Which statement about the mean and the standard deviation of z-scores is true?

Which statement about the mean and the standard deviation of z-scores is true?

• A. The mean is 0 and the standard deviation is 1.
• B. The mean is 1 and the standard deviation is 0.
• C. The mean is the original mean and the SD is 1.
• D. The mean is 0 and the SD is the original SD.

Answer: (A)

Standardizing a variable with z-scores centers and rescales the data. You compute Z as (X − μ) / σ, where μ is the original mean and σ is the original standard deviation. This operation shifts the distribution so its center is at zero and scales it so its spread is one. Mathematically, the mean of Z is (μ − μ)/σ = 0, and the standard deviation of Z is sqrt(Var((X − μ)/σ)) = sqrt(σ^2/σ^2) = 1. Therefore, the transformed scores have a mean of zero and a standard deviation of one. The other descriptions would imply keeping the original mean or spread, or an impossible zero spread after standardization, which isn’t how z-scores work.

Standardizing a variable with z-scores centers and rescales the data. You compute Z as (X − μ) / σ, where μ is the original mean and σ is the original standard deviation. This operation shifts the distribution so its center is at zero and scales it so its spread is one. Mathematically, the mean of Z is (μ − μ)/σ = 0, and the standard deviation of Z is sqrt(Var((X − μ)/σ)) = sqrt(σ^2/σ^2) = 1. Therefore, the transformed scores have a mean of zero and a standard deviation of one. The other descriptions would imply keeping the original mean or spread, or an impossible zero spread after standardization, which isn’t how z-scores work.