Q17
1 markMCQSection A

  1. If the mean and mode of a data are 12 and 21 respectively, then its median is : (a) 6 (b) 13.5 (c) 15 (d) 14

Statistics
Empirical relation between mean, median and mode

Options

(A)6
(B)13.5
(C)15
(D)14
Official Answer

(c) 15, from Mode=3×Median−2×Mean⇒21=3×Median−24⇒Median=453=15\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \Rightarrow 21 = 3 \times \text{Median} - 24 \Rightarrow \text{Median} = \frac{45}{3} = 15.

empirical relationmode = 3 median − 2 meancentral tendencymean 12mode 21median 15statisticsKarl Pearson

Marking Scheme

  • 11 mark for correct option (c) 15.
  • 2Expected working: 21=3 Median−2(12)⇒3 Median=45⇒Median=1521 = 3\,\text{Median} - 2(12) \Rightarrow 3\,\text{Median} = 45 \Rightarrow \text{Median} = 15.
  • 3Accept the rearranged form Median=Mode+2 Mean3=453=15\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3} = \frac{45}{3} = 15.

Hint

Use Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}, i.e. Median=Mode+2×Mean3\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}.

Quick Oral Answer

Using the empirical relation, mode equals three median minus two mean, so 21 equals 3 median minus 24, giving 3 median equal to 45 and median equal to 15.

Analysis & Explanation

Tests the empirical relationship connecting mean, median, and mode for moderately skewed data.


Concept

  • Empirical formula: Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}.
  • Given Mean = 12 and Mode = 21, substitute: 21=3×Median−2(12)=3×Median−2421 = 3 \times \text{Median} - 2(12) = 3 \times \text{Median} - 24.
  • Solving: 3×Median=453 \times \text{Median} = 45, so Median = 15.

Common mistakes

  • Misremembering the formula as Mean=3×Median−2×Mode\text{Mean} = 3 \times \text{Median} - 2 \times \text{Mode} or a similar wrong arrangement.
  • Arithmetic slip while transferring −24-24 to the other side (should add 24, not subtract).

Common Mistakes

  1. 1Taking the median as the simple average of mean and mode, 12+212=13.5\frac{12+21}{2} = 13.5 — this ignores the 3-median, 2-mean weighting.
  2. 2Misremembering the formula as Mean=3 Median−2 Mode\text{Mean} = 3\,\text{Median} - 2\,\text{Mode} and solving for the wrong quantity.
  3. 3Arithmetic slip in 21+24=4521 + 24 = 45 or dividing 45 by 3 incorrectly.

Interesting Facts

The empirical relation Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean} holds approximately for moderately skewed (asymmetrical) distributions and was formulated by statistician Karl Pearson.

For a perfectly symmetric distribution the mean, median and mode all coincide — the empirical formula then trivially gives Median=Mean=Mode\text{Median} = \text{Mean} = \text{Mode}.

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Frequently Asked Questions

What is the empirical relationship between mean, median and mode?

Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}. It can be rearranged as Median=Mode+2×Mean3\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3} and applies to moderately skewed distributions.

Why is the answer not 13.5, the average of 12 and 21?

The median is not the simple midpoint of mean and mode. The empirical formula weights the mean twice, giving Median=21+243=15\text{Median} = \frac{21 + 24}{3} = 15.

When does the empirical relation apply exactly?

It is an approximation for moderately asymmetrical (skewed) data. For symmetric data, mean, median and mode are all equal, and it still holds.