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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 σX2 has a density function.

where

The mean μX and the variance σX2 are the parameters of the normal distribution. The positive square root of is the standard deviation of X. The mean is a measure of location or a measure of central tendency of the random variable (i.e., where the values of the random variable are concentrated), and the variance is a measure of variation of the random variable (i.e., how spread the values of the random variable are). The mean μX might take on any real value, and the variance σX2 might take on any positive value. Each pair of the given values of the mean μX and the variance σX2 determines a specific member in the normal distribution family. A normal random variable X with mean μX and variance σX2 is usually denoted by X ∼ N(μX, σX2).

For a random variable X with mean μX and variance σX2, 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 σX2 = 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 σX2 = 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 σX2 = 25 (Top) and of the Standard Normal Random Variable Z (Bottom)

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