Coefficient of Variation: Formula, Steps and Worked Examples

The coefficient of variation (CV) tells you how big the spread of your data is compared with its average, as a percentage. Because it has no units, you can use it to compare things that are measured differently, such as heights in centimetres and weights in kilograms.

The short version

Find the mean of your data.
Find the standard deviation ( $s$ for a sample, $\sigma$ for a whole population).
Divide the standard deviation by the mean and multiply by 100: $CV = \frac{s}{\bar{x}} \times 100\%$ .

The calculator above is loaded with the first worked example below, set to the sample standard deviation. Replace the numbers with your own, or switch to the population version.

Sample or population CV

The CV uses whichever standard deviation fits your data. Dividing by $n - 1$ gives the sample version, which is the usual case when your numbers are a sample from a bigger group. Dividing by $N$ gives the population version, for when you have every value in the group.

\begin{aligned} CV &= \frac{s}{\bar{x}} \times 100\% \quad \text{(sample)} \\ CV &= \frac{\sigma}{\mu} \times 100\% \quad \text{(population)} \end{aligned}

Worked example 1: test scores

A teacher records the marks of 6 students on a test out of 50:

32, \; 45, \; 38, \; 41, \; 29, \; 35

Each dot is one student. The dashed line is the mean (36.667) and the shaded band runs one standard deviation (5.888) either side of it.

Step 1: Find the mean

\begin{aligned} \bar{x} &= \frac{32 + 45 + 38 + 41 + 29 + 35}{6} \\ &= \frac{220}{6} \approx 36.667 \end{aligned}

Step 2: Find the sample standard deviation

Subtract the mean from each mark, square the result and add them up:

$x_i$	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
32	-4.667	21.778
45	8.333	69.444
38	1.333	1.778
41	4.333	18.778
29	-7.667	58.778
35	-1.667	2.778
Sum		173.333

\begin{aligned} s &= \sqrt{\frac{173.333}{6 - 1}} \\ &= \sqrt{34.667} \approx 5.888 \end{aligned}

Step 3: Divide by the mean

\begin{aligned} CV &= \frac{5.888}{36.667} \times 100 \\ &\approx 16.06\% \end{aligned}

The coefficient of variation is about 16.1%: a typical mark sits roughly 16% of the mean away from the mean. If you treated these 6 students as the whole population, the population standard deviation would give a CV of 14.7%.

Worked example 2: heights against weights

The CV earns its place when the units differ. 5 athletes are measured for height (cm) and weight (kg). Which varies more within the group?

\begin{aligned} &\text{Heights (cm)} \\ &170,\, 175,\, 168,\, 182,\, 177 \\ &\text{Weights (kg)} \\ &68,\, 82,\, 71,\, 95,\, 77 \end{aligned}

	Mean	$\sum (x_i - \bar{x})^2$	$s$	CV
Heights (cm)	174.4	125.2	5.595	3.21%
Weights (kg)	78.6	453.2	10.644	13.54%

Standard deviation divided by the mean, as a percentage. You cannot compare centimetres with kilograms, but you can compare these two bars.

The standard deviations (5.59 cm and 10.64 kg) cannot be compared, because the units differ. The CVs can: weights vary about 4.2 times as much as heights, relative to their means.

Worked example 3: two production lines

A factory measures the diameter of 8 bolts from each of two lines. Line A makes small bolts and line B makes large ones.

	Mean (mm)	$s$ (mm)	CV
Line A	5.00125	0.0203	0.41%
Line B	25.05125	0.1827	0.73%

Line B has the larger standard deviation in millimetres, but its bolts are also bigger. The CV puts the two on the same footing.

Line B's standard deviation (0.183 mm) is about 9 times line A's (0.02 mm), yet its CV is only about 1.8 times as large, because its bolts are about 5 times the size. Judged against size, line B is still the less consistent line.

Try it yourself

Switch the calculator to Compare CVs, choose the sample standard deviation and paste in the two lines, or get the standard deviation first with the Standard Deviation Calculator.

When to use the CV instead of standard deviation

The units differ. A standard deviation of 8 cm and one of 12 kg cannot be compared. Their CVs can.
The means differ a lot. A standard deviation of 5 is large when the mean is 10 and small when the mean is 10,000. The CV takes the size of the mean into account.

If your sets share the same units and have similar means, the standard deviation alone is fine.

Reading the result

There are no universal cut-offs for a high or low CV. What counts as steady depends on the field: a laboratory expects far tighter repeat measurements than a survey of household incomes does. Compare your CV with other values for the same kind of data, as the examples above do. A CV above 100% means the standard deviation is larger than the mean.

Limits of the coefficient of variation

A mean of zero has no CV. The formula divides by the mean, and a mean close to zero makes the CV very large and unstable.
Negative means. The calculator divides by the absolute value of the mean so the CV stays positive, but a CV on data that includes negative values, such as temperatures in °C, says little. It suits data measured from a true zero, like lengths, masses and counts.
Outliers. Both the mean and the standard deviation move when one value is extreme, so the CV does too.

Frequently asked questions

What is the difference between CV and relative standard deviation (RSD)?

Essentially none. Relative standard deviation is the coefficient of variation written as a positive percentage, the absolute value of the CV: $RSD = \left| \frac{s}{\bar{x}} \right| \times 100\%$ .

Can the coefficient of variation be more than 100%?

Yes. It means the standard deviation is bigger than the mean. It often happens with right-skewed data such as incomes, and is a sign to check that the mean and standard deviation are the right summaries.

Should I use the sample or the population standard deviation?

Use the sample version (divide by $n - 1$ ) when your data is part of a larger group, and the population version (divide by $N$ ) when you have every value. In the first example the two give 16.1% and 14.7%.

Related calculators and guides

Coefficient of variation calculator: from a dataset, from a mean and standard deviation, or comparing two sets.
Standard deviation calculator and variance calculator.
How to calculate standard deviation and how to find outliers.