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The empirical relation Mode = 3 Median − 2 Mean is best described as ____.

Class 10StatisticsStatistics and Probability
  • Aan approximation that works well for moderately skewed data
  • Ban exact identity that is true for every set of data
  • Cthe definition of the mode for grouped data
  • Da rule that applies only when all class widths are equal

Step-by-Step Solution

It is an empirical (observed) approximation, not a proved identity. It holds closely for moderately skewed distributions and is used as a quick check, or to estimate the third measure when two are known. It is exact only in the symmetric case, where all three measures coincide.
Answer: A) an approximation that works well for moderately skewed data

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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