],\u00a0 [latex]MIN = 151[\/latex]<\/li>\r\n \t<li>Number of classes: [latex]C = 5[\/latex]<\/li>\r\n \t<li>Width: [latex]W = \\frac{MAX \u2212 MIN}{C} = \\frac{192 \u2212 151}{5} =8.2 [\/latex] (round up to 10 since 'easier' to work with)<\/li>\r\n \t<li>Starting point: [latex]LL_1 = 150[\/latex] (150 is less than min value of 151 and 'easier' to work with)<\/li>\r\n \t<li>Lower Limits:\r\n[latex]\r\n\\begin{align*}\r\nLL_2 &amp;= LL_1 + W = 150 + 10 = 160\\\\\r\nLL_3 &amp;= LL_2 + W = 160 + 10 = 170\\\\\r\n...\r\n\\end{align*}\r\n[\/latex]<\/li>\r\n \t<li>Precision level: Data recorded to nearest 1's<\/li>\r\n \t<li>Upper Limits:\r\n[latex]\r\n\\begin{align*}\r\nUL_1 &amp;= LL_2 - Precision = 160 - 1 = 159\\\\\r\nUL_2 &amp;= LL_3 - Precision = 170 - 1 = 169\\\\\r\n...\r\n\\end{align*}\r\n[\/latex]<\/li>\r\n \t<li>Count of data in each class and input into the \"Frequency\" column (see the table below).<\/li>\r\n \t<li>Divide each frequency by the total number of data to get the relative (%) frequencies.<\/li>\r\n<\/ol>\r\n<span style=\"color: #003366\"><strong>Resulting Frequency Distribution: <\/strong><\/span>Below is the resulting frequency table\/distribution:\r\n<table class=\"grid\" style=\"border-collapse: collapse;width: 36.8452%;height: 173px;margin-left: 40px\" border=\"0\">\r\n<tbody style=\"padding-left: 40px\">\r\n<tr>\r\n<th style=\"text-align: center\">Class<\/th>\r\n<th style=\"text-align: center\">Frequency<\/th>\r\n<th style=\"text-align: center\">% Frequency<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">150 \u2013 159<\/td>\r\n<td style=\"text-align: center\">10<\/td>\r\n<td style=\"text-align: center\">33.33%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">160 \u2013 169<\/td>\r\n<td style=\"text-align: center\">4<\/td>\r\n<td style=\"text-align: center\">13.33%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">170 \u2013 179<\/td>\r\n<td style=\"text-align: center\">10<\/td>\r\n<td style=\"text-align: center\">33.33%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">180 \u2013 189<\/td>\r\n<td style=\"text-align: center\">4<\/td>\r\n<td style=\"text-align: center\">13.33%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">190 \u2013 199<\/td>\r\n<td style=\"text-align: center\">2<\/td>\r\n<td style=\"text-align: center\">3.67%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\"><strong>Totals<\/strong><\/td>\r\n<td style=\"text-align: center\">30<\/td>\r\n<td style=\"text-align: center\">100%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Creating Frequency Tables in Excel (VIDEO)<\/h1>\r\n<h2>Example 9.1.2<\/h2>\r\n<strong>Problem Setup: <\/strong>Let us now revisit the 30 students' heights example from the previous section, but this time, using Excel. <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-2_Excel.xlsx\">Click here<\/a>\u00a0to download the file shown in the video below.\r\n\r\n[embed]https:\/\/youtu.be\/zoV-uWgRzew[\/embed]\r\n\r\n<strong>Solution: <\/strong>See the above video for the solution to this problem. Click below to reveal the steps followed. Also, <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-2_Solutions.xlsx\">click here<\/a>\u00a0to download the solution file.\r\n\r\n[h5p id=\"19\"]\r\n<h1>Creating Frequency Tables using Pivot Tables (VIDEO)<\/h1>\r\n<h2>Example 9.1.3<\/h2>\r\n<strong>Problem Setup: <\/strong>Let us again revisit the 30 students' heights example from the previous section, but this time, using a Pivot Table. <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-3_Excel.xlsx\">Click here<\/a>\u00a0to download the file shown in the video below.\r\n\r\n[embed]https:\/\/youtu.be\/3HXCOeGHEp8[\/embed]\r\n\r\n<strong>Solution: <\/strong>See the above video for the solution to this problem. Click below to reveal the steps followed. Also, <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-3_Solutions.xlsx\">click here<\/a> to download the solution file.\r\n\r\n[h5p id=\"20\"]\r\n<h1>Class Frequencies, Boundaries, Limits, Marks and Width<\/h1>\r\n<h2>Frequencies<\/h2>\r\n<ul>\r\n \t<li>It is important to be able to interpret what this frequency distribution tells us.<\/li>\r\n \t<li>Keep in mind that the FREQUENCY refers to the number of data in each class.<\/li>\r\n \t<li>In a sample of 30 BCIT students, 10 of them have heights between 150 and 159 cms., 4 between 160 and 169 cms., 10 between 170 and 179 cms., etc.<\/li>\r\n<\/ul>\r\n<h2>Relative Frequencies<\/h2>\r\n<ul>\r\n \t<li>The frequency as a percent of the total<\/li>\r\n \t<li>We obtain the relative frequency by dividing the frequency of the classes by the total frequency, i.e., f\/r.t<\/li>\r\n<\/ul>\r\n<h2>Class Boundaries<\/h2>\r\n<ul>\r\n \t<li>The CLASS BOUNDARIES are the mid-points BETWEEN the classes.<\/li>\r\n \t<li>In example 9.1, the U.L. of the first class is 159, and the L.L. of the second class is 160.<\/li>\r\n \t<li>The boundary between the first and second class is then 159.5.<\/li>\r\n \t<li>Similarly, 169.5 is the next boundary.<\/li>\r\n<\/ul>\r\n<h2>Class Marks<\/h2>\r\n<ul>\r\n \t<li>The CLASS MARKS are the mid-point WITHIN the classes.<\/li>\r\n \t<li>The mid-point of the first class in example 9.1 is [latex]\\frac{150 + 159} {2} = 154.5[\/latex].<\/li>\r\n \t<li>The mid-point of the second class is [latex]\\frac{160 + 169} {2} = 164.5[\/latex]. The rest then are 174.5, 184.5, and 194.5.<\/li>\r\n \t<li>There are as many class marks as there are classes.<\/li>\r\n<\/ul>\r\n<h2>Class Width<\/h2>\r\nThe class width is <em>NOT<\/em> the difference between upper and lower limits of the same class. Rather, is can be calculated by taking the difference between the following successive values:\r\n<ul>\r\n \t<li>lower limits (ex: 160 \u2212 150)<\/li>\r\n \t<li>upper limits (ex: 169 \u2212 159)<\/li>\r\n \t<li>boundaries (ex: 159.5 \u2212 149.5)<\/li>\r\n \t<li>marks (ex: 164.5 \u2212 154.5)<\/li>\r\n<\/ul>\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Frequency Distributions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"17\"]\r\n\r\n[h5p id=\"18\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">\u00a0Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Construct and understand frequency distributions<\/p>\n<\/div>\n<\/div>\n<h2>A frequency distribution is:<\/h2>\n<ul>\n<li>An organized tabulation\/graphical representation of the number of data in each category.<\/li>\n<li>It allows us to have a glance at the entire data conveniently<a class=\"footnote\" title=\"https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3117575\/#CIT1\" id=\"return-footnote-138-1\" href=\"#footnote-138-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/li>\n<li>See the sample frequency table below (<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-0-1_Student_Survey_Data.xlsx\">created from student survey data from 2019<\/a>):<\/li>\n<\/ul>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 42.6234%;height: 187px;margin-left: 40px\">\n<tbody style=\"padding-left: 40px\">\n<tr style=\"padding-left: 40px\">\n<th style=\"width: 26.7399%;text-align: center;height: 18px\">Class<\/th>\n<th style=\"width: 1.8315%;text-align: center;height: 18px\">Frequency<\/th>\n<th style=\"width: 16.4835%;text-align: center;height: 18px\">% Frequency<\/th>\n<\/tr>\n<tr style=\"padding-left: 40px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 143 \u2013 149<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">1<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">0.57%<\/td>\n<\/tr>\n<tr style=\"padding-left: 40px\">\n<td style=\"width: 26.7399%;text-align: center;height: 21px\">\u00a0 150 \u2013 156<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 21px\">13<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 21px\">7.43%<\/td>\n<\/tr>\n<tr style=\"padding-left: 40px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 157 \u2013 163<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">32<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">18.29%<\/td>\n<\/tr>\n<tr style=\"padding-left: 40px\">\n<td style=\"width: 26.7399%;text-align: center;height: 22px\">\u00a0 164 \u2013 170<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 22px\">47<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 22px\">26.86%<\/td>\n<\/tr>\n<tr style=\"padding-left: 40px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 171 \u2013 177<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">26<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">14.86%<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 178 \u2013 184<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">44<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">25.14%<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 185 \u2013 191<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">9<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">5.14%<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\">\u00a0 192 \u2013 198<\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">3<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">1.71%<\/td>\n<\/tr>\n<tr style=\"padding-left: 40px\">\n<td style=\"width: 26.7399%;text-align: center;height: 18px\"><strong>Totals<\/strong><\/td>\n<td style=\"width: 1.8315%;text-align: center;height: 18px\">175<\/td>\n<td style=\"width: 16.4835%;text-align: center;height: 18px\">100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Steps to Construct A Frequency Distribution<\/h1>\n<p>To construct a frequency table, follow the following steps:<\/p>\n<ol>\n<li>Calculate the range (MIN and MAX values)<\/li>\n<li>Choose the number of classes\/intervals to divide the range into (more on this below)<\/li>\n<li>Calculate the width of each class (round to the closest &#8216;convenient&#8217; number): \\[ \\text{Width (}W\\text{)} = \\frac{MAX &#8211; MIN}{\\text{#Classes (}C\\text{)}} \\]<\/li>\n<li>Pick an ARBITRARY starting point (less than or equal to the MIN) =\u00a0 first lower limit (LL). Ie: \\[ \\text{First Lower Limit (}LL_1\\text{)} = \\text{Starting Point}\\]<\/li>\n<li>Calculate lower limits. \\[ \\text{Lower Limit (}LL\\text{)} = \\text{Previous Lower Limit} + \\text{Width (}W\\text{)} \\]<\/li>\n<li>Determine the precision level in data. Ex: precision = 2 decimal places, or 0.01, for money.<\/li>\n<li>Subtract precision level from lower limits to calculate upper limits. \\[ \\text{Upper Limit (}LL\\text{)} = \\text{Next Lower Limit } &#8211; \\text{ Precision Level (ex: 0.01)} \\]<\/li>\n<li>Count the number of data in each class. In Excel, use a <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/create-a-pivottable-to-analyze-worksheet-data-a9a84538-bfe9-40a9-a8e9-f99134456576\">Pivot Table<\/a> or use =<a href=\"https:\/\/support.microsoft.com\/en-us\/office\/countif-function-e0de10c6-f885-4e71-abb4-1f464816df34\">COUNTIF<\/a>().<\/li>\n<li>Calculate the percent of data in each class by dividing by the total number of data: \\[\\text{% frequency}= \\frac{\\text{frequency (}f\\text{)}}{\\text{sample size (} n \\text{)}} \\]<\/li>\n<\/ol>\n<h1>Determining number of classes<\/h1>\n<p>If there are too many class intervals:<\/p>\n<ul>\n<li>There is no reduction in the &#8220;bulkiness&#8221; of data<\/li>\n<li>And minor deviations also become noticeable<a class=\"footnote\" title=\"https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3117575\/#CIT2\" id=\"return-footnote-138-2\" href=\"#footnote-138-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/li>\n<\/ul>\n<p>If there are too few class intervals:<\/p>\n<ul>\n<li>The shape of the distribution itself cannot be determined<\/li>\n<li>Generally, 6\u201314 intervals are adequate<a class=\"footnote\" title=\"Dawson B, Trapp RG. 4th ed. New York: McGraw Hill; 2004. Basic and clinical biostatistics. [Google Scholar]\" id=\"return-footnote-138-3\" href=\"#footnote-138-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a>.<\/li>\n<\/ul>\n<p>See the table (using the <a href=\"https:\/\/www.sixsigmadaily.com\/cell-intervals-rules-of-thumb\/#:~:text=As%20a%20rule%20of%20thumb,%2B%201%20intervals)%20as%20well.\">2<sup>k<\/sup> rule<\/a>) below for suggested numbers of classes:<\/p>\n<table class=\"grid aligncenter\" style=\"height: 399px;margin-left: 40px;width: 400px; width: 333px;\">\n<tbody>\n<tr style=\"height: 18px\">\n<th style=\"width: 264.229px;height: 18px;vertical-align: middle;text-align: center\"><b>Sample Size<\/b><\/th>\n<th style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\"><strong>Number of Classes<\/strong><\/th>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">9\u201316<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">4<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">17\u201332<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">5<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">33\u201364<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">6<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">65\u2013128<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">7<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">129\u2013256<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">8<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">257\u2013512<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">9<\/td>\n<\/tr>\n<tr style=\"height: 18px;padding-left: 40px\">\n<td style=\"width: 264.229px;height: 18px;text-align: center\">513\u20131,024<\/td>\n<td style=\"width: 268.271px;height: 18px;vertical-align: middle;text-align: center\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 264.229px;text-align: center\">1,025\u20132,048<\/td>\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">11<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 264.229px;text-align: center\">2,049\u20134,096<\/td>\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">12<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 264.229px;text-align: center\">4,097\u20138,192<\/td>\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">13<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 264.229px;text-align: center\">8,193\u201316,384<\/td>\n<td style=\"width: 268.271px;vertical-align: middle;text-align: center\">14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Creating Frequency Tables &#8216;Manually&#8217;<\/h1>\n<h2>Example 9.1.1<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup: <\/strong><\/span>Suppose we have taken a SAMPLE of 30 BCIT students (<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-1-1.xlsx\">click here to download as Excel sheet<\/a>), and asked them of their heights, in centimeters (below):<a class=\"footnote\" title=\"We used to survey the students and ask 25 questions. See the last survey results from 2019 collected here: https:\/\/docs.google.com\/spreadsheets\/d\/14hmeeeEvaI-3uwUDAwSot0l6UqYlurgwMKCMHb1CSy8\/edit?usp=sharing\" id=\"return-footnote-138-4\" href=\"#footnote-138-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a> (click here for the Excel file with these values):<\/p>\n<table style=\"height: 54px;margin-left: 40px\">\n<tbody style=\"padding-left: 40px\">\n<tr style=\"height: 18px;padding-left: 40px\">\n<td style=\"height: 18px;width: 41.6875px;padding-left: 40px\">173<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">153<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">172<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">191<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">173<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">167<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">156<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">169<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">175<\/td>\n<td style=\"height: 18px;width: 41.9792px;padding-left: 40px\">169<\/td>\n<\/tr>\n<tr style=\"height: 18px;padding-left: 40px\">\n<td style=\"height: 18px;width: 41.6875px;padding-left: 40px\">159<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">163<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">177<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">155<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">152<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">178<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">172<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">188<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">152<\/td>\n<td style=\"height: 18px;width: 41.9792px;padding-left: 40px\">171<\/td>\n<\/tr>\n<tr style=\"height: 18px;padding-left: 40px\">\n<td style=\"height: 18px;width: 41.6875px;padding-left: 40px\">174<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">183<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">192<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">151<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">159<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">184<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">170<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">186<\/td>\n<td style=\"height: 18px;width: 41.1875px;padding-left: 40px\">155<\/td>\n<td style=\"height: 18px;width: 41.9792px;padding-left: 40px\">156<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #003366\"><strong>Question: <\/strong><\/span>What is the frequency table for these heights?<\/p>\n<p><span style=\"color: #003366\"><strong>Steps: <\/strong><\/span>We will follow the steps outlined in the first section:<\/p>\n<ol>\n<li>Range: [latex]MAX = 192[\/latex],\u00a0 [latex]MIN = 151[\/latex]<\/li>\n<li>Number of classes: [latex]C = 5[\/latex]<\/li>\n<li>Width: [latex]W = \\frac{MAX \u2212 MIN}{C} = \\frac{192 \u2212 151}{5} =8.2[\/latex] (round up to 10 since &#8216;easier&#8217; to work with)<\/li>\n<li>Starting point: [latex]LL_1 = 150[\/latex] (150 is less than min value of 151 and &#8216;easier&#8217; to work with)<\/li>\n<li>Lower Limits:<br \/>\n[latex]\\begin{align*}  LL_2 &= LL_1 + W = 150 + 10 = 160\\\\  LL_3 &= LL_2 + W = 160 + 10 = 170\\\\  ...  \\end{align*}[\/latex]<\/li>\n<li>Precision level: Data recorded to nearest 1&#8217;s<\/li>\n<li>Upper Limits:<br \/>\n[latex]\\begin{align*}  UL_1 &= LL_2 - Precision = 160 - 1 = 159\\\\  UL_2 &= LL_3 - Precision = 170 - 1 = 169\\\\  ...  \\end{align*}[\/latex]<\/li>\n<li>Count of data in each class and input into the &#8220;Frequency&#8221; column (see the table below).<\/li>\n<li>Divide each frequency by the total number of data to get the relative (%) frequencies.<\/li>\n<\/ol>\n<p><span style=\"color: #003366\"><strong>Resulting Frequency Distribution: <\/strong><\/span>Below is the resulting frequency table\/distribution:<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 36.8452%;height: 173px;margin-left: 40px\">\n<tbody style=\"padding-left: 40px\">\n<tr>\n<th style=\"text-align: center\">Class<\/th>\n<th style=\"text-align: center\">Frequency<\/th>\n<th style=\"text-align: center\">% Frequency<\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">150 \u2013 159<\/td>\n<td style=\"text-align: center\">10<\/td>\n<td style=\"text-align: center\">33.33%<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">160 \u2013 169<\/td>\n<td style=\"text-align: center\">4<\/td>\n<td style=\"text-align: center\">13.33%<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">170 \u2013 179<\/td>\n<td style=\"text-align: center\">10<\/td>\n<td style=\"text-align: center\">33.33%<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">180 \u2013 189<\/td>\n<td style=\"text-align: center\">4<\/td>\n<td style=\"text-align: center\">13.33%<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">190 \u2013 199<\/td>\n<td style=\"text-align: center\">2<\/td>\n<td style=\"text-align: center\">3.67%<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\"><strong>Totals<\/strong><\/td>\n<td style=\"text-align: center\">30<\/td>\n<td style=\"text-align: center\">100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Creating Frequency Tables in Excel (VIDEO)<\/h1>\n<h2>Example 9.1.2<\/h2>\n<p><strong>Problem Setup: <\/strong>Let us now revisit the 30 students&#8217; heights example from the previous section, but this time, using Excel. <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-2_Excel.xlsx\">Click here<\/a>\u00a0to download the file shown in the video below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"How to create a frequency distribution in Excel\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/zoV-uWgRzew?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><strong>Solution: <\/strong>See the above video for the solution to this problem. Click below to reveal the steps followed. Also, <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-2_Solutions.xlsx\">click here<\/a>\u00a0to download the solution file.<\/p>\n<div id=\"h5p-19\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-19\" class=\"h5p-iframe\" data-content-id=\"19\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 9.1.2 Steps\"><\/iframe><\/div>\n<\/div>\n<h1>Creating Frequency Tables using Pivot Tables (VIDEO)<\/h1>\n<h2>Example 9.1.3<\/h2>\n<p><strong>Problem Setup: <\/strong>Let us again revisit the 30 students&#8217; heights example from the previous section, but this time, using a Pivot Table. <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-3_Excel.xlsx\">Click here<\/a>\u00a0to download the file shown in the video below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Inserting pivots tables in Excel to generate a frequency distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3HXCOeGHEp8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><strong>Solution: <\/strong>See the above video for the solution to this problem. Click below to reveal the steps followed. Also, <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/9-1-3_Solutions.xlsx\">click here<\/a> to download the solution file.<\/p>\n<div id=\"h5p-20\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-20\" class=\"h5p-iframe\" data-content-id=\"20\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 9.1.3 Steps\"><\/iframe><\/div>\n<\/div>\n<h1>Class Frequencies, Boundaries, Limits, Marks and Width<\/h1>\n<h2>Frequencies<\/h2>\n<ul>\n<li>It is important to be able to interpret what this frequency distribution tells us.<\/li>\n<li>Keep in mind that the FREQUENCY refers to the number of data in each class.<\/li>\n<li>In a sample of 30 BCIT students, 10 of them have heights between 150 and 159 cms., 4 between 160 and 169 cms., 10 between 170 and 179 cms., etc.<\/li>\n<\/ul>\n<h2>Relative Frequencies<\/h2>\n<ul>\n<li>The frequency as a percent of the total<\/li>\n<li>We obtain the relative frequency by dividing the frequency of the classes by the total frequency, i.e., f\/r.t<\/li>\n<\/ul>\n<h2>Class Boundaries<\/h2>\n<ul>\n<li>The CLASS BOUNDARIES are the mid-points BETWEEN the classes.<\/li>\n<li>In example 9.1, the U.L. of the first class is 159, and the L.L. of the second class is 160.<\/li>\n<li>The boundary between the first and second class is then 159.5.<\/li>\n<li>Similarly, 169.5 is the next boundary.<\/li>\n<\/ul>\n<h2>Class Marks<\/h2>\n<ul>\n<li>The CLASS MARKS are the mid-point WITHIN the classes.<\/li>\n<li>The mid-point of the first class in example 9.1 is [latex]\\frac{150 + 159} {2} = 154.5[\/latex].<\/li>\n<li>The mid-point of the second class is [latex]\\frac{160 + 169} {2} = 164.5[\/latex]. The rest then are 174.5, 184.5, and 194.5.<\/li>\n<li>There are as many class marks as there are classes.<\/li>\n<\/ul>\n<h2>Class Width<\/h2>\n<p>The class width is <em>NOT<\/em> the difference between upper and lower limits of the same class. Rather, is can be calculated by taking the difference between the following successive values:<\/p>\n<ul>\n<li>lower limits (ex: 160 \u2212 150)<\/li>\n<li>upper limits (ex: 169 \u2212 159)<\/li>\n<li>boundaries (ex: 159.5 \u2212 149.5)<\/li>\n<li>marks (ex: 164.5 \u2212 154.5)<\/li>\n<\/ul>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Frequency Distributions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-17\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-17\" class=\"h5p-iframe\" data-content-id=\"17\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key Takeaways for Frequency Distributions\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-18\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-18\" class=\"h5p-iframe\" data-content-id=\"18\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Takeaways Solutions for Frequency Distributions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-138-1\">https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3117575\/#CIT1 <a href=\"#return-footnote-138-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-138-2\">https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3117575\/#CIT2 <a href=\"#return-footnote-138-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-138-3\">Dawson B, Trapp RG. 4th ed. New York: McGraw Hill; 2004. Basic and clinical biostatistics. [Google Scholar] <a href=\"#return-footnote-138-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-138-4\">We used to survey the students and ask 25 questions. See the last survey results from 2019 collected here: https:\/\/docs.google.com\/spreadsheets\/d\/14hmeeeEvaI-3uwUDAwSot0l6UqYlurgwMKCMHb1CSy8\/edit?usp=sharing <a href=\"#return-footnote-138-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":883,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-138","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/883"}],"version-history":[{"count":26,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/138\/revisions"}],"predecessor-version":[{"id":1942,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/138\/revisions\/1942"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}