The standard deviation of a continuous random variable is

The standard deviation of a continuous random variable is

• A. the square root of the variance
• B. the expected value
• C. the square of the variance
• D. always equal to 1

Answer: (A)

The standard deviation measures how spread out a random variable is and is defined as the square root of the variance. Variance itself is the expected squared deviation from the mean: Var(X) = E[(X − μ)²], where μ is the mean E[X]. Taking the square root gives the standard deviation, which shares the same units as the variable and provides a direct sense of typical deviation from the mean. So the standard deviation is the square root of the variance. The other statements don’t describe this spread measure: the expected value is the mean, not dispersion; squaring the variance gives a much larger quantity (variance squared), not the standard deviation; and the standard deviation is not always 1 (only in special cases where the variable is standardized to have unit variance).

The standard deviation measures how spread out a random variable is and is defined as the square root of the variance. Variance itself is the expected squared deviation from the mean: Var(X) = E[(X − μ)²], where μ is the mean E[X]. Taking the square root gives the standard deviation, which shares the same units as the variable and provides a direct sense of typical deviation from the mean.

So the standard deviation is the square root of the variance. The other statements don’t describe this spread measure: the expected value is the mean, not dispersion; squaring the variance gives a much larger quantity (variance squared), not the standard deviation; and the standard deviation is not always 1 (only in special cases where the variable is standardized to have unit variance).