Coefficient of Variation Formula - Definition and Examples

#Formula
TL;DR
The coefficient of variation formula divides the standard deviation by the mean, CV=sxˉ\text{CV} = \frac{s}{\bar{x}}. This article gives five worked examples and when the measure breaks down.
BT
Bhanzu TeamLast updated on November 4, 20267 min read

Key Points

  • The coefficient of variation formula is CV=sxˉ\text{CV} = \frac{s}{\bar{x}}, the standard deviation divided by the mean.

  • NIST's Dataplot reference documentation states that the coefficient of variation is the standard deviation divided by the mean.

  • Because both parts carry the same units, they cancel, so the coefficient of variation is a unitless ratio and is often reported as a percentage.

  • It exists to compare spread between data sets measured on different scales or in different units, which a raw standard deviation cannot do.

  • The measure is only meaningful when the mean is positive and the scale has a true zero, so it fails for temperatures in Celsius.

What Is the Coefficient of Variation Formula?

The coefficient of variation formula expresses spread relative to the average, and is written

CV=sxˉ\text{CV} = \frac{s}{\bar{x}}

where ss is the standard deviation and xˉ\bar{x} the mean. NIST's Dataplot reference documentation defines it as the standard deviation divided by the mean. Multiplying by 100 gives the same figure as a percentage, which is the more common reporting convention.

Why Dividing Makes the Units Cancel

A standard deviation of 5 means nothing on its own: if the mean is 10 the data is wildly spread, and if the mean is 10,000 it is remarkably tight. The coefficient of variation supplies that missing context.

The standard deviation of heights in centimetres is itself in centimetres, and so is the mean, so dividing one by the other cancels the units and leaves a bare number. That is the entire purpose: the weights of elephants in kilograms and of mice in grams cannot have their standard deviations compared, but their coefficients of variation can.

The same logic rules out some uses. A scale with an arbitrary zero, like Celsius, gives a mean with no fixed meaning to divide by.

Variable Key

Symbol

Meaning

Unit or type

Notes

CV\text{CV}

coefficient of variation

dimensionless

a ratio, often shown as a percentage

ss

standard deviation

same units as the data

use the sample version for sample data

xˉ\bar{x}

mean

same units as the data

must be non-zero, and positive in practice

×100\times 100

percentage conversion

dimensionless

optional reporting convention

Examples of the Coefficient of Variation Formula

Example 1

A sample has xˉ=50\bar{x} = 50 and s=5s = 5. Find the coefficient of variation.

CV=550=0.1\text{CV} = \frac{5}{50} = 0.1

Final answer: CV=0.1\text{CV} = 0.1, or 10 per cent.

Example 2

Two machines fill bottles. Machine A averages 500 ml with s=10s = 10 ml; machine B averages 50 ml with s=4s = 4 ml. Which is less consistent, judging by the standard deviations?

Machine A has the larger standard deviation, 10 ml against 4 ml, so it looks less consistent. Try that reading.

It is the wrong comparison. A 10 ml wobble on a 500 ml bottle is a small proportional error, while 4 ml on a 50 ml bottle is a large one, so dividing by the means settles it:

CVA=10500=0.02,CVB=450=0.08\text{CV}_A = \frac{10}{500} = 0.02, \qquad \text{CV}_B = \frac{4}{50} = 0.08

Final answer: Machine B is four times less consistent in relative terms, despite its smaller standard deviation.

Example 3

A data set of test scores has xˉ=72\bar{x} = 72 and s=9s = 9. Express the coefficient of variation as a percentage.

CV=972×100=12.5\text{CV} = \frac{9}{72} \times 100 = 12.5

Final answer: 12.512.5 per cent

Example 4

Find the coefficient of variation of 2, 4, 4, 4, 5, 5, 7, 9.

The mean is 408=5\frac{40}{8} = 5. The squared deviations sum to 32, so the sample standard deviation is

s=327≈2.138s = \sqrt{\frac{32}{7}} \approx 2.138 CV=2.1385≈0.428\text{CV} = \frac{2.138}{5} \approx 0.428

Final answer: CV≈0.428\text{CV} \approx 0.428, or about 42.8 per cent.

Example 5

Daily temperatures average 10 degrees Celsius with a standard deviation of 4 degrees. Is a coefficient of variation of 0.4 meaningful?

The arithmetic works, giving 410=0.4\frac{4}{10} = 0.4. But converting the same temperatures to Fahrenheit changes the mean to 50 while the standard deviation becomes 7.2, so

7.250=0.144\frac{7.2}{50} = 0.144

Final answer: The two figures describe identical weather yet differ by nearly a factor of three, so the coefficient of variation is not meaningful on either temperature scale. The measure needs a scale where zero means none of the quantity.

Don't Confuse the CV With the Standard Deviation or the Z-Score

Quantity

Formula

What it tells you

Coefficient of variation

sxˉ\frac{s}{\bar{x}}

spread relative to the mean, unitless

Standard deviation

∑(xi−xˉ)2n−1\sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

spread in the data's own units

Z-score

xi−xˉs\frac{x_i - \bar{x}}{s}

how far one value sits from the mean

Variance

∑(xi−xˉ)2n−1\frac{\sum (x_i - \bar{x})^2}{n-1}

spread in squared units

The CV describes a whole data set, while a Z-score describes a single observation inside it.

Where Solutions Go Off the Rails

Mistake 1: Using it on a scale with an arbitrary zero

Where it slips in: The formula only needs a mean and a standard deviation, both of which exist for any numeric data.

Don't do this: Report a coefficient of variation for temperatures in Celsius, pH, or calendar years.

The correct way: Use it only where zero means the absence of the quantity, such as mass, length, time or counts. If changing the unit would change the answer, the measure does not apply.

Students who ask what zero means on the scale before dividing avoid this entirely, and it is the same check that decides whether any ratio of measurements is meaningful.

Mistake 2: Dividing by a mean close to zero

Where it slips in: The data is centred near zero, perhaps because it records changes rather than amounts.

Don't do this: Compute a CV when the mean is near zero, which inflates the ratio without limit.

The correct way: Report the standard deviation instead. A tiny denominator makes the ratio unstable, so a small change in the mean swings the CV wildly.

Mistake 3: Mixing sample and population standard deviations in a comparison

Where it slips in: One figure comes from a calculator's n−1n-1 key and another from its nn key.

Don't do this: Compare a CV built from ss against one built from σ\sigma.

The correct way: Use the same version throughout. On small samples the two standard deviations differ enough to change which data set looks more variable.

The Five-Bullet Summary

  • The coefficient of variation formula is CV=sxˉ\text{CV} = \frac{s}{\bar{x}}, and multiplying by 100 reports it as a percentage.

  • Dividing spread by mean cancels the units, which is what makes different scales comparable.

  • A larger standard deviation does not mean less consistency once the means differ.

  • The measure requires a true zero and a mean well away from zero, so it fails for Celsius and for change data.

  • To work through statistics with a teacher, explore a high school math tutor or math tuition.

Three Problems to Cement the Coefficient of Variation

Find the CV for xˉ=80\bar{x} = 80, s=12s = 12. Then for two crops averaging 3.2 and 48 tonnes per hectare with standard deviations 0.6 and 7.2, decide which yield is more variable in relative terms.

Want a live Bhanzu trainer to walk through more statistics problems? Book a free demo class.

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Frequently Asked Questions

What is the coefficient of variation formula?
CV=sxˉ\text{CV} = \frac{s}{\bar{x}}, the standard deviation divided by the mean, often multiplied by 100 to give a percentage.
Why use the coefficient of variation formula instead of the standard deviation?
Because a standard deviation cannot be compared across different units or very different scales. Dividing by the mean cancels the units, turning spread into a relative figure.
Can the coefficient of variation be greater than 1?
Yes. Any data set whose standard deviation exceeds its mean gives a CV above 1, or above 100 per cent, which is common for skewed data such as incomes.
When should the coefficient of variation not be used?
When the mean is zero or near zero, or when the measurement scale has an arbitrary zero point rather than a true one.
✍️ Written By
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Bhanzu Team
Content Creator and Editor
Bhanzu’s editorial team, known as Team Bhanzu, is made up of experienced educators, curriculum experts, content strategists, and fact-checkers dedicated to making math simple and engaging for learners worldwide. Every article and resource is carefully researched, thoughtfully structured, and rigorously reviewed to ensure accuracy, clarity, and real-world relevance. We understand that building strong math foundations can raise questions for students and parents alike. That’s why Team Bhanzu focuses on delivering practical insights, concept-driven explanations, and trustworthy guidance-empowering learners to develop confidence, speed, and a lifelong love for mathematics.
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