Coefficient of Variation (CV)
The coefficient of variation is standard deviation divided by the mean. How to calculate it, compare spread across scales, and when not to use it.
Coefficient of Variation (CV)
You need to compare the spread of two datasets that use different units or have very different averages. The coefficient of variation solves this by expressing standard deviation as a percentage of the mean, giving you a unitless measure of relative variability. Everitt & Skrondal's Cambridge Dictionary of Statistics defines it as the standard deviation divided by the mean, and the NIST e-Handbook of Statistical Methods calls it the relative standard deviation (RSD).
CV Formula: CV = s / x̄ (× 100%)
The coefficient of variation formula is simple: for a sample, CV = s / x̄ (× 100%); for a population, CV = σ / μ (× 100%). The NIST e-Handbook gives both forms. Multiplying by 100% expresses the result as a percentage. A CV of 15% means the standard deviation is 15% of the mean. The key is that the coefficient of variation is a unitless ratio, so you can compare it across datasets with different units or scales.
How to Calculate Coefficient of Variation: A Worked Example
You have two datasets. Dataset A: monthly sales in dollars: amounts around $12,000. Dataset B: customer satisfaction scores (out of 10): 7, 8, 7, 9, 8. Compare their relative variability.
Step 1: Calculate Mean and Standard Deviation
Dataset A: mean = $12,000; sample standard deviation (s) = $1,581.14 (using n-1). Dataset B: mean = 7.8; s = 0.84.
Step 2: Apply the CV Formula
Dataset A CV = ($1,581.14 / $12,000) × 100% = 13.18%. Dataset B CV = (0.84 / 7.8) × 100% = 10.77%.
Step 3: Interpret
Sales have a higher relative spread (13.18%) than satisfaction scores (10.77%), even though the absolute standard deviations are vastly different. Without the coefficient of variation, you could not compare them directly because one is in dollars and the other is in score units.
| Dataset | Mean | Standard Deviation (s) | CV (%) | Interpretation |
|---|---|---|---|---|
| A: Monthly Sales ($) | 12,000 | 1,581.14 | 13.18 | Higher relative variability |
| B: Satisfaction Scores (0–10) | 7.8 | 0.84 | 10.77 | Lower relative variability |
When the Coefficient of Variation Is Meaningful
The coefficient of variation is only valid for ratio-scale data with a positive mean. Ratio scales have a true zero point, income, weight, length, time, where a value of zero means none of the thing exists. The NIST e-Handbook states that CV is undefined when the mean is zero and unstable for means near zero. Use it when you need to compare variability across datasets with different units or widely different means, as in quality control, lab work, or comparing investment risk profiles. In finance, it helps compare the risk per unit of return across assets with different expected returns.
When the Coefficient of Variation Misleads
Do not use the coefficient of variation on interval-scale data. Interval scales like temperature in Celsius or Fahrenheit have no true zero; 0°C does not mean 'no temperature'. A mean near zero produces a CV that approaches infinity, making it meaningless. The same applies to negative means, dividing by a negative number flips the sign of the CV, which destroys any relative interpretation. The NIST e-Handbook warns that CV is unstable for means near zero. Another failure case: using CV when the mean is small but the data are positive, such as daily rainfall in a desert. A tiny mean and moderate standard deviation yield a very large CV that suggests high variability, but the absolute variation might be trivial.
CV in Lab and Quality Work: Relative Standard Deviation
In laboratory science and quality control, the relative standard deviation (RSD) is the standard term for the coefficient of variation multiplied by 100%. A typical RSD threshold for an analytical method might be 5% or less, if repeated measurements of the same sample yield an RSD above that, the method is considered too imprecise. The NIST e-Handbook uses RSD and CV interchangeably. The RSD is also used in inter-laboratory comparisons, where different labs measure the same material and you assess the relative spread in their results. Unlike variance or variance vs standard deviation, which deal in absolute units, the CV is unitless and lets you compare precision across different analytes, concentrations, or measurement techniques.
Common Questions
What is the coefficient of variation used for?
The coefficient of variation compares the relative variability of datasets that have different units or different means. It is the ratio of standard deviation to mean, expressed as a percentage.
How do I interpret a coefficient of variation of 20%?
A CV of 20% means the standard deviation is 20% of the mean. Lower values indicate less relative spread. Values above 50% may suggest high relative variability, but this depends on the context.
Can I calculate CV for negative values?
No. If the mean is negative, the CV becomes negative, which is meaningless. The coefficient of variation requires a positive mean to be interpretable.
What is the difference between CV and standard deviation?
Standard deviation is an absolute measure of spread in the original units. The coefficient of variation is a unitless relative measure that divides standard deviation by the mean, allowing comparison across different scales.
Is CV the same as relative standard deviation?
Yes. The NIST e-Handbook and Everitt & Skrondal both treat coefficient of variation and relative standard deviation (RSD) as synonyms, with RSD typically expressed as a percentage.
What happens to CV when the mean is zero?
The CV is undefined when the mean is zero because you divide by zero. For means near zero, the CV becomes very large and unstable, making it unreliable.
How do I calculate CV in Excel?
Calculate the mean with AVERAGE() and the sample standard deviation with STDEV.S(). Then divide standard deviation by the mean and multiply by 100: = (STDEV.S(range) / AVERAGE(range)) * 100.