MathsCBSE Class 11 · Statistics

Variance and Standard Deviation: Methods, Shortcuts and Properties (Class 11)

Updated 17 October 2026 · Fact-checked

Variance σ² = Σ f(x − x̄)²/N = Σ f x²/N − x̄² and standard deviation σ = √variance. For class data with coded values y = (x − A)/h, σ² = h²[Σ f y²/N − (Σ f y/N)²]. Adding a constant to every observation leaves σ unchanged; multiplying by k multiplies σ by |k|. If an observation is replaced, recompute Σx and Σx² from N, x̄ and σ.

Each arrow is a deviation from the mean; squaring gives areas 36, 1, 1, 4, 16, and their average is the variance.

Illustration drawn for QuestionBuddy.

Understand Variance and Standard Deviation: Methods and Properties

Variance and standard deviation. For observations $x_{i}$ with frequencies $f_{i}$ ( $N = \sum f_{i}$ ) and mean $\overset{x}{ˉ}$ ,
$ $σ^{2} = \frac{\sum f _{i} ( x _{i} - x ˉ ) ^{2}}{N} = \frac{\sum f _{i} x _{i}^{2}}{N} - \overset{x}{ˉ}^{2}, σ = σ^{2} .$ $
The standard deviation has the same units as the data. The second form (mean of squares minus square of the mean) is quicker for calculation.

Raw data. $σ^{2} = \frac{1}{n} \sum (x_{i} - \overset{x}{ˉ})^{2}$ . For $4, 9, 11, 12, 14$ : $\overset{x}{ˉ} = 10$ , squared deviations $36, 1, 1, 4, 16$ , $σ^{2} = \frac{58}{5} = 11.6$ , $σ \approx 3.41$ .

Frequency distributions. Make a table: $x$ , $f$ , $f x$ , $f x^{2}$ . For continuous data use class mid-points.

Shortcut (step-deviation) method. Choose an assumed mean $A$ (usually a mid-point near the middle) and class width $h$ ; put $y_{i} = \frac{x _{i} - A}{h}$ . Then
$ $\overset{x}{ˉ} = A + h \frac{\sum f _{i} y _{i}}{N}, σ^{2} = h^{2} [\frac{\sum f _{i} y _{i}^{2}}{N} - (\frac{\sum f _{i} y _{i}}{N})^{2}] .$ $
Do not forget the factors $h$ and $h^{2}$ at the end.

Properties.

Adding a constant $q$ to every observation: mean $+ q$ , $σ$ unchanged.
Multiplying every observation by $p$ : mean $\times p$ , $σ \times ∣ p ∣$ , variance $\times p^{2}$ .
Hence for $p x + q$ : $σ_{new} = ∣ p ∣ σ$ .
$σ = 0$ if and only if all observations are equal.

Replacing or finding observations. From the given $N$ , $\overset{x}{ˉ}$ and $σ$ : $\sum x = N \overset{x}{ˉ}$ and $\sum x^{2} = N (σ^{2} + \overset{x}{ˉ}^{2})$ . To replace an observation $a$ by $b$ , adjust: $\sum x \to \sum x - a + b$ and $\sum x^{2} \to \sum x^{2} - a^{2} + b^{2}$ , then recompute the mean and the variance. To find missing observations, subtract the known values from the totals, giving the sum and the sum of squares of the unknowns; solve with $(a + b)^{2} = a^{2} + b^{2} + 2 ab$ .

Combining two groups (sizes $n_{1}, n_{2}$ , means $m_{1}, m_{2}$ , standard deviations $s_{1}, s_{2}$ ): the combined mean is $\frac{n _{1} m _{1} + n _{2} m _{2}}{n _{1} + n _{2}}$ ; for the variance, add up the sums of squares $n_{i} (s_{i}^{2} + m_{i}^{2})$ , divide by $n_{1} + n_{2}$ and subtract the square of the combined mean.

The analysis of frequency distributions (coefficient of variation) is not in the 2026-27 syllabus.

Variance and Standard Deviation: Methods and Properties: formulas to remember

Variance: $σ^{2} = Σ f x^{2} / N - \overset{x}{ˉ}^{2}$; Equivalent to Σ f(x − x̄)²/N.
Standard deviation: $σ = \sqrt (σ^{2})$; Same units as the data.
Step-deviation method: $σ^{2} = h^{2} [Σ f y^{2} / N - (Σ f y / N)^{2}], y = (x - A) / h$; Mean x̄ = A + h Σ f y / N.
Change of scale: $N e w σ = ∣ p ∣ σ f or o b ser v a t i o n s p x + q$; The shift q does not matter.
From totals: $Σ x = N \overset{x}{ˉ} an d Σ x^{2} = N (σ^{2} + \overset{x}{ˉ}^{2})$; Used for replaced or missing values.

How to solve Variance and Standard Deviation: Methods and Properties questions

Use these steps for variance and standard deviation.

1Find $N$ and the mean (use the step-deviation method for class data).
2Compute $\sum f x^{2}$ (or $\sum f y^{2}$ ) from the table.
3Use $σ^{2} = \frac{\sum f x ^{2}}{N} - \overset{x}{ˉ}^{2}$ , or the shortcut formula with $h^{2}$ .
4Take the square root to get $σ$ and state the units.
5For changes to the data, work with $\sum x$ and $\sum x^{2}$ and recompute.
6For missing values, form the sum and the sum of squares of the unknowns and solve the pair of equations.

Variance and Standard Deviation: Methods and Properties in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Change of units

Suppose the daily temperatures in a town have a mean of $2 0^{\circ}$ C and a standard deviation of $5^{\circ}$ C. They are converted to Fahrenheit using $F = 1.8 C + 32$ .

Find the new mean and standard deviation.

Try it first, then show the solutionSolution

New mean $= 1.8 \times 20 + 32 = 6 8^{\circ}$ F.
New standard deviation $= ∣1.8∣ \times 5 = 9^{\circ}$ F; the shift $32$ does not change the spread.

Answer: Mean $6 8^{\circ}$ F; SD $9^{\circ}$ F.

Memory hook: Shift moves the mean; scale stretches the spread.

Spread of marks

Suppose five students scored 40, 45, 50, 55 and 60 in a test.

Find the variance and standard deviation.

Try it first, then show the solutionSolution

Mean $= \frac{250}{5} = 50$ .
Squared deviations: $100, 25, 0, 25, 100$ ; sum $= 250$ .
$σ^{2} = \frac{250}{5} = 50$ and $σ = 50 \approx 7.07$ .

Answer: Variance 50; SD about 7.07 marks.

Memory hook: Square the gaps, average, then take the root.

Tricks to remember Variance and Standard Deviation: Methods and Properties

Short memory aids. Say each one aloud once, then use it on a question the same day.

Mean of squares minus square of the mean
σ² = Σx²/N − x̄²: no need to subtract the mean from each value.
Use it when: Quick variance.
h squared at the end
In the shortcut method the variance carries h² and the SD carries h.
Use it when: Class-interval data.
Totals from N, mean and SD
Σx = N x̄ and Σx² = N(σ² + x̄²).
Use it when: Replacing or missing observations.
Shifts are free
Adding a constant never changes SD.
Use it when: Property questions.

Variance and Standard Deviation: Methods and Properties: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Raw data · 2 marks · Very short answerBoard level

Find the variance and standard deviation of $4, 9, 11, 12, 14$ .

Try it first, then show the solutionSolution

$\overset{x}{ˉ} = \frac{50}{5} = 10$ ; squared deviations $36, 1, 1, 4, 16$ sum to $58$ . (1 mark)
$σ^{2} = \frac{58}{5} = 11.6$ , $σ = 11.6 \approx 3.41$ . (1 mark)

Answer: Variance 11.6; SD about 3.41

Example 2: Discrete table · 3 marks · Short answerBoard level

Find the variance and standard deviation for $x = 3, 5, 7, 9, 11$ with $f = 2, 4, 6, 5, 3$ .

Try it first, then show the solutionSolution

$N = 20$ ; $\sum f x = 6 + 20 + 42 + 45 + 33 = 146$ , so $\overset{x}{ˉ} = 7.3$ . (1 mark)
$\sum f x^{2} = 18 + 100 + 294 + 405 + 363 = 1180$ . (1 mark)
$σ^{2} = \frac{1180}{20} - 7. 3^{2} = 59 - 53.29 = 5.71$ ; $σ \approx 2.39$ . (1 mark)

Answer: Variance 5.71; SD about 2.39

Example 3: Step-deviation method · 3 marks · Short answerHarder board level

For the classes $10$ – $20$ , $20$ – $30$ , $30$ – $40$ , $40$ – $50$ , $50$ – $60$ with frequencies $3, 8, 14, 10, 5$ , find the mean and standard deviation using $A = 35$ , $h = 10$ .

Try it first, then show the solutionSolution

Mid-points $15, 25, 35, 45, 55$ ; $y = - 2, - 1, 0, 1, 2$ ; $N = 40$ . $\sum f y = - 6 - 8 + 0 + 10 + 10 = 6$ ; $\sum f y^{2} = 12 + 8 + 0 + 10 + 20 = 50$ . (1 mark)
$\overset{x}{ˉ} = 35 + 10 \times \frac{6}{40} = 36.5$ . (1 mark)
$σ^{2} = 100 [\frac{50}{40} - (\frac{6}{40})^{2}] = 100 (1.25 - 0.0225) = 122.75$ ; $σ \approx 11.08$ . (1 mark)

Answer: Mean 36.5; SD about 11.08

Example 4: Replacing an observation · 4 marks · Short answerHarder board level

The mean and standard deviation of 10 observations are 20 and 4. One observation, 15, is wrongly recorded and should be 25. Find the correct mean and standard deviation.

Try it first, then show the solutionSolution

$\sum x = 200$ ; corrected: $200 - 15 + 25 = 210$ , so the mean is $21$ . (1 mark)
$\sum x^{2} = 10 (16 + 400) = 4160$ . (1 mark)
Corrected: $4160 - 225 + 625 = 4560$ . (1 mark)
$σ^{2} = \frac{4560}{10} - 2 1^{2} = 456 - 441 = 15$ , so $σ = 15 \approx 3.87$ . (1 mark)

Answer: Mean 21; SD $15 \approx 3.87$

Example 5: Why the mean minimises squared deviations · 5 marks · Long answerHOTS / competency

For observations $x_{1}, \dots, x_{n}$ with mean $\overset{x}{ˉ}$ , prove that $\sum (x_{i} - a)^{2} = \sum (x_{i} - \overset{x}{ˉ})^{2} + n (\overset{x}{ˉ} - a)^{2}$ for every number $a$ , and deduce that $\sum (x_{i} - a)^{2}$ is least when $a = \overset{x}{ˉ}$ . Verify for $2, 4, 6, 8$ and $a = 3$ .

Try it first, then show the solutionSolution

Write $x_{i} - a = (x_{i} - \overset{x}{ˉ}) + (\overset{x}{ˉ} - a)$ and square: $(x_{i} - \overset{x}{ˉ})^{2} + 2 (x_{i} - \overset{x}{ˉ}) (\overset{x}{ˉ} - a) + (\overset{x}{ˉ} - a)^{2}$ . (1 mark)
Sum over $i$ : the middle term is $2 (\overset{x}{ˉ} - a) \sum (x_{i} - \overset{x}{ˉ}) = 0$ because deviations from the mean add to zero. (1 mark)
So $\sum (x_{i} - a)^{2} = \sum (x_{i} - \overset{x}{ˉ})^{2} + n (\overset{x}{ˉ} - a)^{2}$ . (1 mark)
The last term is $\geq 0$ and is $0$ only for $a = \overset{x}{ˉ}$ , so the sum is least at the mean. (1 mark)
Check: $\overset{x}{ˉ} = 5$ ; $\sum (x - 3)^{2} = 1 + 1 + 9 + 25 = 36$ ; $\sum (x - 5)^{2} = 9 + 1 + 1 + 9 = 20$ ; $n (\overset{x}{ˉ} - a)^{2} = 4 \times 4 = 16$ ; $20 + 16 = 36$ . (1 mark)

Answer: Proved; least at $a = \overset{x}{ˉ}$ ; check $36 = 20 + 16$

Where students lose marks in Variance and Standard Deviation: Methods and Properties

The slips examiners see most often, and what to write instead.

Mistake: Forgetting the square of the mean
σ² = Σx²/N − x̄² has two parts.
Wrong working
$σ^{2} = \frac{1180}{20} = 59$ .
Right working
$59 - 7. 3^{2} = 5.71$ .
Do this: Write "mean of squares − square of mean" as a line.
Usually costs: The answer.
Mistake: Omitting h in the shortcut method
The coded variance must be scaled back.
Wrong working
$σ^{2} = 1.2275$ for coded values.
Right working
$σ^{2} = h^{2} \times 1.2275 = 122.75$ .
Do this: Write h² in front of the bracket.
Usually costs: The answer.
Mistake: Reporting variance as the SD
The SD is the square root of the variance.
Wrong working
SD $= 11.6$ .
Right working
SD $= 11.6 \approx 3.41$ .
Do this: Underline the quantity asked.
Usually costs: 1 mark.
Mistake: Updating only the mean after a correction
Σx² also changes.
Wrong working
New SD kept at 4.
Right working
Recompute $\sum x^{2}$ with the corrected value.
Do this: Adjust both totals.
Usually costs: The answer.

Exam tips

Tabulate $x$ , $f$ , $f x$ , $f x^{2}$ and add each column before substituting.
Keep the mean unrounded until the end; rounding the mean early spoils the variance.
For the shortcut method write down $A$ , $h$ and the coded column $y$ .
In replacement problems recompute both $\sum x$ and $\sum x^{2}$ .
State the standard deviation with units and sensible decimals.

Practice questions on Variance and Standard Deviation: Methods and Properties

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The variance of $1, 2, 3, 4, 5$ is:
(a) 2
(b) 3
(c) $2$
(d) 10
Hint
Mean 3; squared deviations $4, 1, 0, 1, 4$ .
Answer
(a). $\frac{10}{5} = 2$ .
Q2 · 1 mark · MCQHarder board level
If the standard deviation of a data set is 5, then after multiplying every observation by $- 3$ and adding 4, the standard deviation is:
(a) 5
(b) 15
(c) $- 15$
(d) 11
Hint
Use $∣ p ∣ σ$ .
Answer
(b). $∣ - 3∣ \times 5 = 15$ .
Q3 · 3 marks · Short answerHarder board level
Find the mean and standard deviation of the first eight odd natural numbers $1, 3, 5, \dots, 15$ .
Hint
Use the symmetry about the mean.
Answer
$\overset{x}{ˉ} = \frac{64}{8} = 8$ . Deviations $- 7, - 5, - 3, - 1, 1, 3, 5, 7$ ; squares sum to $2 (49 + 25 + 9 + 1) = 168$ . $σ^{2} = \frac{168}{8} = 21$ , $σ = 21 \approx 4.58$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Adding 7 to every observation does not change the standard deviation.
Reason (R): The standard deviation measures spread, and a common shift leaves all deviations from the mean unchanged.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
The mean also increases by 7.
Answer
(a). Each $x_{i} - \overset{x}{ˉ}$ stays the same.
Q5 · 4 marks · Short answerHOTS / competency
Group A has 10 observations with mean 20 and standard deviation 3; group B has 15 observations with mean 30 and standard deviation 4. Find the mean and standard deviation of the combined 25 observations.
Hint
Use $\sum x^{2} = n (s^{2} + m^{2})$ for each group.
Answer
Combined mean $= \frac{200 + 450}{25} = 26$ . $\sum x^{2} = 10 (9 + 400) + 15 (16 + 900) = 4090 + 13740 = 17830$ . $σ^{2} = \frac{17830}{25} - 2 6^{2} = 713.2 - 676 = 37.2$ ; $σ \approx 6.10$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the variance formula that uses the mean of squares?Show answer
Σ f x²/N − x̄².
2.How does the SD change if every observation is multiplied by k?Show answer
It is multiplied by |k|.
3.What does adding a constant do to the SD?Show answer
Nothing; it stays the same.
4.How do you get Σx² from N, mean and SD?Show answer
Σx² = N(σ² + x̄²).
5.When is the SD zero?Show answer
When all observations are equal.

Variance and Standard Deviation: Methods and Properties: frequently asked questions

Why is the variance divided by N and not N − 1?

In this course the data are treated as a whole population, so N is used. N − 1 appears in later work on samples.

Why do we square the deviations?

Squaring removes signs, avoids cancelling, and gives a quantity with convenient algebra.

What is the coefficient of variation?

It compares the spread of two data sets by dividing the SD by the mean. The chapter page notes that this analysis is removed from the 2026-27 syllabus.

Which is bigger, MD or SD?

For the same data, the standard deviation is never smaller than the mean deviation about the mean.