Z Scores and the Normal Distribution
The z score of a random variable X that is normally distributed with parameters m and s is computed as follows:
The random variable zX is a standard normal random variable, i.e., the z score of a normal random variable with parameters m and s is a random variable that is also normally distributed, but with parameters m = 0 and s = 1. Since every normal random variable can be converted (using z scores) into a standard normal random variable, probabilities associated with a normal random variable assuming values within an interval (a, b) can be easily calculated using the probability values in a Standard Distribution Table. That is, for X ~ N(0,1),
P(a < X < b) = P(X < b) - P(X < a) for a < b
Thus, the z score is a very convenient way of standardizing the computations of probabilities involving normal random variables with different values for m and s, without involving tables for each combination of m and s or tedious integrations of complex functions.
The following table shows the value of P(X < z), where X is a standard normal random variable, for selected values of z:
| z | p(X < z) |
| -3.0 | 0.0013 |
| -2.0 | 0.0228 |
| -1.0 | 0.1587 |
| 0 | 0.5000 |
| 1.0 | 0.8413 |
| 2.0 | 0.9772 |
| 3.0 | 0.9987 |
EX. A random variable Y is normally distributed with m = 20 and s = 2. Y has a z score that is calculated as
Using a Standard Distribution Table, the probability that Y takes on a value greater than 23 is
The probability that Y takes on a value between 19 and 22 is
