{"id":24723,"date":"2021-08-11T02:51:04","date_gmt":"2021-08-11T02:51:04","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=24723"},"modified":"2026-01-24T18:41:31","modified_gmt":"2026-01-24T18:41:31","slug":"summarizing-data-using-frequency-distributions","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/summarizing-data-using-frequency-distributions\/","title":{"rendered":"Summarizing Data Using Frequency Distributions"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Relative Frequency Definition\",\n    \"text\": \"The class frequency divided by the total number of observations is most likely called:\\n\\nA. Relative frequency.\\n\\nB. Percentage frequency.\\n\\nC. Cumulative relative frequency.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Relative frequency.\",\n      \"explanation\": \"Relative frequency refers to the proportion or percentage of observations that fall within a given class. It is calculated by dividing the absolute frequency of each class by the total number of observations. Cumulative relative frequency, by contrast, sums relative frequencies up to a given class, and the term percentage frequency is not formally used.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Frequency in a Tally Sheet\",\n    \"text\": \"The number of tally sheet count for each value or a group is most likely known as:\\n\\nA. Class limit.\\n\\nB. Frequency.\\n\\nC. Class width.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Frequency.\",\n      \"explanation\": \"The frequency of a class refers to the number of data entries or observations that fall within that class. Class limits define the boundaries of a class, while class width measures the distance between the lower limits of consecutive classes.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/XCKuqfoNqOo\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Frequency Distribution<\/strong><strong>\u00a0<\/strong><\/h2>\n<p>A frequency distribution refers to the presentation of statistical data in a tabular format to simplify data analysis. In a frequency distribution, data is subdivided into groups or intervals.<\/p>\n<p>The standard procedure for constructing a frequency distribution involves the following steps:<\/p>\n<ol>\n<li>Determine the number of classes one wishes to have. 5 \u2013 20 is always a good number.<\/li>\n<li>Determine the interval size. To do this, the range and the number of classes should guide an analyst. The range is the difference between the smallest and the largest observations. In the case of a fractional result, an analyst should take the next higher whole number as the size of the interval.<\/li>\n<li>Determine the starting point. It could be the lower limit of the lowest observation or a convenient value just below the lower limit.<\/li>\n<li>Add the class interval to the starting point to get the second lower limit. This process should be repeated.<\/li>\n<li>List the lower limits in a vertical column alongside the upper-class limits.<\/li>\n<li>Complete the table by counting the number of observations that fall under each class.<\/li>\n<\/ol>\n<p>Points to note:<\/p>\n<ul>\n<li>Classes should be mutually exclusive and have similar widths.<\/li>\n<li>The last class includes the maximum value.<\/li>\n<li>All class intervals should be tabulated even if they have zero observations.<\/li>\n<li>The sum of the frequencies should be equal to the number of observations.<\/li>\n<li>Tally bars offer a convenient tool for the visual presentation of the number of observations in each.<\/li>\n<\/ul>\n<h4>Example: Frequency Distribution Table<\/h4>\n<p>You have been given the following data showing the percentage returns that certain classes of investment offer in a year. Use the data to construct a frequency distribution table.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\text{-10%} &amp; \\text{2%} &amp; \\text{32%} &amp; \\text{-28%} &amp; \\text{25%} \\\\ \\hline \\text{-25.60%} &amp; \\text{4%} &amp; \\text{11%} &amp; \\text{-14%} &amp; \\text{15%} \\\\ \\hline \\text{23%} &amp; \\text{13%} &amp; \\text{6%} &amp; \\text{-2.70%} &amp; \\text{8%} \\\\ \\hline \\text{12%} &amp; \\text{28%} &amp; \\text{17.50%} &amp; \\text{5.80%} &amp; \\text{20%} \\\\ \\hline \\text{4.60%} &amp; \\text{17%} &amp; \\text{-3.90%} &amp; \\text{22.40%} &amp; \\text{15%} \\\\ \\end{array} $$<\/p>\n<p><strong>Solution<\/strong><\/p>\n<ol>\n<li>We have 25 observations in total. First of all, we will sort the data in ascending order.<\/li>\n<li>After this, we will calculate the range of the data, where Range = Maximum value \u2212 Minimum value = 32% &#8211; (-28%) = 60%.<\/li>\n<li>We will then determine an interval width of 10%.<\/li>\n<li>The lowest return intervals will be -30% \u2264 Rt &lt; -20%, while the highest one will be 30% \u2264 Rt &lt; 40%.<\/li>\n<li>Then, we will count the observations in each interval.<\/li>\n<\/ol>\n<p>$$ \\begin{array}{c|c|c} \\textbf{Interval} &amp; \\textbf{Tally} &amp; \\textbf{Frequency} \\\\ \\hline -30\\% \\leq R_t &lt; -20\\% &amp; \\text{II} &amp; \\text{2} \\\\ -20\\% \\leq R_t &lt; -10\\% &amp; \\text{I} &amp; \\text{1} \\\\ -10\\% \\leq R_t &lt; 0\\% &amp; \\text{III} &amp; \\text{3} \\\\ 0\\% \\leq R_t &lt;10\\% &amp; \\text{IIIIII} &amp; \\text{6} \\\\ 10\\% \\leq R_t &lt; 20\\% &amp; \\text{IIIIIII} &amp; \\text{7} \\\\ 20\\% \\leq R_t &lt; 30\\% &amp; \\text{IIIII} &amp; \\text{5} \\\\ 30\\% \\leq R_t &lt; 40\\% &amp; \\text{I} &amp; \\text{1} \\\\ \\textbf{Total} &amp; \\text{} &amp; \\textbf{25} \\\\ \\end{array} $$<\/p>\n<h3>Absolute versus Relative Frequency<\/h3>\n<p><strong>Absolute frequency<\/strong> is the actual number of observations in a given interval.<\/p>\n<p><strong>Relative frequency<\/strong> refers to the percentage of observations falling within a given class. It is calculated by dividing the absolute frequency of each return interval by the total number of observations. Using our earlier example when we introduced the frequency distribution table, we could come up with the relative frequency for each interval using the formula below:<\/p>\n<p>$$\\text{Relative Frequency}=\\frac{\\text{Absolute frequency}}{\\text{Total frequency}}$$<\/p>\n<p>Where \\(\\text{Total frequency}\\) is the total number of observations.<\/p>\n<h4 style=\"margin-top: .05pt;\"><strong><span style=\"color: #282d34;\">Example: Relative Frequency<\/span><\/strong><\/h4>\n<p>$$<br \/>\n\\begin{array}{c|c|c|c}<br \/>\n\\textbf { Interval } &amp; \\textbf { Tally } &amp; \\textbf { Frequency } &amp; \\textbf { Relative Frequency } \\\\<br \/>\n\\hline-30 \\% \\leq \\mathrm{R}_{\\mathrm{t}} \\leq-20 \\% &amp; \\text { II } &amp; 2 &amp; \\frac{2}{25}=8 \\% \\\\<br \/>\n-20 \\% \\leq \\mathrm{R}_{\\mathrm{t}} \\leq-10 \\% &amp; \\text { I } &amp; 1 &amp; \\frac{1}{25}=4 \\% \\\\<br \/>\n-10 \\% \\leq \\mathrm{R}_{t} \\leq 0 \\% &amp; \\text { III } &amp; 3 &amp; \\frac{3}{25}=12 \\% \\\\<br \/>\n0 \\% \\leq \\mathrm{R}_{t} \\leq 10 \\% &amp; \\text { IIIII } &amp; 6 &amp; \\frac{6}{25}=24 \\% \\\\<br \/>\n10 \\% \\leq \\mathrm{R}_{t} \\leq 20 \\% &amp; \\text { IIIIII } &amp; 7 &amp; \\frac{7}{25}=28 \\% \\\\<br \/>\n20 \\% \\leq \\mathrm{R}_{t} \\leq 30 \\% &amp; \\text { IIII } &amp; 5 &amp; \\frac{5}{25}=20 \\% \\\\<br \/>\n30 \\% \\leq \\mathrm{R}_{t} \\leq 40 \\% &amp; \\text { I } &amp; 1 &amp;\\frac{1}{25}=4 \\% \\\\<br \/>\n\\text { Total } &amp; &amp; 25 &amp; \\frac{25}{25}=100 \\%<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>In the above table, the absolute frequency of the 1<sup>st<\/sup> interval is 2. Similarly, the relative frequency of the 1<sup>st<\/sup> interval is 8%. The same applies to other intervals.<\/p>\n<p><strong>Cumulative absolute frequency<\/strong> is the sum of the absolute frequencies, including the given interval.<\/p>\n<p><strong>Cumulative relative frequency<\/strong> similarly sums up the relative frequencies up to and including the given relative frequency.<\/p>\n<h4><strong>Example<\/strong>:<strong> Cumulative Frequencies<\/strong><\/h4>\n<p>$$<br \/>\n\\begin{array}{c|c|c|c|c|c}<br \/>\n\\textbf { Interval } &amp; \\textbf { Tally } &amp; \\textbf { Frequency } &amp; \\textbf { Relative Frequency } &amp; \\begin{array}{c}<br \/>\n\\textbf { Cumulative Absolute } \\\\<br \/>\n\\textbf { Frequency }<br \/>\n\\end{array} &amp; \\begin{array}{c}<br \/>\n\\textbf { Cumulative Relative } \\\\<br \/>\n\\textbf { Frequency }<br \/>\n\\end{array} \\\\<br \/>\n\\hline-30 \\% \\leq R \\leq-20 \\% &amp; \\text { II } &amp; 2 &amp; \\frac{2 }{25}=8 \\% &amp; 2 &amp; 8 \\% \\\\<br \/>\n-20 \\% \\leq R_{4} \\leq-10 \\% &amp; \\text { I } &amp; 1 &amp; \\frac{1 }{25}=4 \\% &amp; 3 &amp; 12 \\% \\\\<br \/>\n-10 \\% \\leq R_{1} \\leq 0 \\% &amp; \\text { III } &amp; 3 &amp; \\frac{3}{ 25}=12 \\% &amp; 6 &amp; 24 \\% \\\\<br \/>\n0 \\% \\leq R_{1} \\leq 10 \\% &amp; \\text { IIIII } &amp; 6 &amp; \\frac{6}{25}=24 \\% &amp; 12 &amp; 48 \\% \\\\<br \/>\n10 \\% \\leq R \\leq 20 \\% &amp; \\text { IIIII } &amp; 7 &amp; \\frac{7}{25}=28 \\% &amp; 19 &amp; 76 \\% \\\\<br \/>\n20 \\% \\leq R \\leq 30 \\% &amp; \\text { IIII } &amp; 5 &amp; \\frac{5}{25}=8 \\% &amp; 24 &amp; 96 \\% \\\\<br \/>\n30 \\% \\leq R \\leq 40 \\% &amp; \\text { I } &amp; 1 &amp; \\frac{1}{25}=4 \\% &amp; 25 &amp; 100 \\% \\\\<br \/>\n\\text { Total } &amp; &amp; 25 &amp; \\frac{25}{25}=100 \\% &amp; &amp;<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>In the above table, cumulative absolute frequency is the sum of the absolute frequencies up to and including the given interval. The cumulative relative frequency similarly sums up the relative frequencies up to and including the given relative frequency.<\/p>\n<blockquote>\n<h2><span style=\"font-size: revert; color: initial;\"><b>Question 1<\/b><\/span><\/h2>\n<p><span style=\"font-size: revert; color: initial;\">The class frequency divided by the total number of observations is <em>most likely<\/em> called:<\/span><\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>Relative frequency.<\/li>\n<li>Percentage frequency.<\/li>\n<li>Cumulative relative frequency.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>Relative frequency refers to the percentage of observations falling within a given class. It is calculated by dividing the absolute frequency of each return interval by the total number of observations.<\/p>\n<p><strong>C is incorrect<\/strong>. Cumulative relative frequency sums up the relative frequencies up to and including the given relative frequency.<\/p>\n<p><strong>B is incorrect.<\/strong> There is no such term as percentage frequency.<\/p>\n<h2><strong>Question 2<\/strong><\/h2>\n<p>The number of tally sheet count for each value or a group is <em>most likely<\/em> known as:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>Class limit.<\/li>\n<li>Frequency.<\/li>\n<li>Class width.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>The frequency of a class is the number of data entries in the class.<\/p>\n<p><strong>A is incorrect<\/strong>. Each class will have a \u201clower-class limit\u201d and an \u201cupper-class limit\u201d, which are the lowest and highest numbers in each class.<\/p>\n<p><strong>C is incorrect<\/strong>. The \u201cclass width\u201d is the distance between the lower limits of consecutive classes.<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Frequency Distribution\u00a0 A frequency distribution refers to the presentation of statistical data in a tabular format to simplify data analysis. In a frequency distribution, data is subdivided into groups or intervals. 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