The coefficient of variation measures how large the standard deviation is compared with the mean. It is useful because the same standard deviation can mean very different things for small and large averages. By converting spread into a relative amount, it helps compare variability between data sets with different units or different scales.
Scientists, engineers, and analysts use it when they want to know which process, measurement, or group is more consistent.
Key Facts
- Coefficient of variation: CV = s / x̄ for a sample, or CV = σ / μ for a population.
- Percent form: CV% = (s / x̄) × 100%.
- A smaller CV means the data are more consistent relative to the mean.
- A larger CV means the data are more variable relative to the mean.
- CV is unitless because the units in the standard deviation and mean cancel.
- CV is most meaningful when the mean is positive and not close to zero.
Vocabulary
- Coefficient of Variation
- A unitless measure of relative variability found by dividing the standard deviation by the mean.
- Standard Deviation
- A measure of how far data values typically are from the mean.
- Mean
- The arithmetic average of a data set, found by adding all values and dividing by the number of values.
- Relative Variability
- The amount of spread in a data set compared with the size of its typical value.
- Unitless Measure
- A quantity with no physical unit because the units cancel during calculation.
Common Mistakes to Avoid
- Comparing standard deviations alone, which is wrong when the means or units are different because the same spread can be small or large relative to the typical value.
- Forgetting to multiply by 100 for CV%, which gives the decimal form instead of the percent form and can make the answer look 100 times too small.
- Using CV when the mean is near zero, which is wrong because dividing by a very small mean can produce a huge or unstable value.
- Mixing sample and population formulas, which is wrong because sample calculations use s and x̄ while population calculations use σ and μ.
Practice Questions
- 1 A data set has mean x̄ = 80 and sample standard deviation s = 12. Find the coefficient of variation as a decimal and as a percent.
- 2 Machine A fills bottles with mean 500 mL and standard deviation 8 mL. Machine B fills bottles with mean 250 mL and standard deviation 6 mL. Which machine has the larger coefficient of variation?
- 3 Two classes have the same standard deviation on a test, but Class 1 has a mean of 40 and Class 2 has a mean of 80. Explain which class has greater relative variability and why.