Class 11 Statistics Notes · GSEB

Coefficient of Variation

Coefficient of Variation — compare the consistency of two groups in any unit with the percentage form of standard deviation. GSEB Class 11 Statistics notes with solved comparisons, the means caveat and gauges.

Last updated: 5 Oct 2026

Notes

Coefficient of Variation — the Percentage Spread

Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.

“Uncontrolled variation is the enemy of quality.”— Edward Deming

Coefficient of Variation (CV)

The relative measure of the standard deviation — the standard deviation expressed as a percentage of the mean. The standard deviation says “how big is the spread?”; the CV says “how big is that spread compared with the average it came from?”

Coefficient of variation

CV = \frac{s}{\bar{x}} \times 100

Why Karl Pearson multiplied it by 100

The ratio s/x̄ — the coefficient of standard deviation — often comes out as a fraction (0.2973, 0.4594 …). Karl Pearson suggested the percentage form so that it can be understood easily: multiplying by 100 turns 0.2973 into 29.73%.Its single job is to compare variability across groups that sit at different levels or use different units — 44 runs against 53 runs, 56.2 kg against 62.5 inch, a chai stall's ₹8,000 day against a cloud kitchen's ₹2,00,000 day.

The Consistency Rule

⭐ Lower CV means more consistent

A smaller CV means less dispersion, more stability and more consistency; a larger CV means more dispersion and less stability. The verdict is carried by the percentage alone — never by the raw standard deviation.

Judged on the SD alone

A Wow! Momo franchisee lines up two outlets: outlet X takes ₹8,000 a day with SD ₹3,200, outlet Y takes ₹1,20,000 a day with SD ₹14,400. The smaller rupee swing looks “steadier”, so Y is written off — a decision made on raw spread across two very different scales.

Judged on the CV

X: 3,200/8,000 = 40% · Y: 14,400/1,20,000 = 12%. Y's daily takings wobble least relative to their own size — the CV, not the rupee spread, settles the argument.

Two Groups, One Scale

Two gauges on the same 0–50% rail — emerald on the left is the steady end, rose on the right is the wild end. Underneath each gauge the raw observations sit on their own number line, so the width of the dot cloud and the position of the marker always agree.

One rail, 0–50% — the marker is the CV, the dots underneath are the observations.

Batsman A

29.73%

mean 44 · SD 13.08 runs · CV = 13.08 ÷ 44 × 100

0% steady25%50% wild

x̄ = 440100 runs

Lower CV — the steadier group

Batsman B

45.94%

mean 53 · SD 24.35 runs · CV = 24.35 ÷ 53 × 100

0% steady25%50% wild

x̄ = 530100 runs

Higher CV — the wilder group

29.73% sits left of 45.94% — batsman A is the more consistent scorer.

Reading the rail

The marker shows the CV; the dots show the data. When the cloud spreads wider around its mean the marker slides to the right, and the chip under the card flips from “steadier” to “wilder” — the picture and the percentage never disagree.

When the Means Are Not the Same

Switch the toggle. With both averages pinned to the same point, the CV comparison is a straight verdict; move the averages apart and a caveat appears.

Store P

10%

mean 12 · SD 1.2

Store Q

22%

mean 12 · SD 2.64

Same mean — the CV verdict stands on its own

Both stores average 12 minutes per order, so relative spread is the only thing left to compare: P = 10%, Q = 22%. This is the shape of comparison Zomato's operations teams use before copying one dark store's packing flow into another city — same average, different steadiness, one clear winner.

⭐ The book's Additional Information

The consistency claim holds mainly when the two means are the same or approximately the same. Where the averages differ, the book is careful: a consistent group is the better one only when the means match — “the same cannot be said” once they do not.

Working Backwards from a CV

The formula can be run in reverse. Two rearrangements cover every backwards question:

Standard deviation from a CV

\sigma = \bar{x} \times \frac{CV}{100}

Mean from a CV

\bar{x} = \frac{\sigma}{CV/100}

Mean 10, CV 60%

σ = 10 × 60/100 = 6

variance = σ² = 36

x̄ = 25, CV 20%

σ = 25 × 20/100 = 5

variance = σ² = 25

Exercise 4.5, Q3 — given the coefficient and the standard deviation, the mean falls out directly
Series	CV	SD	Mean = SD ÷ (CV/100)
Series 1	30%	15	15 ÷ 0.30 = 50
Series 2	25%	9	9 ÷ 0.25 = 36

Illustration 19 — Batsman A vs Batsman B

Runs in the last 10 innings
Aspect	Batsman A	Batsman B
Mean (x̄)	44 runs	53 runs
Standard deviation (s)	13.08 runs	24.35 runs
CV = (s/x̄) × 100	29.73%	45.94%
Verdict	Batsman A is more consistent	Higher CV — the scores swing wider

Solved Example

Problem

Runs in the last 10 innings — Batsman A: 25, 50, 45, 30, 70, 42, 36, 48, 34, 60 Batsman B: 10, 70, 50, 20, 95, 55, 42, 60, 48, 80 Decide who is the more consistent batsman.

Solution

A: x̄ = 44, s = 13.08, CV = 29.73% · B: x̄ = 53, s = 24.35, CV = 45.94% → batsman A is more consistent.

Adding a Constant to Every Observation

What moves and what does not when a constant is added

Every observation shifts by the same amount, so every deviation (x − x̄) stays exactly where it was — the standard deviation is unchanged. The mean itself, however, moves: 45 → 45 + 5 = 50.The CV is therefore read off the new mean: 5/50 × 100 = 10% (before the shift it was 5/45 × 100 = 11.11%).

Original set

mean 45 · SD 5

CV = 5/45 × 100 = 11.11%

After adding 5 to each observation

mean 50 · SD 5 (unchanged)

CV = 5/50 × 100 = 10%

Worked Example Gallery

Coefficient of Variation — solved practicals

4 problems

Two groups, same unit — batsmen (Illustration 19)

Full question

From the following data of runs scored by two batsmen A and B in last 10 innings, decide who is more consistent.

Runs by batsman A	25	50	45	30	70	42	36	48	34	60
Runs by batsman B	10	70	50	20	95	55	42	60	48	80

Group	Mean	SD	CV = (s/x̄) × 100
Batsman A	44	13.08	29.73%
Batsman B	53	24.35	45.94%

A 29.73% < B 45.94% → batsman A is more consistent

CV of A = (13.08/44) × 100 = 29.73%.
CV of B = (24.35/53) × 100 = 45.94%.
29.73% < 45.94% → the group with the lower CV is the more consistent one.

Given mean and SD — workers (Illustration 20)

Full question

The following information is available for two workers of a factory :

	Workers A	Workers B
Mean time of completing job (minutes)	30	25
Standard Deviation (minutes)	6	4

Which worker has more relative variation or fluctuation in the time taken to complete the job?

Worker	Mean	SD	CV
A	30 min	6 min	20%
B	25 min	4 min	16%

A 20% vs B 16% → worker B is more stable

CV of A = (6/30) × 100 = 20%.
CV of B = (4/25) × 100 = 16%.
Higher CV means more fluctuation — worker A varies more.

Different units — weights vs heights (Illustration 21)

Full question

The means and standard deviations of heights and weights of 50 students of a class are as follows :

	Weight	Height
Mean	56.2 kg	62.5 inch
Standard Deviation	4.8 kg	9.3 inch

Where do you find more variation, among heights or weights?

Measurement	Mean	SD	CV
Weight	56.2 kg	4.8 kg	8.54%
Height	62.5 inch	9.3 inch	14.88%

Weight 8.54% vs height 14.88% → height varies more

Kilograms and inches cannot be compared directly — use the unit-free CV.
CV of weight = (4.8/56.2) × 100 = 8.54%.
CV of height = (9.3/62.5) × 100 = 14.88%.

Firms — relative variation in wages (Illustration 26, part 2)

Full question

The following results are obtained on the basis of daily wages (in ₹) paid to workers of two firms A and B :

	Firm A	Firm B
No. of workers	20	30
Mean daily wages (₹)	250	400
Standard deviation of wages (₹)	10	12

Answer the following questions using the above information.

2. Determine which firm shows more relative variation in wages paid to its workers.

Firm	Mean wage	SD	CV
A	₹250	₹10	4%
B	₹400	₹12	3%

Firm 4% vs 3% → firm A has more variation

CV of firm A = (10/250) × 100 = 4%.
CV of firm B = (12/400) × 100 = 3%.
4% > 3% → firm A has more variation.

Quick Reference

Idea	Statement
Formula	CV = (s/x̄) × 100 — a unit-free percentage
From a CV	σ = mean × CV/100, and variance = σ²
Reading it	Lower CV = more stable and consistent · higher CV = less stable
Unit-free	A percentage, so groups measured in different units (kg vs inch) can still be ranked
Validity	Compare CVs freely mainly when the means are same or approximately same
Adding a constant	The SD stays 5 while the mean moves to 50 → CV = 10%

Key Takeaways

CV = (s/x̄) × 100 — Karl Pearson's percentage form of s/x̄, offered because the raw ratio usually lands as a fraction. So what: a percentage is the one number two differently sized groups can actually be compared on.
⭐ Lower CV → more stable, less dispersed, more consistent; higher CV → less stable. So what: every "who is more consistent" question in this chapter is settled by this one line.
The consistency claim is honest mainly when the means are the same or approximately the same — with means like 44 and 53 the book itself says "the same cannot be said". So what: quote the two means next to the two CVs whenever they differ.
Backwards working: σ = mean × CV/100 and variance = σ² — mean 10 with CV 60% gives variance 36, and x̄ = 25 with CV 20% gives variance 25.
Adding a constant leaves the deviations (and so the SD) untouched but shifts the mean: 45 → 50, so the CV is recomputed as 5/50 × 100 = 10%.
Anchors worth carrying: batsmen 29.73% vs 45.94% → A more consistent · workers 20% vs 16% → B steadier · weight 8.54% vs height 14.88% → height varies more · firms 4% vs 3% → A more variation. So what: one formula, and it works whether the two groups speak in runs, minutes, kilograms or rupees.