A student reports mean = 30, median = 32 and mode = 40 for one and the same distribution. What does the empirical relation say?

Class 10StatisticsStatistics and Probability

AThe mode should be about 34, so the report is fine
BThe mode should be about 40, so the report is consistent
CThe mode should be about 36, so the reported mode looks wrong
DThe mean should be about 40, so the reported mean looks wrong

Step-by-Step Solution

Mode ≈ 3 Median − 2 Mean = 3(32) − 2(30) = 96 − 60 = 36. The relation is only an approximation, so a small gap would be acceptable, but a reported 40 against an expected 36 is a clear signal to re-check the working.

Answer: C) The mode should be about 36, so the reported mode looks wrong

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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