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Original
2026-01-01
Modified
2026-02-28
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<p><strong>Median of ungrouped data:</strong> To find the median of ungrouped data, follow these steps.</p>
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<p><strong>Median of ungrouped data:</strong> To find the median of ungrouped data, follow these steps.</p>
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<p><strong>Step 1:</strong> Sort the data in<a>ascending</a>or<a>descending order</a>.</p>
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<p><strong>Step 1:</strong> Sort the data in<a>ascending</a>or<a>descending order</a>.</p>
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<p><strong>Step 2:</strong>Consider n to be the total number of observations. If the n is an<a></a><a>odd number</a>, then the median \(=\frac{n+1}{2}\)th observation in the sorted list. If n is even, then the median is the average of the \(\frac{n}{2}\)th and the \(\frac{n}{2}+1\)th observation. </p>
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<p><strong>Step 2:</strong>Consider n to be the total number of observations. If the n is an<a></a><a>odd number</a>, then the median \(=\frac{n+1}{2}\)th observation in the sorted list. If n is even, then the median is the average of the \(\frac{n}{2}\)th and the \(\frac{n}{2}+1\)th observation. </p>
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<p>Example 1: Find the median of the data 56, 67, 54, 34, 78, 43, 23. </p>
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<p>Example 1: Find the median of the data 56, 67, 54, 34, 78, 43, 23. </p>
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<p>Solution: </p>
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<p>Solution: </p>
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<p>Sort the data : 23, 34, 43, 54, 56, 67, 78. Here, the number of observations n = 7.</p>
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<p>Sort the data : 23, 34, 43, 54, 56, 67, 78. Here, the number of observations n = 7.</p>
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<p>Then, the median = \(\frac{n+1}{2}\) \(=\frac{7+1}{2}\) \(=\frac{8}{2}=4\)</p>
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<p>Then, the median = \(\frac{n+1}{2}\) \(=\frac{7+1}{2}\) \(=\frac{8}{2}=4\)</p>
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<p>Therefor, the 4th observation is the median. That is 54.</p>
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<p>Therefor, the 4th observation is the median. That is 54.</p>
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<p>Example 2: Find the median of the data 50, 67, 24, 34, 78, 43.</p>
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<p>Example 2: Find the median of the data 50, 67, 24, 34, 78, 43.</p>
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<p>Solution: </p>
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<p>Solution: </p>
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<p>By<a>sorting</a>the data: 24, 34, 43, 50, 67, 78. Here, the number of observations n = 6 \(\frac{n}{2}=\frac{6}{2}=\) 3rd<a>term</a>.</p>
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<p>By<a>sorting</a>the data: 24, 34, 43, 50, 67, 78. Here, the number of observations n = 6 \(\frac{n}{2}=\frac{6}{2}=\) 3rd<a>term</a>.</p>
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<p>\(\frac{n}{2}+1=\frac{6}{2}+1=4\)th term.</p>
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<p>\(\frac{n}{2}+1=\frac{6}{2}+1=4\)th term.</p>
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<p>Median \(=\frac{43+50}{2}\)</p>
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<p>Median \(=\frac{43+50}{2}\)</p>
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<p>\(=46.5\)</p>
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<p>\(=46.5\)</p>
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<p><strong>Median of grouped data: </strong>When data are grouped into<a>class intervals</a>(with frequencies), use these steps:</p>
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<p><strong>Median of grouped data: </strong>When data are grouped into<a>class intervals</a>(with frequencies), use these steps:</p>
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<p><strong>Step 1:</strong>Compute the<a>cumulative frequency</a>of each class, to determine where \(\frac{n}{2}\) lies. Here, \(n=∑fi (n = \sum f_i) \) (total number of observations). </p>
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<p><strong>Step 1:</strong>Compute the<a>cumulative frequency</a>of each class, to determine where \(\frac{n}{2}\) lies. Here, \(n=∑fi (n = \sum f_i) \) (total number of observations). </p>
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<p><strong>Step 2:</strong>Identify the median class: the<a>class interval</a>in which \(\frac{n}{2}\) falls.</p>
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<p><strong>Step 2:</strong>Identify the median class: the<a>class interval</a>in which \(\frac{n}{2}\) falls.</p>
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<p><strong>Step 3:</strong>Use the formula, Median \(= l+\frac{(\frac{n }{2}-c )}{f}×h\)</p>
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<p><strong>Step 3:</strong>Use the formula, Median \(= l+\frac{(\frac{n }{2}-c )}{f}×h\)</p>
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<p>Where, l = lower limit of the median class. c = cumulative frequency of the class just before the median class. f = frequency of the median class. h = class width (size of the class interval). </p>
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<p>Where, l = lower limit of the median class. c = cumulative frequency of the class just before the median class. f = frequency of the median class. h = class width (size of the class interval). </p>
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<p>Example: Classes: 0-10, 10-20, 20-30, 30-40, 40-50 frequencies : 2, 12, 22, 8, 6 Total n = 50 So, \(\frac{n}{2}\) = 25.</p>
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<p>Example: Classes: 0-10, 10-20, 20-30, 30-40, 40-50 frequencies : 2, 12, 22, 8, 6 Total n = 50 So, \(\frac{n}{2}\) = 25.</p>
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<p>The cumulative frequencies are: 0-10 → 2 10-20 → 14 20-30 → 36, ……</p>
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<p>The cumulative frequencies are: 0-10 → 2 10-20 → 14 20-30 → 36, ……</p>
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<p>Since 25 lies in the class 20-30, that’s the median class. Here, l = 20, c = 14, f = 22, h = 10. </p>
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<p>Since 25 lies in the class 20-30, that’s the median class. Here, l = 20, c = 14, f = 22, h = 10. </p>
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<p>By applying the formula: </p>
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<p>By applying the formula: </p>
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<p>Median \( = 20 + \left(\frac{25 - 14}{22}\right) \times 10 = 20 + \left(\frac{11}{22}\right) \cdot 10 = 20 + 5 = 25 \)</p>
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<p>Median \( = 20 + \left(\frac{25 - 14}{22}\right) \times 10 = 20 + \left(\frac{11}{22}\right) \cdot 10 = 20 + 5 = 25 \)</p>
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