Which statement correctly describes the units of variance and standard deviation?

Which statement correctly describes the units of variance and standard deviation?

• A. They both have the same units as the data.
• B. Variance has squared units while standard deviation has the same units as the data.
• C. Variance is dimensionless.
• D. Standard deviation has squared units, variance has original units.

Answer: (B)

The units question hinges on what variance and standard deviation actually do. Variance is the average of squared deviations from the mean. If your data are measured in meters, each deviation is in meters, and squaring them gives square meters. After averaging, the variance retains those square units. Standard deviation, on the other hand, is the square root of the variance, so taking the square root brings the units back to the original data units—in this case, meters. So the correct statement is that variance has squared units while standard deviation has the same units as the data. For other units, like dollars, the same logic applies: variance in dollars squared, standard deviation in dollars. The other options fail because they either ignore the squaring (A), claim variance is dimensionless (C) except in special dimensionless data, or swap the units (D).

The units question hinges on what variance and standard deviation actually do. Variance is the average of squared deviations from the mean. If your data are measured in meters, each deviation is in meters, and squaring them gives square meters. After averaging, the variance retains those square units. Standard deviation, on the other hand, is the square root of the variance, so taking the square root brings the units back to the original data units—in this case, meters. So the correct statement is that variance has squared units while standard deviation has the same units as the data. For other units, like dollars, the same logic applies: variance in dollars squared, standard deviation in dollars. The other options fail because they either ignore the squaring (A), claim variance is dimensionless (C) except in special dimensionless data, or swap the units (D).