If a distribution is standardized, which statement is true about its mean and standard deviation?

If a distribution is standardized, which statement is true about its mean and standard deviation?

• A. The mean becomes 0 and the SD becomes 1.
• B. The mean becomes 1 and the SD becomes 0.
• C. The mean remains the original mean and the SD becomes 1.
• D. The mean becomes 0 and the SD remains unchanged.

Answer: (C)

Standardizing a distribution means converting each value to a z-score by subtracting the original mean and dividing by the original standard deviation. This process centers the data at zero and scales the spread to unit variance. If X has mean μ and standard deviation σ, then the standardized value is Z = (X − μ)/σ. The mean of Z becomes 0 because E[Z] = (E[X] − μ)/σ = (μ − μ)/σ = 0, and the standard deviation of Z becomes 1 because Var(Z) = Var(X)/σ^2 = σ^2/σ^2 = 1. So the correct statement is that the mean becomes 0 and the standard deviation becomes 1. The other descriptions don’t match what standardization does to the center and the spread.

Standardizing a distribution means converting each value to a z-score by subtracting the original mean and dividing by the original standard deviation. This process centers the data at zero and scales the spread to unit variance. If X has mean μ and standard deviation σ, then the standardized value is Z = (X − μ)/σ. The mean of Z becomes 0 because E[Z] = (E[X] − μ)/σ = (μ − μ)/σ = 0, and the standard deviation of Z becomes 1 because Var(Z) = Var(X)/σ^2 = σ^2/σ^2 = 1. So the correct statement is that the mean becomes 0 and the standard deviation becomes 1. The other descriptions don’t match what standardization does to the center and the spread.