CDS Mathematics · Statistics

Choosing a Measure of Central Tendency

Each average suits a different job: the mean uses every value, the median resists extreme values, the mode gives the most common value, and the harmonic mean averages rates.

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Why this matters

Eight PYQs, most of them EASY. They test which average fits a situation and the properties behind the choice. One table and the empirical relation Mode = 3 Median − 2 Mean answer all of them.

Concept 1 of 2: Which average to use

Ask what the average must do. If extreme values should not distort it, use the median. If you want the most common item — a shoe size to stock — use the mode. If you are averaging rates with a fixed numerator, use the harmonic mean.

Definition

Mean: uses every value; pulled by extremes; needs every value (fails with open-ended classes).
Median: a positional average; unaffected by extremes; can be found with open-ended classes and read off an ogive (where the 'less than' and 'more than' curves cross).
Mode: the most frequent value; may not be unique, or may not exist; suits categories like sizes.
Harmonic mean: averages rates like speed or price per unit when the numerator is fixed.
Geometric mean: averages growth rates and ratios.

Diagram · mean, median & mode under skew

With a long right tail (positive skew) the mean is dragged toward it, giving Mode < Median < Mean (the order reverses for a left tail). This is the basis of the empirical relation Mode ≈ 3·Median − 2·Mean. For any data, Σ(xᵢ − x̄) = 0 — deviations above and below the mean always cancel.

Measure	Best for	Weakness
Mean	Balanced data, further algebra	Pulled by extreme values
Median	Skewed data, open-ended classes	Ignores the size of the other values
Mode	Most common item (sizes, categories)	May not be unique
Harmonic mean	Averaging rates (km/h, Rs./unit)	Only for positive values
Geometric mean	Growth rates, ratios	Needs positive values

The median is the positional average and the one least affected by extreme observations.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q91Easy

Example 1 · Statistics · Choosing a Measure of Central Tendency

Which measure of central tendency is least affected by the presence of extreme observations in the data?

Positional does not mean 'the mode'

The median is fixed by rank alone, which is why it is called the positional average. Some books also call the mode positional; if both are offered, the median is the standard answer.

Drill 6 more on which average to use

Concept 2 of 2: The empirical relation

In a moderately skewed distribution the median sits between the mean and the mode, about a third of the way from the mean. That gives a rule to find any one of the three from the other two.

Definition

For a moderately asymmetrical distribution: Mode $= 3\,$ Median $- 2\,$ Mean.
Equivalently Mean $-$ Mode $= 3($ Mean $-$ Median $)$ .
In a symmetrical distribution all three are equal.
If the mean exceeds the median, the distribution is skewed to the right and the mode is lower still.

Empirical relation

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Worked example

In a moderately skewed distribution the mode is

40

and the median is

46

. Estimate the mean.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q55Easy

Example 2 · Statistics · Choosing a Measure of Central Tendency

In an asymmetrical distribution, if the mean and median of the distribution are 270 and 220 respectively, then the mode of the data is

Three median, two mean

It is

3\,

Median

- 2\,

Mean, not the other way round. Swapping the coefficients gives a 'mode' outside the range of the other two.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The empirical relation
Empirical relation
$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$

Reference tables (1)

Which average to use5 rows

Measure	Best for	Weakness
Mean	Balanced data, further algebra	Pulled by extreme values
Median	Skewed data, open-ended classes	Ignores the size of the other values
Mode	Most common item (sizes, categories)	May not be unique
Harmonic mean	Averaging rates (km/h, Rs./unit)	Only for positive values
Geometric mean	Growth rates, ratios	Needs positive values

The median is the positional average and the one least affected by extreme observations.

Watch out for (2)

Positional does not mean 'the mode'
→ Which average to use
Three median, two mean
→ The empirical relation

Test yourself on Statistics

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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