EconomicsISC Class 11 · Measures of Dispersion

Standard Deviation, Variance and Coefficient of Variation (ISC Class 11 Economics)

Updated 17 October 2026 · Fact-checked

The standard deviation (SD, sigma) is the square root of the mean of the squared deviations from the arithmetic mean: sigma = sqrt[Sum d squared / n] = sqrt[Sum x squared / n - (mean) squared]. Variance = sigma squared. The coefficient of variation CV = sigma / mean x 100 is a relative measure; the series with the smaller CV is more consistent (less variable).

Understand Standard Deviation, Variance and Coefficient of Variation

Standard deviation. The most widely used measure of dispersion; it uses all items and is the base of advanced statistics. It is measured in the units of the data.

Individual series.

Deviation method: sigma = sqrt[Sum (X - mean)^2 / n].
Direct method: sigma = sqrt[Sum X^2 / n - (Sum X / n)^2].
Assumed mean: sigma = sqrt[Sum d^2 / n - (Sum d / n)^2], d = X - A.

Discrete series. sigma = sqrt[Sum f X^2 / N - (Sum f X / N)^2].

Continuous series (step deviation). d' = (m - A) / h: sigma = h x sqrt[Sum f d'^2 / N - (Sum f d' / N)^2].

Variance = sigma squared.

Properties. Unaffected by a change of origin (adding a constant). Multiplied by k when every value is multiplied by k (k positive). Always non-negative. Gives more weight to extreme items than MD does.

Coefficient of variation. CV = (sigma / mean) x 100. It is a relative measure: use it to compare variability or consistency of series with different units or means. Smaller CV: more consistent (or more stable).

Use. SD summarises spread; CV compares two series (stability of prices, consistency of players, uniformity of production).

Standard Deviation, Variance and Coefficient of Variation: key facts

Standard deviation (individual): sigma = sqrt[Sum X^2 / n - (Sum X / n)^2] | Variance = sigma^2; Direct method.
Continuous (step deviation): sigma = h x sqrt[Sum f d'^2 / N - (Sum f d' / N)^2] | d' = (m - A) / h; Short-cut.
Coefficient of variation: CV = sigma / mean x 100 | Smaller CV = more consistent; Per cent.

How to answer Standard Deviation, Variance and Coefficient of Variation questions

Set out columns for deviations and squares.

1Find the mean (or choose an assumed mean).
2Compute d and d squared (or fd and fd squared).
3Substitute into the formula and take the square root.
4Find CV when two series are to be compared.

Standard Deviation, Variance and Coefficient of Variation in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Consistency

Suppose cricketer A has a mean score of 50 with SD 10, and cricketer B has a mean of 80 with SD 12.

Who is more consistent?

Try it first, then show the solutionSolution

CV of A = 10 / 50 x 100 = 20%.
CV of B = 12 / 80 x 100 = 15%.

Answer: B is more consistent (smaller CV).

Memory hook: Lower CV, steadier performer.

Tricks to remember Standard Deviation, Variance and Coefficient of Variation

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

Root of mean of squares minus square of mean
sigma = sqrt(mean of X squared - mean squared).
Use it when: Direct method.
Origin free, scale bound
Adding a number leaves SD; multiplying scales it.
Use it when: Properties.
CV: lower is steadier
Smaller CV means more consistent.
Use it when: Comparison.

Standard Deviation, Variance and Coefficient of Variation: solved examples

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

Example 1: Variance · 1 mark · MCQBoard level

If the standard deviation of a series is 6, its variance is:
(a) 6 (b) 12 (c) 36 (d) 3

Try it first, then show the solutionSolution

Variance = sigma squared = 36.

Answer: (c) 36

Example 2: SD of an individual series · 4 marks · Short answerBoard level

Find the standard deviation and the coefficient of variation of 4, 8, 10, 12, 16. (4 marks)

Try it first, then show the solutionSolution

Mean = 10; deviations -6, -2, 0, 2, 6; squares 36, 4, 0, 4, 36; Sum = 80.
sigma = sqrt(80/5) = sqrt(16) = 4.
CV = 4 / 10 x 100 = 40%.
Check by the direct method: Sum X squared = 580; 580/5 - 100 = 16.

Answer: sigma = 4; CV = 40%.

Example 3: SD of a continuous series · 6 marks · Long answerHarder board level

For classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 4, 8, 15, 9, 4 (mean 25.25), find the standard deviation and the coefficient of variation by the step deviation method with A = 25, h = 10. (6 marks)

Try it first, then show the solutionSolution

d' = -2, -1, 0, 1, 2; fd' = -8, -8, 0, 9, 8; Sum fd' = 1.
fd'^2 = 16, 8, 0, 9, 16; Sum = 49; N = 40.
sigma = 10 x sqrt(49/40 - (1/40)^2) = 10 x sqrt(1.225 - 0.000625) = 10 x 1.1065 = 11.07.
CV = 11.07 / 25.25 x 100 = 43.8%.

Answer: sigma = 11.07; CV = 43.8%.

Example 4: Discrete series and consistency · 5 marks · Long answerHarder board level

Find the SD of x = 2, 4, 6, 8, 10 with f = 3, 5, 8, 6, 3 (mean 6.08) and its CV. A second series has mean 8 and SD 2.2; which is more consistent? (5 marks)

Try it first, then show the solutionSolution

Sum fx squared = 12 + 80 + 288 + 384 + 300 = 1,064; N = 25.
Variance = 1,064/25 - 6.08^2 = 42.56 - 36.9664 = 5.5936; sigma = 2.365.
CV = 2.365 / 6.08 x 100 = 38.9%.
Second series: CV = 2.2 / 8 x 100 = 27.5%.
The second series has the smaller CV, so it is more consistent.

Answer: sigma = 2.365; CV = 38.9%; the second series (27.5%) is more consistent.

Example 5: Correcting a mistake · 6 marks · Long answerHOTS / competency

For 10 observations the mean is 20 and the SD is 4. It is found that one observation 12 was wrongly copied as 22. Find the correct mean and SD. If every observation is then doubled and 5 added, find the new mean and SD. (6 marks)

Try it first, then show the solutionSolution

Wrong Sum X = 200; correct Sum X = 190; correct mean = 19.
Wrong Sum X squared = 10 x (16 + 400) = 4,160; correct = 4,160 - 484 + 144 = 3,820.
Variance = 3,820/10 - 19^2 = 382 - 361 = 21; SD = sqrt(21) = 4.58.
New mean = 2 x 19 + 5 = 43; new SD = 2 x 4.58 = 9.17 (the added 5 does not change the SD).

Answer: Correct mean 19, SD 4.58; new mean 43, new SD 9.17.

Where students lose marks in Standard Deviation, Variance and Coefficient of Variation

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

Mistake: Forgetting to take the square root
Stopping at variance.
Wrong working
sigma = 16
Right working
sigma = 4
Do this: Take the root.
Usually costs: 2 marks.
Mistake: Forgetting h in the step deviation formula
Missing factor.
Wrong working
sigma = 1.1065
Right working
sigma = 11.07
Do this: Multiply by h.
Usually costs: 2 marks.
Mistake: Reading a larger SD as more consistent
Reversed meaning.
Wrong working
Larger CV = steadier
Right working
Smaller CV = steadier
Do this: Lower variability means consistency.
Usually costs: 1-2 marks.

Exam tips

Set out the table clearly and total each column.
Check the direct method against the deviation method.
State which series is more consistent in words.

Practice questions on Standard Deviation, Variance and Coefficient of Variation

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 coefficient of variation is:
(a) sigma / mean x 100 (b) mean / sigma (c) sigma squared (d) range / mean
Hint
A relative measure.
Answer
(a) sigma / mean x 100
Q2 · 2 marks · Short answerBoard level
State two properties of the standard deviation. (2 marks)
Hint
Origin and scale.
Answer
It is unaffected by a change of origin, and it is multiplied by k when each value is multiplied by k (k positive).
Q3 · 4 marks · Short answerHarder board level
Find the SD and the CV of 5, 7, 8, 10, 15. (4 marks)
Hint
Mean = 9.
Answer
Mean = 9; deviations -4, -2, -1, 1, 6; squares 16, 4, 1, 1, 36 = 58; variance 11.6; sigma = 3.406; CV = 3.406 / 9 x 100 = 37.8%.
Q4 · 4 marks · Short answerHarder board level
Series A has mean 120 and SD 18; series B has mean 75 and SD 12. Find the CV of each, say which is more consistent, and find the SD of B if all values are tripled and 10 is added. (4 marks)
Hint
15% and 16%.
Answer
CV(A) = 18 / 120 x 100 = 15%; CV(B) = 12 / 75 x 100 = 16%. A is more consistent. New SD of B = 3 x 12 = 36 (the addition of 10 has no effect).
Q5 · 4 marks · Short answerHOTS / competency
Two plants make bulbs. Plant X: mean life 1,200 hours, SD 96. Plant Y: mean life 900 hours, SD 63. A buyer prefers longer life but also uniform quality. Compare using CV and recommend. (4 marks)
Hint
8% and 7%.
Answer
CV(X) = 96 / 1,200 x 100 = 8%; CV(Y) = 63 / 900 x 100 = 7%. Y is slightly more uniform, but X has a longer life. If uniformity matters most choose Y; otherwise X gives longer life with only a slightly higher variability.

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.SD formula (direct)?Show answer
sqrt(Sum X squared / n - mean squared).
2.Variance?Show answer
SD squared.
3.CV formula?Show answer
sigma / mean x 100.
4.SD unaffected by?Show answer
Change of origin.
5.Smaller CV means?Show answer
More consistent.

Standard Deviation, Variance and Coefficient of Variation: frequently asked questions

Why square deviations?

To remove signs and stress large deviations.

Can SD be negative?

No.

Is SD in same units as data?

Yes.

What is variance for?

Further statistical analysis.