What is the standard normal form?

• A. X ~ N(0,1); z = X / sigma
• B. Z ~ N(0,1); z = (X - mu)/sigma
• C. Z ~ N(mu, sigma^2)
• D. Z ~ N(0,1); z = mu - X / sigma

Answer: (B)

Standard normal form is the normal distribution with mean zero and standard deviation one. To convert a normal variable X that has mean mu and standard deviation sigma to this form, you subtract the mean and divide by the standard deviation: Z = (X - mu)/sigma, which makes Z follow a standard normal distribution Z ~ N(0,1). This is exactly what the statement describes: Z ~ N(0,1) and z = (X - mu)/sigma.

Using X/sigma would shift the mean to mu/sigma, not zero, so it isn’t standard normal. Saying Z ~ N(mu, sigma^2) would simply restate the original distribution of X, not the standardized form. And mu - X over sigma reverses the order and changes the sign, not producing the standardization you want.