What is the standard normal form?

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.

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.