Coefficient of variation calculator
Enter numbers, get the CV.
What is the coefficient of variation?
The standard deviation divided by the mean, usually shown as a percentage: CV = s ÷ x̄ × 100%. It measures spread relative to the size of the numbers. NIST’s Dataplot reference defines it for data on a ratio scale, that is, continuous values with a meaningful zero, and notes that the relative standard deviation is the same ratio in percent.
Example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The population SD is 2, so the CV is 40%; the sample SD is 2.14, giving 42.8%.
Why use it instead of the standard deviation?
It has no units, so it compares spread between data on different scales — for example, the variability of weights in grams and in kilograms, or of two lab methods. Multiply the eight numbers in the example by 1,000 (grams instead of kilograms): the mean becomes 5000 and the SD 2138, but the CV is still 42.8%. The SD changed with the unit; the CV did not.
Sample or population?
Use the sample CV when your numbers are a sample from a larger group, and the population CV when they are the whole group. The calculator shows both. The two differ by the factor √[(n − 1) ÷ n], which matters only for small lists.
Worked example: which method is more precise?
Method A reads 9.8, 10.1, 10.0, 9.9 and 10.2 units; method B, on a different scale, reads 98, 104, 101, 97 and 105. The SDs, 0.16 and 3.54, cannot be compared directly. The CVs can: 1.6% for A against 3.5% for B, so A is more consistent relative to its level. In analytical laboratories this ratio is usually called the relative standard deviation (RSD).
How do I compute it by hand?
Take the same eight numbers. Step 1: the mean is 40 ÷ 8 = 5. Step 2: subtract the mean from each value and square the result; the squares add up to 32. Step 3: divide by n = 8 for the population variance (4) or by n − 1 = 7 for the sample variance (4.571). Step 4: take the square root to get the SD, 2 or 2.138. Step 5: divide by the mean and multiply by 100 to get 40.0% or 42.8%.
Is the CV the same as the standard error?
No. The coefficient of variation describes how spread out the individual values are relative to their mean. The standard error, s ÷ √n, describes how precisely the mean itself is known and shrinks as the sample grows. For the example, the standard error is 2.138 ÷ √8 = 0.756, but the CV stays near 43% however many more values of the same kind you collect.
When is the CV misleading?
When the mean is near zero, or for data that can be negative (such as temperatures in °F or °C) — the ratio then has no sensible meaning. Temperatures show why. Readings of 10, 20 and 30 °C have a sample CV of 50%, but the same three temperatures in °F, 50, 68 and 86, give 26.5%. The temperature did not change; only the arbitrary zero did. Ratio-scale quantities such as mass, length, time, concentration and price do not have this problem.
Common mistakes with the coefficient of variation
First, using it on interval-scale data or on values with a mean close to zero, where a tiny change in the mean makes the CV explode. Second, comparing a sample CV with a population CV: use the same version for every data set you compare. Third, expecting it to say whether a spread is acceptable: a CV that is routine for biological measurements can be far too large for a machined part, and the right limit comes from the field. Fourth, computing it for a handful of values: with 3 or 4 numbers the SD, and so the CV, is very uncertain.
Sources: NIST Dataplot reference manual, “Coefficient of variation” (definition and ratio-scale requirement). Related: standard deviation calculator, standard error calculator, average calculator, mean absolute deviation calculator and percent error calculator.