Empirical Relation Between Mean Median and Mode Explained

Empirical Relation Between Mean Median and Mode connects the three measures of central tendency for a moderately skewed distribution. The standard formula is Mode = 3 Median − 2 Mean. This page explains the formula, its rearrangements, calculation methods, sign-based interpretation and numerical examples used in quantitative aptitude.

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What Is the Empirical Relation Between Mean, Median and Mode?

The empirical relation is Mode = 3 Median − 2 Mean. Equivalently, Mean − Mode = 3(Mean − Median).

This approximate relation is used when the values of mean, median and mode belong to a moderately skewed distribution. If any two measures are known, the third can be calculated. For example, when Mean = 24 and Median = 26, Mode = 3(26) − 2(24) = 78 − 48 = 30.

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Empirical Relation of Mean, Median and Mode

Learn the empirical relationship between mean, median and mode, including the formula Mode = 3 Median − 2 Mean and how these measures are connected.

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Empirical Relation Between Mean Median and Mode Formula & Tricks

Important Formulas

Standard empirical relation
Mode = 3 Median − 2 Mean

Use this form when mean and median are given and mode is required.

Relation using deviations
Mean − Mode = 3(Mean − Median)

This is an equivalent form of the standard relation and helps compare the deviations of the three measures.

Finding the median
Median = (Mode + 2 Mean) ÷ 3

Rearrange the standard formula when mode and mean are known.

Finding the mean
Mean = (3 Median − Mode) ÷ 2

Rearrange the standard formula when median and mode are known.

Shortcut form
Mode = Median + 2(Median − Mean)

The mode is the median increased by twice the difference between median and mean.

Quick Tricks

Use the shortest rearrangement

For mode, calculate the difference between median and mean, double it, and add it to the median.

Example: If Mean = 18 and Median = 21, Mode = 21 + 2(21 − 18) = 27.
Check the order from skewness

For a positively skewed distribution, Mean > Median > Mode. For a negatively skewed distribution, Mean < Median < Mode. The calculated values should follow the expected order when the distribution type is known.

Example: If Mean = 30 and Median = 26, the formula gives Mode = 3(26) − 2(30) = 18, giving Mean > Median > Mode.
Check the result by substitution

Substitute the calculated value into Mean − Mode = 3(Mean − Median). Equal values on both sides confirm the arithmetic.

Example: For Mean = 24, Median = 26 and Mode = 30: 24 − 30 = −6 and 3(24 − 26) = −6.

Empirical Relation Between Mean Median and Mode Concepts

Calculating Mode from Mean and Median

Mode is found by subtracting twice the mean from three times the median.

Apply Mode = 3 Median − 2 Mean. Keep the coefficients and signs unchanged while substituting the values.

Example: If Mean = 32 and Median = 35, Mode = 3(35) − 2(32) = 105 − 64 = 41.

Calculating Median from Mean and Mode

Median is calculated as one-third of the sum of mode and twice the mean.

Use Median = (Mode + 2 Mean) ÷ 3. The numerator must be divided by 3 after adding both terms.

Example: If Mean = 30 and Mode = 42, Median = (42 + 2 × 30) ÷ 3 = 102 ÷ 3 = 34.

Calculating Mean from Median and Mode

Mean is calculated by subtracting mode from three times the median and dividing the result by 2.

Use Mean = (3 Median − Mode) ÷ 2. This form follows directly from Mode = 3 Median − 2 Mean.

Example: If Median = 21 and Mode = 18, Mean = (3 × 21 − 18) ÷ 2 = 45 ÷ 2 = 22.5.

Relation with Distribution Skewness

The relative order of mean, median and mode indicates the direction of skewness.

In a positively skewed distribution, the longer right tail pulls the mean upward, so Mean > Median > Mode. In a negatively skewed distribution, the longer left tail pulls the mean downward, so Mean < Median < Mode. In a symmetric distribution, all three measures are equal.

Example: Mean = 30, Median = 26 and Mode = 18 satisfy Mean > Median > Mode, which is consistent with positive skewness.

Approximate Nature of the Relation

The empirical relation is an approximate formula, not an exact identity for every dataset.

It is generally applied to a unimodal, moderately skewed distribution. For a perfectly symmetric distribution, Mean = Median = Mode, and the formula also gives the same value.

Example: If Mean = Median = 15, then Mode = 3(15) − 2(15) = 15.

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Practice Empirical Relation Between Mean Median and Mode Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Quick Revision Points

Core formulas and rules for the empirical relation between mean, median and mode.

  • Mode = 3 Median − 2 Mean.
  • Median = (Mode + 2 Mean) ÷ 3.
  • Mean = (3 Median − Mode) ÷ 2.
  • Equivalent relation: Mean − Mode = 3(Mean − Median).
  • Shortcut: Mode = Median + 2(Median − Mean).
  • Positive skewness generally follows Mean > Median > Mode.
  • Negative skewness generally follows Mean < Median < Mode.
  • The relation is approximate and is mainly used for moderately skewed distributions.

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Empirical Relation Between Mean Median and Mode FAQs

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

The relation is Mode = 3 Median − 2 Mean. Its equivalent form is Mean − Mode = 3(Mean − Median).

If the mean is 20 and the median is 24, what is the mode?

Mode = 3(24) − 2(20) = 72 − 40 = 32.

If the mean is 30 and the mode is 42, what is the median?

Median = (42 + 2 × 30) ÷ 3 = 102 ÷ 3 = 34.

If the median is 25 and the mode is 19, what is the mean?

Mean = (3 × 25 − 19) ÷ 2 = 56 ÷ 2 = 28.

Which measure is greatest in a positively skewed distribution?

The usual order is Mean > Median > Mode. The right tail increases the mean relative to the median and mode.

What happens to the empirical relation in a symmetric distribution?

Mean, median and mode are equal. If each is k, then 3k − 2k = k, so the relation holds.

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