What are the mean and standard deviation of a distribution of z-scores?

What are the mean and standard deviation of a distribution of z-scores?

• A. Mean 0 and standard deviation 1.
• B. Mean 1 and standard deviation 0.
• C. Mean 0 and standard deviation equal to the original standard deviation.
• D. Mean equal to the original mean and SD equal to 1.

Answer: (A)

Standardizing a variable makes its distribution centered at zero and scaled to have unit spread. If you transform X by Z = (X − μ) / σ, where μ is the mean and σ is the standard deviation of X, then Z has a mean of 0 and a standard deviation of 1. This comes from E[Z] = E[(X − μ)/σ] = (E[X] − μ)/σ = 0 and Var(Z) = Var[(X − μ)/σ] = Var(X)/σ^2 = σ^2/σ^2 = 1, so the SD is 1. So the mean of z-scores is 0 and their standard deviation is 1. (If you use sample estimates for μ and σ, you’ll get values very close to 0 and 1, respectively, in large samples.)

Standardizing a variable makes its distribution centered at zero and scaled to have unit spread. If you transform X by Z = (X − μ) / σ, where μ is the mean and σ is the standard deviation of X, then Z has a mean of 0 and a standard deviation of 1. This comes from E[Z] = E[(X − μ)/σ] = (E[X] − μ)/σ = 0 and Var(Z) = Var[(X − μ)/σ] = Var(X)/σ^2 = σ^2/σ^2 = 1, so the SD is 1. So the mean of z-scores is 0 and their standard deviation is 1. (If you use sample estimates for μ and σ, you’ll get values very close to 0 and 1, respectively, in large samples.)