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Assertion (A): The mean, median and mode of a symmetric distribution are all equal. Reason (R): In a perfectly symmetric distribution the empirical relation Mode = 3 Median − 2 Mean gives Mode = Median = Mean. Choose the correct option.

Class 10StatisticsStatistics and Probability
  • ABoth A and R are true, and R is the correct explanation of A.
  • BBoth A and R are true, but R is not the correct explanation of A.
  • CA is true but R is false.
  • DA is false but R is true.

Step-by-Step Solution

If Mean = Median = M, then 3M − 2M = M, so Mode = M too. R correctly explains why all three coincide in a symmetric distribution.
Answer: A) Both A and R are true, and R is the correct explanation of A.

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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