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For the data 0–10: 8, 10–20: 12, 20–30: 20, 30–40: 10 a student wrote Median = 20 + 10 × (25 − 40)/20 = 12.5. What is the mistake?

Class 10StatisticsStatistics and Probability
  • AThe cf of the median class itself was used instead of the cf of the class before it
  • BThe wrong class width h was used
  • CThe modal class was used in place of the median class
  • Dn was not halved before being put into the formula

Step-by-Step Solution

n = 50 and n/2 = 25; the cumulative frequencies are 8, 20, 40, 50, so the median class 20–30 is right. But cf in the formula is the cumulative frequency of the class BEFORE the median class, which is 20 — not 40. The correct value is 20 + 10(25 − 20)/20 = 22.5. A negative bracket is the tell-tale sign of this slip.
Answer: A) The cf of the median class itself was used instead of the cf of the class before it

Key idea: Median for grouped data

**Step 1**: build cumulative frequency (cf) column.
**Step 2**: find n/2 (half the total frequency).
**Step 3**: locate the **median class** — the FIRST class whose cumulative frequency ≥ n/2.
**Step 4**: Apply formula.

**Median formula**:
**Median = l + h × (n/2 − cf) / f**

Where:
- l = lower limit of median class
- h = class width
- n = total frequency
- cf = cumulative frequency of class BEFORE median class
- f = frequency of median class

The formula uses linear interpolation within the median class to find where the n/2-th item falls.

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