Which definition correctly describes variance and its relationship to standard deviation?

• A. Variance is the square root of standard deviation.
• B. Standard deviation is the average of squared deviations.
• C. Variance is the cube of the standard deviation.
• D. Variance is the average squared deviation from the mean; standard deviation is the square root of variance.

Answer: (D)

The idea being tested is how variance and standard deviation quantify spread and how they relate to each other. Variance measures spread by averaging the squared deviations of each value from the mean, which keeps all deviations positive and gives more weight to larger differences. Standard deviation is obtained by taking the square root of that variance, which puts the measure back in the original units of the data and makes it easier to interpret as a typical distance from the mean. So, the correct description is that variance is the average squared deviation from the mean, and standard deviation is the square root of that variance. The other statements mix up these relationships: variance is not the square root of standard deviation; standard deviation is not the average of squared deviations; and variance is not the cube of the standard deviation.