Question 171 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.

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

Marking scheme

  1. 1.1 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.
  2. 2.No 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.

Common mistakes

  1. 1.Misremembering 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. 2.Sign 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. 3.Arithmetic slip in division, e.g. dividing 45 incorrectly to get a non-integer distractor like 13.5.

Analysis and 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.

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.

Question details

Q17 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Statistics
Topic
Empirical Relationship Between Mean, Median and Mode
Bloom's level
apply
Syllabus topic
Unit VII: Statistics and Probability — Statistics: Empirical relation 3 Median=Mode+2 Mean3\,\text{Median} = \text{Mode} + 2\,\text{Mean}
Author
CBSE
Reviewed by
Padhantu Academic Team

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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.