6.1 The standard normal distribution

The standard normal distribution is a normal distribution of standardized values called z -scores . A z -score is measured in units of the standard deviation. For example, if the mean of a normal distribution is five and the standard deviation is two, the value x = 11 is three standard deviations above (or to the right of) the mean. The calculation is as follows:

x = μ + ( z )( σ ) = 5 + (3)(2) = 11

The z -score is three.

The mean for the standard normal distribution is zero, and the standard deviation is one. What this does is dramatically simplify the mathematical calculation of probabilities. Take a moment and substitute zero and one in the appropriate places in the above formula and you can see that the equation collapses into one that can be much more easily solved using integral calculus. The transformation z = $\frac{x-\mu }{\sigma }$ produces the distribution Z ~ N (0, 1). The value x comes from a known normal distribution with known mean μ and known standard deviation σ . The z -score tells how many standard deviations a particular x is away from the mean.

Z -scores

If X is a normally distributed random variable and X ~ N(μ, σ) , then the z -score is:

The z -score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ . Values of x that are larger than the mean have positive z -scores, and values of x that are smaller than the mean have negative z -scores. If x equals the mean, then x has a z -score of zero.

Suppose the random variables X and Y have the following normal distributions: X ~ N (5, 6) and Y ~ N (2, 1). If x = 17, then z = 2. (This was previously shown.) If y = 4, what is z ?

z = $\frac{y-\mu }{\sigma }$ = $\frac{4-2}{1}$ = 2 where µ = 2 and σ = 1.

The z -score for y = 4 is z = 2. This means that four is z = 2 standard deviations to the right of the mean. Therefore, x = 17 and y = 4 are both two (of their own ) standard deviations to the right of their respective means.

The z -score allows us to compare data that are scaled differently. To understand the concept, suppose X ~ N (5, 6) represents weight gains for one group of people who are trying to gain weight in a six week period and Y ~ N (2, 1) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since x = 17 and y = 4 are each two standard deviations to the right of their means, they represent the same, standardized weight gain relative to their means .