NDA Maths · Statistics

Dispersion — Standard Deviation, Variance, Mean Deviation

How spread out the data is around its centre — mean deviation, variance and standard deviation each measure spread on a different scale.

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Why this matters

44 PYQs across 2017–2026. 22 EASY + 18 MODERATE + 4 HARD — almost every paper has one. The favourite shapes are linear-transformation effects on SD, the computational identity that links mean-of-squares to mean-squared and variance, and coefficient-of-variation comparisons. Master the six concepts below and dispersion becomes formulaic, not intimidating.

Concept 1 of 6

Mean Deviation

Intuition

Average distance between each observation and a chosen centre — almost always the mean or the median. It is the simplest measure of spread, but its absolute-value formula makes algebra clunky, which is why SD is preferred in higher statistics.

Definition

For $n$ observations $x_1,\ldots,x_n$ and a reference value $A$ , the mean deviation about $A$ is the average of the absolute deviations from $A$ . When $A$ is the median, the mean deviation is minimum among all possible $A$ .

Mean Deviation about A

\text{MD}(A) = \dfrac{1}{n}\sum_{i=1}^{n}|x_i - A|

$A$ reference point (typically mean or median)
$|x_i - A|$ absolute deviation of $x_i$ from $A$

Diagram · mean deviation = average distance from the centre

MD = (3 + 1 + 0 + 4) / 4 = 2

Take each value's distance from the centre (the red segments, signs dropped) and average them. Mean deviation can be taken about the mean or the median; about the median it is smallest.

Worked example

Find the mean deviation of

2, 4, 6, 8, 10

about the mean.

Compute the mean: $\bar{x} = \dfrac{2+4+6+8+10}{5} = 6$ .
Compute absolute deviations from 6: $|2-6|, |4-6|, |6-6|, |8-6|, |10-6| = 4, 2, 0, 2, 4$ .
Sum the absolute deviations: $4+2+0+2+4 = 12$ .
Divide by $n$ : $\text{MD} = \dfrac{12}{5} = 2.4$ .

Answer:

\text{MD} = 2.4

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the mean deviation of

3, 6, 9, 12

about the mean.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Mean deviation of $2, 4, 6$ about the mean $(\bar{x}=4)$ ?
2.
Mean deviation of $1, 2, 3$ about the mean?
3.
Mean deviation of $10, 20$ about the mean?
4.
Mean deviation of $5, 5, 5$ ?

From the bank · past-year question

Example 1StatisticsEASY

What is the mean deviation of the first 10 natural numbers?

[Q110 · Sep · 2024]

Mean deviation about median is always $\leq$ about the mean

Among all reference points

A

, the median minimises

\sum|x_i - A|

. If a PYQ asks for the minimum mean deviation, the answer uses the median, not the mean.

Drill 10 more on mean deviation

Concept 2 of 6

Variance

Intuition

Average of the squared distances from the mean. Squaring kills the absolute-value bracket and gives variance much better algebraic properties — but the unit becomes the data's unit squared (cm becomes cm², rupees becomes rupees²), which is why we usually report its square root, the standard deviation.

Definition

For $n$ observations with mean $\bar{x}$ , the variance is the average of the squared deviations from $\bar{x}$ . It can also be computed using the identity $\sigma^2 = \overline{x^2} - \bar{x}^2$ , where $\overline{x^2}$ is the mean of the squares.

Variance — two equivalent forms

\sigma^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2

$\sigma^2$ variance
$\bar{x}$ arithmetic mean
$\sum x_i^2$ sum of squares of observations

Visualization · move the points, watch the squared deviations

x1: 2.0x2: 4.0x3: 6.0x4: 8.0x5: 10.0

Variance σ² = 8.00SD σ = 2.83Σ(xᵢ − x̄)² = 40.00

Each red square has side |xᵢ − x̄|, so its AREA is the squared deviation. Variance is the AVERAGE area. Pull all points toward the mean — every square shrinks. Pull them apart — they grow.

Worked example

Find the variance of

2, 4, 6, 8, 10

Compute the mean: $\bar{x} = 6$ .
Compute squared deviations: $(2-6)^2, (4-6)^2, (6-6)^2, (8-6)^2, (10-6)^2 = 16, 4, 0, 4, 16$ .
Sum the squared deviations: $16 + 4 + 0 + 4 + 16 = 40$ .
Divide by $n$ : $\sigma^2 = \dfrac{40}{5} = 8$ .

Answer:

\sigma^2 = 8

Practice this conceptself-check · 4 quick reps

Try it yourself

For 5 observations,

\sum x_i = 30

and

\sum x_i^2 = 220

. Find the variance.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Variance of $1, 2, 3$ $(\bar{x}=2)$ ?
2.
Variance of $2, 4, 6$ $(\bar{x}=4)$ ?
3.
Variance of $5, 5, 5$ ?
4.
Variance of $0, 10$ $(\bar{x}=5)$ ?

From the bank · past-year question

Example 2StatisticsMODERATE

\sum_{i=1}^{10}x_{i}=110

and

\sum_{i=1}^{10}x_{i}^{2}=1540

, then what is the variance?

[Q116 · Sep · 2021]

Computational form saves time on $\Sigma x_i^2$ -style PYQs

When you are given

\sum x_i

and

\sum x_i^2

directly, use

\sigma^2 = \overline{x^2} - \bar{x}^2

— not the original definition. NDA papers favour this shape because it tests whether you remember the identity.

Drill 6 more on variance

Concept 3 of 6

Standard Deviation

Intuition

Square root of the variance — brings the unit back to the data's original unit. SD is what statisticians actually report when they talk about "spread". If the data is in centimetres, SD is in centimetres; if in rupees, SD is in rupees. Variance lives in squared units.

Definition

The standard deviation is the non-negative square root of the variance. It has the same unit as the original observations.

Standard Deviation

\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}

$\sigma$ standard deviation (always $\geq 0$ )

Worked example

Find the standard deviation of

2, 4, 6, 8, 10

From the variance example above, $\sigma^2 = 8$ .
Take the square root: $\sigma = \sqrt{8} = 2\sqrt{2}$ .
Numerically, $\sigma \approx 2.83$ .

Answer:

\sigma = 2\sqrt{2} \approx 2.83

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the standard deviation of

1, 2, 3, 4, 5

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Variance $16$ . Standard deviation?
2.
Variance $49$ . Standard deviation?
3.
Variance $2$ . Standard deviation?
4.
SD of $3, 3, 3$ ?

From the bank · past-year question

Example 3StatisticsMODERATE

If the mean and sum of squares of 10 observations are 40 and 16160 respectively, then what is the standard deviation?

[Q103 · Sep · 2023]

SD and mean deviation share units; variance does not

If the data is measured in cm, SD and MD are also in cm but variance is in cm². A PYQ asks "which has the same unit as the mean?" — answer is SD or MD, not variance.

Drill 5 more on standard deviation

Concept 4 of 6

Linear Transformation of SD and Variance

Intuition

Adding a constant shifts every value but the spread between them is unchanged — so shift has zero effect on SD or variance. Multiplying by

a

stretches all distances by

|a|

, so SD scales by

|a|

and variance by

a^2

Definition

If $Y = aX + b$ is a linear transformation of $X$ , then the variance of $Y$ is $a^2$ times the variance of $X$ , and the standard deviation of $Y$ is $|a|$ times the SD of $X$ . The shift $b$ has no effect on either.

Variance and SD under Y = aX + b

\text{Var}(Y) = a^2\,\text{Var}(X) \qquad \sigma_Y = |a|\,\sigma_X

$a$ scale factor
$b$ shift — irrelevant for dispersion

Worked example

The SD of a dataset

X

is 4. Find the SD of

Y = 3X + 7

Identify the transformation: $a = 3,\ b = 7$ .
The shift $b = 7$ has no effect on SD.
The multiplier $a = 3$ scales SD by $|3| = 3$ .
Therefore $\sigma_Y = 3 \times 4 = 12$ .

Answer:

\sigma_Y = 12

Practice this conceptself-check · 4 quick reps

Try it yourself

If the SD of

X

6

, find the SD of

Y = -2X + 10

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
SD is $5$ . SD of $2X$ ?
2.
SD is $4$ . SD of $X + 100$ ?
3.
Variance is $9$ . Variance of $3X$ ?
4.
SD is $6$ . SD of $-X$ ?

From the bank · past-year question

Example 4StatisticsEASY

The standard deviation of 100 observations is 10. If 5 is multiplied to each of the observations, then what is the new standard deviation?

[Q115 · Apr · 2026]

Squaring $a$ for variance, taking absolute value for SD

Students often write

\sigma_Y^2 = a\,\sigma_X^2

(forgetting the square) or

\sigma_Y = a\,\sigma_X

(forgetting the modulus). If

a

is negative,

|a|

is the correct scale for SD — SD is non-negative by definition.

Drill 8 more on linear transformation of sd and variance

Concept 5 of 6

Coefficient of Variation (CV)

Intuition

SD measures absolute spread, but a SD of 10 means different things on a salary scale (small) versus a marks scale (large). CV normalises SD by the mean and reports a unitless percentage, so you can compare variability across totally different datasets.

Definition

The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage. The dataset with the higher CV is considered more variable.

Coefficient of Variation

\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100 \%

$\sigma$ standard deviation
$\bar{x}$ arithmetic mean

Worked example

A dataset has mean

50

and standard deviation

10

. Find its coefficient of variation.

Apply the formula: $\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100$ .
Substitute: $\text{CV} = \dfrac{10}{50} \times 100$ .
Compute: $\text{CV} = 0.2 \times 100 = 20\%$ .

Answer:

\text{CV} = 20\%

Practice this conceptself-check · 4 quick reps

Try it yourself

Dataset A has mean

100

, SD

15

. Dataset B has mean

40

, SD

8

. Which is more variable?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Mean $50$ , SD $5$ . CV?
2.
Mean $20$ , SD $4$ . CV?
3.
Mean $100$ , SD $25$ . CV?
4.
Mean $40$ , SD $0$ . CV?

From the bank · past-year question

Example 5StatisticsEASY

If the mean of a frequency distribution is 100 and the coefficient of variation is 45%, then what is the value of the variance?

[Q109 · Apr · 2021]

CV is unitless — that's the entire point

Some answers give CV with a unit attached. Wrong. CV is a percentage. If the question compares two datasets with different units (e.g. height in cm vs weight in kg), only CV makes a fair comparison — not SD.

Drill 4 more on coefficient of variation (cv)

Concept 6 of 6

Computational Identity & Minimum-SSE Property

Intuition

Two identities that NDA papers exploit relentlessly. First: the mean of the squares equals the variance plus the square of the mean. Second: among all reference points, the sum of squared deviations is minimised when the reference is the mean (the median minimises absolute deviations; the mean minimises squared deviations).

Definition

From $\sigma^2 = \overline{x^2} - \bar{x}^2$ follows the identity $\sum x_i^2 = n(\sigma^2 + \bar{x}^2)$ . Also, the function $f(a) = \sum (x_i - a)^2$ is minimised when $a = \bar{x}$ .

Two load-bearing identities

\dfrac{\sum x_i^2}{n} = \bar{x}^2 + \sigma^2 \qquad \text{and} \qquad \arg\min_{a}\sum_{i}(x_i - a)^2 = \bar{x}

Worked example

Given

\sum x_i = 50

and

\sum x_i^2 = 530

for

n = 10

observations, find the variance.

Compute the mean: $\bar{x} = \dfrac{50}{10} = 5$ .
Use the identity $\sigma^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2$ .
Substitute: $\sigma^2 = \dfrac{530}{10} - 5^2 = 53 - 25$ .
Compute: $\sigma^2 = 28$ .

Answer:

\sigma^2 = 28

Practice this conceptself-check · 4 quick reps

Try it yourself

If the mean of 4 observations is 5 and the variance is 4, find

\sum x_i^2

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Mean of squares $30$ , mean $5$ . Variance?
2.
$n=5$ , $\sum x^2 = 100$ , $\bar{x}=4$ . Variance?
3.
Variance $9$ , mean $4$ . Mean of squares?
4.
Mean of squares $50$ , mean $7$ . Variance?

From the bank · past-year question

Example 6StatisticsMODERATE

Let

\displaystyle\sum_{i=1}^{9}x_i^2=855

. If

M

is the mean and

\sigma

is the standard deviation of

x_1,x_2,\ldots,x_9

, then what is the value of

M^2+\sigma^2

[Q111 · Sep · 2024]

$\overline{x^2} \neq (\bar{x})^2$ — they differ by exactly $\sigma^2$

Mean of the squares is NOT the square of the mean. Their difference is the variance:

\overline{x^2} - \bar{x}^2 = \sigma^2 \geq 0

. PYQs plant this trap by asking for "mean of squares" or "

M^2 + \sigma^2

" and expecting you to recognise it as

\sum x_i^2/n

Scaling inside the deviation moves the minimiser too

For

S(a) = \sum (c\,x_i - a)^2

, expand and minimise: the minimum is at

a = c\,\bar{x}

, NOT

a = \bar{x}

. PYQs commonly use

c = 2

(e.g.

S = \sum (2x_i - a)^2

) and expect you to identify the minimiser as

2\bar{x}

— twice the mean, not the mean.

Drill 5 more on computational identity & minimum-sse property

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 (6)

Mean Deviation
Mean Deviation about A
$\text{MD}(A) = \dfrac{1}{n}\sum_{i=1}^{n}|x_i - A|$
Variance
Variance — two equivalent forms
$\sigma^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2$
Standard Deviation
Standard Deviation
$\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}$
Linear Transformation of SD and Variance
Variance and SD under Y = aX + b
$\text{Var}(Y) = a^2\,\text{Var}(X) \qquad \sigma_Y = |a|\,\sigma_X$
Coefficient of Variation (CV)
Coefficient of Variation
$\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100 \%$
Computational Identity & Minimum-SSE Property
Two load-bearing identities
$\dfrac{\sum x_i^2}{n} = \bar{x}^2 + \sigma^2 \qquad \text{and} \qquad \arg\min_{a}\sum_{i}(x_i - a)^2 = \bar{x}$

Watch out for (7)

Mean deviation about median is always $\leq$ about the mean
→ Mean Deviation
Computational form saves time on $\Sigma x_i^2$ -style PYQs
→ Variance
SD and mean deviation share units; variance does not
→ Standard Deviation
Squaring $a$ for variance, taking absolute value for SD
→ Linear Transformation of SD and Variance
CV is unitless — that's the entire point
→ Coefficient of Variation (CV)
$\overline{x^2} \neq (\bar{x})^2$ — they differ by exactly $\sigma^2$
→ Computational Identity & Minimum-SSE Property
Scaling inside the deviation moves the minimiser too
→ Computational Identity & Minimum-SSE Property

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1StatisticsEASY

Consider the following statements: Statement 1: Range is not a good measure of dispersion. Statement 2: Range is highly affected by the existence of extreme values. Which one of the following is correct in respect of the above statements?

[Q110 · Apr · 2017]

Example 2StatisticsMODERATE

The mean and variance of five observations are 14 and 13.2 respectively. Three of the five observations are 11, 16 and 20. What are the other two observations?

[Q101 · Apr · 2023]

Example 3StatisticsEASY

Consider the following statements: I. Mean and variance have the same unit of measurement. II. Mean deviation and standard deviation have the same unit of measurement. Which of the statements given above is/are correct?

[Q120 · Sep · 2025]

Example 4StatisticsEASY

The variance of 25 observations is 4. If 2 is added to each observation, then the new variance of the resulting observations is

[Q110 · Sep · 2018]

Example 5StatisticsEASY

Mean=5, SD=2. If 5 added to each value, what is CV of new set?

[Q112 · Apr · 2018]

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