We often need to estimate the probability that an event we are observing will have a certain outcome. If the possible outcomes of that event are normally distributed, we can do this using the standard normal distribution curve. Since it is a continuous distribution, the probability of obtaining any single outcome is exactly 0, which is why the probability of the outcome falling within a certain interval is always calculated instead.

If we assume that the random variable Z, which has a standard normal distribution, contains all possible outcomes of the observed event, then the probability of a score falling within a given interval equals the area under the normal curve bounded by that interval and the X axis.

Since calculating the area under the curve requires calculations with integrals, statistics textbooks usually include a table in the appendix from which you can read the areas under the standard normal curve for different values of the standardized score.

There are different types of standard normal distribution tables.

One common table format of the standard normal distribution gives the probability, i.e. the area under the curve, for the interval from the arithmetic mean to a certain value (0 to z). The rows of the table give the values of z to the first decimal place, and the columns give the values for the second decimal place. The size of the area, i.e. the probability that the value of the random variable Z is in the interval from 0 to z, is read at the intersection of the row and the column. Since the normal distribution is symmetrical, the area on both the left and right sides of the curve is equal. Therefore, the value read is also the probability that the value of the random variable Z is in the interval -z to 0. The following figure illustrates the area read in this type of table.

Graphical display of values from standard normal distribution tables type 1

Another table format of the standard normal distribution gives the area under the curve for the interval -∞ to z. This means that the values at the intersection of a row and a column in the table represent the probability that the value of the random variable Z is less than or equal to z. The area that we read in this type of table is shown in the following image.

Graphical representation of values from standard normal distribution tables type 2

Further examples of calculating the area under the curve will show how these two types of tables are used.