Median from a Discrete Frequency Table

Quick answer: Sort the values and use cumulative frequencies to locate the central observation. For odd N use position (N + 1)/2; for even N average values at positions N/2 and N/2 + 1.

Worked example

Values 10, 20, 30 have frequencies 2, 3, 1. Find the median.

1. Total N = 6, so use positions 3 and 4.
2. Cumulative frequencies are 2, 5, 6.
3. Positions 3 and 4 both fall in the value-20 group.
4. Median = (20 + 20)/2 = 20.

Check the result

Expanded data are 10, 10, 20, 20, 20, 30, confirming the two middle values.

Common mistake to avoid

Do not take the median of the frequency column. Frequencies tell you positions; the original values provide the median.

Try it yourself

1. Values 1, 3, 8 have frequencies 1, 2, 2. Find median. 2. Find median of 1, 1, 3, 7.

Answers and reasoning

1. N = 5; position 3 is value 3. 2. Average positions 2 and 3: (1 + 3)/2 = 2.

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