Q17
1 markMCQSection A

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 Relationship Between Mean, Median and Mode

Options

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

(c) 15 — using Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}: 21=3×Median−2(12)  ⟹  Median=453=1521 = 3 \times \text{Median} - 2(12) \implies \text{Median} = \frac{45}{3} = 15.

empirical relationshipmean median mode3 median = mode + 2 meanKarl Pearson formulastatistics class 10measures of central tendency

Marking Scheme

  • 11 mark: correctly selecting option (c) 15 using the empirical relation Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean} with Mean=12,Mode=21\text{Mean} = 12, \text{Mode} = 21.
  • 2No partial marks for MCQs; working may be shown for verification only.

Hint

Use the empirical formula: Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}, i.e. 3 Median=Mode+2 Mean3\,\text{Median} = \text{Mode} + 2\,\text{Mean}.

Quick Oral Answer

Using the empirical formula Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}, with Mean=12\text{Mean} = 12 and Mode=21\text{Mode} = 21, we get 21=3 Median−2421 = 3\,\text{Median} - 24, so Median=453=15\text{Median} = \frac{45}{3} = 15.

Analysis & Explanation

Applies the empirical relationship between mean, median and mode for moderately skewed data.


Concept

  • CBSE Statistics uses: Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}, i.e., 3 Median=Mode+2 Mean3\,\text{Median} = \text{Mode} + 2\,\text{Mean}.

Key points

  • Given Mean=12,Mode=21\text{Mean} = 12, \text{Mode} = 21: 21=3×Median−24  ⟹  3×Median=45  ⟹  Median=1521 = 3 \times \text{Median} - 24 \implies 3 \times \text{Median} = 45 \implies \text{Median} = 15.

Common mistakes

  • Misremembering the formula order, e.g., writing Mean = 3 Median − 2 Mode or Mode = 3 Mean − 2 Median instead of the correct relation.

Real-world

  • This empirical relation (Karl Pearson) is used for quick estimation of one central tendency measure when the other two are known, e.g., in preliminary analysis of income or exam score distributions.

Common Mistakes

  1. 1Misremembering the formula, e.g. writing Mean=3 Median−2 Mode\text{Mean} = 3\,\text{Median} - 2\,\text{Mode} instead of the correct Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}.
  2. 2Sign errors while rearranging the equation, e.g. writing 3 Median=Mode−2 Mean3\,\text{Median} = \text{Mode} - 2\,\text{Mean} instead of Mode+2 Mean\text{Mode} + 2\,\text{Mean}.
  3. 3Arithmetic slip in division, e.g. dividing 45 incorrectly to get a non-integer distractor like 13.5.

Interesting Facts

The empirical relationship Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean} was proposed by statistician Karl Pearson as an approximation for moderately skewed distributions, not an exact identity.

This formula is only an approximation; for a perfectly symmetric (normal) distribution the mean, median, and mode all coincide at the same value.

Pearson also developed the related 'Pearson's coefficient of skewness,' which uses the same mean-median-mode relationship to measure how asymmetric a data set is.

Spotted a mistake or something unclear?

Tell us — we fix reported answers fast.

Frequently Asked Questions

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

For a moderately skewed distribution: Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}, equivalently 3 Median=Mode+2 Mean3\,\text{Median} = \text{Mode} + 2\,\text{Mean}.

Who proposed this empirical formula?

The empirical relationship between mean, median, and mode was proposed by the statistician Karl Pearson.