How to Calculate Coefficient of Variation (Step by Step)

The coefficient of variation (CV) is a standardised measure of dispersion that expresses variability as a percentage of the mean. Unlike standard deviation, which is tied to the units of measurement, the CV is dimensionless. This makes it invaluable when you need to compare the spread of datasets that use different units or have very different means.

In this guide you will learn the coefficient of variation formula, the difference between population and sample CV, when to use CV instead of standard deviation, and how to interpret your results. We include three fully worked examples so you can follow every step of the calculation.

What Is the Coefficient of Variation?

The coefficient of variation is the ratio of the standard deviation to the mean, multiplied by 100 to give a percentage. It answers a simple question: how large is the spread relative to the average?

A dataset with a CV of 10% has relatively low variability compared to its mean, while a CV of 80% indicates that the data points are widely scattered. Because the result is a percentage, you can directly compare the variability of two datasets even if one measures height in centimetres and the other measures weight in kilograms.

The Coefficient of Variation Formula

Population CV

When you have data for an entire population, the coefficient of variation is calculated using the population standard deviation and the population mean :

Sample CV

When working with a sample drawn from a larger population (the more common scenario), use the sample standard deviation and sample mean :

Where is calculated with Bessel's correction (dividing by rather than ).

When to Use CV Instead of Standard Deviation

Standard deviation is an excellent measure of spread, but it has a limitation: it is expressed in the same units as the data. This creates problems in two situations.

• Comparing datasets with different units. If you want to know whether heights vary more than weights in a group of people, comparing a standard deviation of 8 cm to one of 12 kg is meaningless. The CV converts both to percentages so you can compare them directly.
• Comparing datasets with very different means. A standard deviation of 5 is large if the mean is 10 but trivial if the mean is 10,000. The CV accounts for this by normalising against the mean.

Use standard deviation when your datasets share the same units and have similar means. Use the coefficient of variation when units differ, means differ substantially, or you need a single dimensionless number to describe relative variability.

Step-by-Step Calculation

Calculating the CV requires three steps: find the mean, find the standard deviation, then divide and multiply by 100.

1. Calculate the mean ( for a sample or for a population).
2. Calculate the standard deviation ( or ).
3. Divide the standard deviation by the mean and multiply by 100.

Worked Example 1: Test Scores

A teacher records the scores of 6 students on a maths test (out of 50):

Step 1: Find the Mean

Step 2: Find the Sample Standard Deviation

Calculate each squared deviation from the mean:

Step 3: Calculate the CV

The coefficient of variation is approximately 16.1%. This tells us the typical deviation from the mean is about 16% of the mean score itself, indicating moderate variability in student performance.

Worked Example 2: Comparing Heights and Weights

This is where the CV truly shines. Suppose you measure both the heights and weights of 5 athletes and want to know which measurement varies more within the group.

Heights (cm): 170, 175, 168, 182, 177

Computing the sample standard deviation (details abbreviated for clarity):

Weights (kg): 68, 82, 71, 95, 77

Interpretation

The CV for heights is 3.21% while the CV for weights is 13.54%. Even though we cannot directly compare centimetres with kilograms, the CV tells us that weights vary about four times more (relative to their mean) than heights do in this group of athletes. This kind of cross-unit comparison is impossible with standard deviation alone.

Worked Example 3: Manufacturing Quality Control

A factory produces two types of bolts. Quality control measures the diameter of a sample from each production line.

Line A: Small bolts (mm): 5.02, 4.98, 5.01, 4.99, 5.03, 5.00, 4.97, 5.01

Line B: Large bolts (mm): 25.12, 24.85, 25.30, 24.92, 25.08, 25.21, 24.78, 25.15

Interpretation

Line A has a CV of 0.40% while Line B has a CV of 0.72%. Although Line B has a much larger absolute standard deviation (0.18 mm vs 0.02 mm), the CV reveals that Line B is also proportionally less consistent. If both lines need the same relative precision, Line B may need attention. Without the CV, you might mistakenly assume Line B is fine simply because a 0.18 mm deviation sounds small.

Interpreting CV Values

There are no universal thresholds for what counts as a "good" or "bad" CV, as this depends entirely on the context. However, some general guidelines are useful.

• CV below 10%: Low variability. The data points are tightly clustered around the mean. Common in well-controlled manufacturing processes and laboratory measurements.
• CV between 10% and 30%: Moderate variability. Typical of many biological and social science measurements.
• CV above 30%: High variability. The data is widely dispersed. This may be expected (e.g. income distributions) or may signal a problem (e.g. inconsistent manufacturing).

In fields like analytical chemistry, a CV above 5% for repeated measurements of the same sample may indicate poor method precision. In ecology, a CV of 50% or more for population counts is perfectly normal. Always interpret the CV within the context of your discipline.

Limitations of the Coefficient of Variation

The CV has some important limitations to keep in mind.

• The mean must not be zero or near zero. Since the CV divides by the mean, a mean of zero makes the formula undefined. A mean very close to zero can produce an artificially inflated CV.
• Negative means cause problems. If the mean is negative (for example, with temperature data in Celsius), the CV can be negative or misleading. The CV works best with ratio-scale data that has a true zero point.
• Sensitive to outliers. Both the mean and standard deviation are affected by extreme values, so the CV inherits this sensitivity.
• Not suitable for all distributions. For heavily skewed data, the mean and standard deviation may not adequately summarise the distribution, making the CV less informative.

CV in Different Fields

The coefficient of variation appears across many disciplines under slightly different names.

• Finance: CV is used to compare the risk (volatility) of investments relative to their expected return. A lower CV indicates a better risk-return tradeoff.
• Laboratory science: Repeated measurements of the same sample should produce a low CV. This is often reported as part of method validation.
• Engineering: In quality control, the CV helps compare process consistency across products of different sizes, as we saw in the bolt example above.
• Biology and ecology: The CV is used to describe natural variation in population sizes, body measurements, and other biological quantities.

Frequently Asked Questions

What is a good coefficient of variation?

There is no single answer. In laboratory settings, a CV below 5% is often considered good precision. In social sciences, a CV of 15 to 25% is common and acceptable. In finance, investors compare CVs across assets rather than judging a single CV in isolation. Always interpret the CV relative to the norms of your field and the nature of your data.

Can the coefficient of variation be greater than 100%?

Yes. A CV exceeding 100% means the standard deviation is larger than the mean, indicating very high variability. This can occur with right-skewed distributions such as income data or insurance claims. It is not inherently wrong, but it does suggest you should check whether the mean and standard deviation are appropriate summaries for your data.

Why not just use standard deviation?

Standard deviation is an absolute measure of spread, reported in the same units as the data. It cannot tell you whether a spread of 10 is large or small without knowing the mean. The CV normalises by the mean, giving a relative measure. This is essential when comparing datasets with different units or vastly different magnitudes.

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

They are the same thing. Relative standard deviation (RSD) is simply another name for the coefficient of variation. Some fields, particularly chemistry and pharmacology, prefer the term RSD. The formula is identical: .

Can I calculate CV for a population and a sample?

Yes. Use the population standard deviation and population mean for a population CV, or the sample standard deviation and sample mean for a sample CV. The choice between them follows the same logic as choosing between population and sample standard deviation.