MathsISC Class 11 · Statistics

Variance and Standard Deviation: Direct, Assumed Mean and Step-Deviation Methods

Updated 17 October 2026 · Fact-checked

The variance is σ² = Σfᵢ(xᵢ − x̄)²/N = Σfx²/N − (Σfx/N)², and the standard deviation is σ = √σ². The short-cut and step-deviation methods use a convenient assumed mean A and class width h. If each observation is multiplied by k and increased by c, the mean becomes k·x̄ + c and the SD becomes |k|σ. The coefficient of variation is (σ/x̄) × 100.

Both sets have the same mean, but the longer orange deviation arrows in the lower set give a larger σ, so the data is less consistent.

Illustration drawn for QuestionBuddy.

Understand Variance and Standard Deviation

Definition. The variance is the mean of the squared deviations from the mean: $σ^{2} = \frac{\sum f _{i} ( x _{i} - x ˉ ) ^{2}}{N}$ and the standard deviation $σ = σ^{2}$ . It has the same unit as the data. An equivalent form: $σ^{2} = \frac{\sum f _{i} x _{i}^{2}}{N} - \overset{x}{ˉ}^{2}$ .

Short-cut (assumed mean) method. With $d_{i} = x_{i} - A$ : $σ^{2} = \frac{\sum f _{i} d _{i}^{2}}{N} - (\frac{\sum f _{i} d _{i}}{N})^{2}$ and $\overset{x}{ˉ} = A + \frac{\sum f _{i} d _{i}}{N}$ .

Step-deviation method. With $u_{i} = \frac{x _{i} - A}{h}$ : $σ^{2} = h^{2} [\frac{\sum f _{i} u _{i}^{2}}{N} - (\frac{\sum f _{i} u _{i}}{N})^{2}]$ and $\overset{x}{ˉ} = A + h \frac{\sum f _{i} u _{i}}{N}$ . Convenient for class intervals of equal width.

Effect of change. Adding a constant c to every value changes the mean by c and leaves σ unchanged. Multiplying every value by k multiplies the mean by k and σ by |k| (variance by k²).

Coefficient of variation. $C V = \frac{σ}{x ˉ} \times 100$ ; the series with the smaller CV is more consistent.

Combined group. For groups with $n_{1}, \overset{x}{ˉ}_{1}, σ_{1}$ and $n_{2}, \overset{x}{ˉ}_{2}, σ_{2}$ : combined mean $\overset{x}{ˉ} = \frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}$ and $σ^{2} = \frac{n _{1} ( σ _{1}^{2} + d _{1}^{2} ) + n _{2} ( σ _{2}^{2} + d _{2}^{2} )}{n _{1} + n _{2}}$ with $d_{i} = \overset{x}{ˉ}_{i} - \overset{x}{ˉ}$ .

Variance and Standard Deviation: formulas to remember

Variance and SD: $σ^2 = \frac{\sum f(x - \bar{x})^2}{N} = \frac{\sum fx^2}{N} - \bar{x}^2$ | $σ = \sqrt{σ^2}$; Same unit as the data.
Step deviation: $σ^2 = h^2[\frac{\sum fu^2}{N} - (\frac{\sum fu}{N})^2]$; u = (x − A)/h.
Linear change: $Mean: $k\bar{x} + c$ | SD: $|k|σ$$; Adding c does not change SD.
Coefficient of variation: $CV = \frac{σ}{\bar{x}} × 100$; Smaller CV: more consistent.

Variance and Standard Deviation 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.

Consistency of two cricketers

Suppose player A scores 40, 45, 50, 55, 60 and player B scores 30, 40, 50, 60, 70 in five matches.

Find the mean, SD and coefficient of variation for each, and decide who is more consistent.

Try it first, then show the solutionSolution

A: mean 50; deviations −10, −5, 0, 5, 10; squares sum $= 250$ ; $σ^{2} = 50$ , $σ = 7.07$ ; CV $= 14.1$ .
B: mean 50; deviations −20, −10, 0, 10, 20; squares sum $= 1000$ ; $σ^{2} = 200$ , $σ = 14.14$ ; CV $= 28.3$ .
Same mean but A has the smaller CV, so A is more consistent.

Answer: A: σ = 7.07 (CV 14.1%); B: σ = 14.14 (CV 28.3%); A is more consistent.

Memory hook: Equal means: compare the SDs directly.

Tricks to remember Variance and Standard Deviation

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

Mean of squares minus square of mean
σ² = Σx²/n − x̄².
Use it when: Quick computation.
Shift does not change spread
Adding the same number to all values leaves σ unchanged.
Use it when: Change of origin.

Variance and Standard Deviation: solved examples

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

Example 1: SD and variance · 1 mark · MCQBoard level

A data set has standard deviation 7. Its variance is:
(a) 3.5
(b) 14
(c) 49
(d) 343

Try it first, then show the solutionSolution

Variance $= σ^{2} = 7^{2} = 49$ .

Answer: (c) 49

Example 2: Variance by the direct method · 3 marks · Short answerBoard level

Find the variance and standard deviation of 6, 7, 10, 12, 13, 4, 8, 12.

Try it first, then show the solutionSolution

Mean $= 9$ . (0.5 mark)
Squared deviations: 9, 4, 1, 9, 16, 25, 1, 9; sum $= 74$ . (1.5 marks)
$σ^{2} = \frac{74}{8} = 9.25$ ; $σ = 3.04$ . (1 mark)

Answer: Variance 9.25; SD 3.04.

Example 3: Step-deviation method · 4 marks · Short answerHarder board level

Find the mean and the standard deviation for the classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 3, 7, 12, 5, 3, using A = 25 and h = 10.

Try it first, then show the solutionSolution

$u = - 2, - 1, 0, 1, 2$ ; $f u = - 6, - 7, 0, 5, 6$ , $\sum f u = - 2$ ; $f u^{2} = 12, 7, 0, 5, 12$ , $\sum f u^{2} = 36$ ; $N = 30$ . (1.5 marks)
Mean $= 25 + 10 \times (- \frac{2}{30}) = 24.33$ . (1 mark)
$σ^{2} = 100 [\frac{36}{30} - (\frac{2}{30})^{2}] = 100 [1.2 - 0.00444] = 119.56$ . (1 mark)
$σ = 10.93$ . (0.5 mark)

Answer: Mean 24.33; SD 10.93.

Example 4: Effect of a linear change · 4 marks · Short answerHarder board level

The mean and SD of a set of marks are 40 and 4. Each mark is multiplied by 3 and then 5 is added. Find the new mean, variance and SD. The marks are then scaled so that the new mean is 100: what is the new CV?

Try it first, then show the solutionSolution

New mean $= 3 \times 40 + 5 = 125$ . (1 mark)
New SD $= 3 \times 4 = 12$ (adding 5 does not change it); variance $= 144$ . (1.5 marks)
CV $= \frac{12}{125} \times 100 = 9.6$ . (1 mark)
CV does not change when all values are multiplied by a constant: rescaling to mean 100 leaves it 9.6%. (0.5 mark)

Answer: Mean 125; SD 12; variance 144; CV 9.6%.

Example 5: Combined mean and SD · 5 marks · Long answerHOTS / competency

Group A has 50 students with mean 60 and SD 4; group B has 100 students with mean 80 and SD 5. Find the mean and standard deviation of the combined group.

Try it first, then show the solutionSolution

Combined mean $= \frac{50 \times 60 + 100 \times 80}{150} = \frac{11000}{150} = 73.33$ . (1.5 marks)
$d_{1} = 60 - 73.33 = - 13.33$ ( $d_{1}^{2} = 177.78$ ); $d_{2} = 80 - 73.33 = 6.67$ ( $d_{2}^{2} = 44.44$ ). (1.5 marks)
$σ^{2} = \frac{50 ( 16 + 177.78 ) + 100 ( 25 + 44.44 )}{150} = \frac{9688.9 + 6944.4}{150} = 110.89$ . (1.5 marks)
$σ = 10.53$ . (0.5 mark)

Answer: Mean 73.33; SD 10.53.

Where students lose marks in Variance and Standard Deviation

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

Mistake: Dividing by n − 1
Using the sample variance formula.
Do this: Here divide by N (population formula).
Usually costs: 1 mark.
Mistake: Square root forgotten
Giving the variance as the SD.
Do this: SD = √variance.
Usually costs: 1 to 2 marks.
Mistake: Not multiplying by h²
Forgetting the class width in the step-deviation method.
Do this: σ² = h²[...].
Usually costs: 2 marks.

Exam tips

Tabulate d, fd, fd².
Show the mean, then the variance, then the SD.
Use the CV for consistency questions.
Remember that a shift does not change the SD but a scale does.

Practice questions on Variance and Standard Deviation

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
Adding a constant 10 to every observation changes the standard deviation by:
(a) +10
(b) ×10
(c) no change
(d) +100
Hint
Shift of origin.
Answer
(c) no change
Q2 · 2 marks · Short answerBoard level
Find the SD of 2, 4, 6, 8.
Hint
Mean 5.
Answer
Squared deviations 9, 1, 1, 9 sum 20; variance 5; SD $5 \approx 2.24$ .
Q3 · 3 marks · Short answerHarder board level
Find the variance of the first 5 natural numbers.
Hint
Mean 3.
Answer
Deviations −2, −1, 0, 1, 2; squares sum 10; $σ^{2} = 2$ .
Q4 · 3 marks · Short answerHarder board level
Five observations have mean 6 and variance 8. Three of them are 2, 6 and 10. Find the other two.
Hint
Use the sum and the sum of squares.
Answer
Sum $= 30$ , so the other two add to $30 - 18 = 12$ . $\sum x^{2} = 5 (8 + 36) = 220$ ; known squares $= 4 + 36 + 100 = 140$ , so the other two have squares summing to 80. With $a + b = 12$ and $a^{2} + b^{2} = 80$ : $ab = 32$ , so the numbers are 4 and 8.
Q5 · 3 marks · Short answerHarder board level
Two series have means 120 and 150 and SDs 12 and 18. Which is more consistent?
Hint
Compare CVs.
Answer
CVs: $\frac{12}{120} \times 100 = 10$ and $\frac{18}{150} \times 100 = 12$ . The first is more consistent.
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a factory tests the lifetimes (in hundreds of hours) of 5 bulbs: 10, 12, 14, 16, 18.
(i) Find the mean. (1 mark)
(ii) Find the variance. (1 mark)
(iii) Find the SD. (1 mark)
(iv) Find the CV. (1 mark)
Hint
Mean is 14.
Answer
(i) 14. (ii) Squared deviations 16, 4, 0, 4, 16 = 40; variance $8$ . (iii) $σ = 2.83$ . (iv) $C V = \frac{2.83}{14} \times 100 = 20.2$ .

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.Variance formula?Show answer
Σf x²/N − x̄².
2.SD unit?Show answer
Same as the data.
3.Effect of multiplying by k?Show answer
SD × |k|.
4.More consistent series?Show answer
Smaller CV.

Variance and Standard Deviation: frequently asked questions

Why square the deviations?

Squaring removes the signs and gives extra weight to larger deviations.

What does a large SD mean?

The data are widely spread about the mean.

Is the SD ever negative?

No, it is the positive square root.