Which statement correctly distinguishes variance from standard deviation?

Which statement correctly distinguishes variance from standard deviation?

• A. Standard deviation is the square of variance.
• B. Variance is the average squared deviation; standard deviation is its square root.
• C. Variance is always smaller than standard deviation.
• D. Standard deviation is independent of units.

Answer: (B)

Understanding how spread is measured helps here. Variance captures spread by averaging the squared deviations from the mean. Because the deviations are squared, variance ends up in squared units, which can be hard to interpret on the original scale. The standard deviation is the square root of that variance, bringing the measure back to the same units as the data. This direct link—variance equals the average of squared deviations, and standard deviation is its square root—explains why this pair of numbers is so closely related and why the standard deviation is often preferred for interpretation. For example, with data like 2, 4, 6, the mean is 4; deviations are -2, 0, 2; squared deviations are 4, 0, 4; the average of those is about 2.667, and the standard deviation is the square root of that, about 1.633. You can see the standard deviation shares the data’s units, while the variance would be in squared units. The other statements don’t hold in general. The standard deviation is not the square of the variance; it is its square root. It isn’t universally true that the variance is smaller than the standard deviation, since their relative sizes depend on the data. And the standard deviation is not independent of units—it's measured in the same units as the data.

Understanding how spread is measured helps here. Variance captures spread by averaging the squared deviations from the mean. Because the deviations are squared, variance ends up in squared units, which can be hard to interpret on the original scale.

The standard deviation is the square root of that variance, bringing the measure back to the same units as the data. This direct link—variance equals the average of squared deviations, and standard deviation is its square root—explains why this pair of numbers is so closely related and why the standard deviation is often preferred for interpretation.

For example, with data like 2, 4, 6, the mean is 4; deviations are -2, 0, 2; squared deviations are 4, 0, 4; the average of those is about 2.667, and the standard deviation is the square root of that, about 1.633. You can see the standard deviation shares the data’s units, while the variance would be in squared units.

The other statements don’t hold in general. The standard deviation is not the square of the variance; it is its square root. It isn’t universally true that the variance is smaller than the standard deviation, since their relative sizes depend on the data. And the standard deviation is not independent of units—it's measured in the same units as the data.