z Score

The Normal Distribution and Standardization

The normal distribution is a family of normal random variables. A normal random variable X with mean or expected value μX and variance σ2X has a density function,

where

The mean μX and the variance σ2X are the parameters of the normal distribution. The positive square root of σ2X,

For a random variable X with mean μX and variance σ2X, the transformation

is called standardization and Z is called the standardized random variable. The standardized random variable has a mean μZ = 0 and a variance σ2Z = 1. If X is normally distributed, then Z is a standard normal random variable. The standard normal random variable Z has a density function

and is denoted by Z~N(0,1). The standard normal distribution is a special case of the normal distribution. Any normal random variable in the normal distribution family can be transformed into the standard normal random variable through standardization. If X is not normally distributed, Z is still called the standardized random variable but does not have a standard normal distribution.

When plotted, the density function of the normal random variable X and that of the standard normal random variable Z have the same shape but X and Z are on different scales. Figure 1 shows the plot of a normal random variable X with a mean μX = 10 and a variance σ2X = 25 and the plot of the standard normal random variable Z. The curve of the normal distribution is symmetrical about the mean and has its peak at the mean. The normal curve is usually called the bell curve or bell-shaped curve because of its appearance.

Figure 1 Plots of the Density Functions of a Normal Random Variable X With μX =10 and σ2X =25 (Top) and of the Standard Normal Random Variable Z (Bottom)

z Score

Suppose x is the value of an observation, sometimes called the raw score, drawn from the distribution of X. Then the z score, or the standard score, of this observation is determined through standardization; that

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