Coefficient of Variation Formula
Calculate the coefficient of variation: CV = (standard deviation / mean) x 100.
Compare variability between datasets with different units or scales.
The Formula
The coefficient of variation (CV) measures relative variability by expressing the standard deviation as a percentage of the mean. Unlike the standard deviation alone, which is in the same units as the data, the CV is a dimensionless number. This makes it ideal for comparing the spread of datasets that have different units or vastly different means.
For example, comparing the variability of stock returns (measured in percentages) to the variability of house prices (measured in dollars) would be meaningless using standard deviation alone. The CV solves this problem by normalizing the spread relative to the average, giving a fair comparison on the same scale.
A lower CV indicates less variability relative to the mean, while a higher CV indicates more variability. In general, a CV below 15% is considered low variability, 15-30% is moderate, and above 30% is high. However, these thresholds depend heavily on the field of study and the type of data being analyzed.
The CV is widely used in analytical chemistry (to validate the precision of measurement methods), finance (to compare the risk-adjusted variability of different investments), biology (to assess measurement reproducibility), and manufacturing (for quality control purposes). Karl Pearson first defined the coefficient of variation in the late 1800s in England as part of his broader work on statistical measures.
One important limitation: the CV is only meaningful when the data is measured on a ratio scale with a true zero point. It should not be used with data measured on an interval scale (like temperature in Celsius or Fahrenheit) because the mean can be zero or negative, which makes the ratio meaningless.
Variables
| Symbol | Meaning |
|---|---|
| CV | Coefficient of variation (expressed as a percentage) |
| σ | Standard deviation of the dataset |
| μ | Mean (average) of the dataset |
Example 1
Problem
A set of test scores has a mean of 75 and a standard deviation of 12. What is the coefficient of variation?
CV = (12 / 75) × 100
CV = 0.16 × 100
CV = 16%. This indicates moderate variability in the test scores relative to the average.
Example 2
Problem
Investment A has a mean return of 8% with a standard deviation of 3%. Investment B has a mean return of 15% with a standard deviation of 7%. Which is more consistent?
CVA = (3 / 8) × 100 = 37.5%
CVB = (7 / 15) × 100 = 46.7%
Investment A has a lower CV (37.5% vs 46.7%), meaning it is relatively more consistent despite having a lower return. Investment B has higher variability relative to its mean.
When to Use It
The coefficient of variation is the right tool when you need to compare variability across datasets with different scales or units.
- Comparing risk between investments with different expected returns
- Evaluating measurement precision in laboratory or manufacturing quality control
- Assessing biological variability across species or experiments with different magnitudes
- Determining which process or method is more consistent when means differ significantly