Which statistic is useful for comparing variability across datasets with different units or means?

Which statistic is useful for comparing variability across datasets with different units or means?

• A. Coefficient of variation
• B. Standard deviation
• C. Mean
• D. Median

Answer: (A)

You compare variability across datasets with different units or means by using a unitless measure of dispersion. The coefficient of variation does exactly that: it takes the standard deviation and divides it by the mean, often expressed as a percentage. Because it normalizes dispersion by the scale of the data, you can directly compare how spread out the data are even when the datasets use different units or have different average values. For example, if one dataset has a mean of 100 with a standard deviation of 20, its coefficient of variation is 0.20 (20%). If another dataset has a mean of 50 with a standard deviation of 15, its coefficient of variation is 0.30 (30%), indicating greater relative variability despite the larger absolute dispersion in the first dataset. Cautions: the coefficient of variation is not appropriate if the mean is near zero or if data can be negative, since the division by a small or negative mean can give misleading results. In those cases, or when the data include zero, other dispersion measures may be preferable. The standard deviation stays in the original units and cannot be directly compared across datasets with different scales. The mean and median describe central tendency, not how spread out the values are, so they don’t address the question of comparing variability.

You compare variability across datasets with different units or means by using a unitless measure of dispersion. The coefficient of variation does exactly that: it takes the standard deviation and divides it by the mean, often expressed as a percentage. Because it normalizes dispersion by the scale of the data, you can directly compare how spread out the data are even when the datasets use different units or have different average values.

For example, if one dataset has a mean of 100 with a standard deviation of 20, its coefficient of variation is 0.20 (20%). If another dataset has a mean of 50 with a standard deviation of 15, its coefficient of variation is 0.30 (30%), indicating greater relative variability despite the larger absolute dispersion in the first dataset.

Cautions: the coefficient of variation is not appropriate if the mean is near zero or if data can be negative, since the division by a small or negative mean can give misleading results. In those cases, or when the data include zero, other dispersion measures may be preferable.

The standard deviation stays in the original units and cannot be directly compared across datasets with different scales. The mean and median describe central tendency, not how spread out the values are, so they don’t address the question of comparing variability.