Coefficient of Variation: Formula and Calculation

The coefficient of variation (CV) calculator measures the size of a standard deviation relative to a dataset’s mean. Enter the numbers in your data set to get the CV as a percentage, along with the mean and sample standard deviation.

Formula

For a sample with a positive, nonzero mean, the coefficient of variation is:

CV = (s / x̄) × 100%

• CV is the coefficient of variation, expressed as a percentage.
• s is the sample standard deviation, in the same units as the observations.
• x̄ is the arithmetic mean of the observations, in the same units.

This calculator uses the sample standard deviation, which divides the sum of squared deviations from the mean by n − 1 before taking the square root. The input list has no fixed physical unit, so the mean and standard deviation retain whatever unit your measurements use; their ratio, and therefore the CV, is unitless.

Worked Example

Using the calculator’s default data values, 8, 10, and 12:

1. Find the mean: x̄ = (8 + 10 + 12) / 3 = 10.
2. Find the sample standard deviation: s = √[((8 − 10)² + (10 − 10)² + (12 − 10)²) / (3 − 1)] = √(8 / 2) = 2.
3. Divide the standard deviation by the mean and multiply by 100: CV = (2 / 10) × 100% = 20.00%.

The calculator therefore returns a CV of 20.00%, a mean of 10.00, and a sample standard deviation of 2.00.

What the Coefficient of Variation Tells You

The CV describes variability in relation to the average, rather than in the original measurement units. A CV of 20% means the standard deviation is one-fifth of the mean. It does not mean that every observation lies within 20% of the mean, nor does it describe the full range of the data.

Consider two sets of measurements: one has a mean of 10 and a standard deviation of 2; another has a mean of 100 and a standard deviation of 20. Both have a CV of 20%, even though their values and absolute spreads differ by a factor of ten. This relative comparison can be useful when data sets use the same meaningful scale but have substantially different averages.

Because the calculation divides by the mean, the CV is not appropriate when the mean is zero or close to zero. A mean near zero can make the percentage extremely large or unstable even when the standard deviation is modest.

When to Use It

Use the CV when you want a unitless summary of relative spread and the observations have a meaningful zero. Examples include comparing the relative variation of positive measurements recorded in the same units, or describing how variable repeated measurements are compared with their average.

• Compare CVs only when the data are reasonably comparable and the mean is positive and not near zero.
• Use the standard deviation when the absolute spread in the original units is what matters.
• For strongly skewed data, outliers, or data with a changing spread, inspect a graph and other summaries as well; the mean and standard deviation can be heavily influenced by extreme values.
• For measurements that can be negative, a CV may be misleading because the mean can be negative or its sign can change the interpretation.

The CV is often treated as a ratio-scale statistic: multiplying all observations by a positive conversion factor leaves the CV unchanged. For example, changing a positive length from centimeters to inches changes both the mean and standard deviation by the same factor, so their ratio stays the same. Adding a constant to every observation does not preserve the CV.

Sample Versus Population

This calculator uses the sample standard deviation, with denominator n − 1. That is a common choice when the entered observations are a sample used to estimate variation in a larger population. If the values make up the entire population of interest, the population standard deviation instead divides the squared deviations by n.

The choice affects the CV. The population version is calculated as population standard deviation divided by the population mean, multiplied by 100%. For the same nonempty data and a nonzero mean, the population standard deviation is no greater than the sample standard deviation, so the population CV is no greater than the sample CV. Choose the version that matches the question and state which one you used when reporting results.

Common Mistakes and Limitations

• Dividing by the wrong mean: Use the arithmetic mean of the same observations used to calculate the standard deviation.
• Mixing sample and population formulas: The two standard deviations have different denominators. Make the choice explicit.
• Using a zero or near-zero mean: The formula is undefined at zero and can be misleading close to zero.
• Assuming a high CV proves poor quality: A CV is descriptive, not a pass/fail rule. Whether variation is acceptable depends on the purpose and context.
• Rounding too early: Keep full precision through the calculation and round the final result. The displayed percentage is rounded to two decimal places.
• Comparing unlike data: A similar CV does not make two processes, populations, or measurement scales equivalent.

A CV alone does not show the shape of a distribution, identify outliers, or indicate whether observations are independent. It can also be unsuitable for interval-scale quantities whose zero point is arbitrary, such as temperature expressed in Celsius or Fahrenheit. In those cases, a ratio to the mean does not have a stable, meaningful interpretation.

Frequently Asked Questions

How do you calculate the coefficient of variation?
Divide the standard deviation by the mean, then multiply by 100 to express the result as a percentage. Specify whether you used the sample or population standard deviation.

Can the coefficient of variation be greater than 100%?
Yes. It can exceed 100% when the standard deviation is larger than the positive mean. That describes relative spread; it is not automatically a calculation error.

Can the coefficient of variation be negative?
A standard deviation is nonnegative, but a negative mean makes the signed formula negative. Such a result is generally not useful as a conventional measure of relative variability; check whether CV is appropriate for the data.

What does a coefficient of variation of 20% mean?
It means the standard deviation is 20% of the mean. It does not mean that all, or any particular proportion, of the observations are within 20% of the mean.

Should I use sample or population standard deviation?
Use the sample version when your data are a sample intended to represent a larger population. Use the population version when the observations are the complete population you want to describe.

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list x | Data values | 8, 10, 12 | 0
out cv | Coefficient of variation | % | 100 * sd(x) / mean(x) | 2
out avg | Mean | - | mean(x) | 2
out s | Sample standard deviation | - | sd(x) | 2