17.6 Exact areas from normal distributions

Areas under normal distributions can be found using:

17.6.1 Using the hard-copy tables

To demonstration the use of the normal distribution tables, consider the percentage of observations smaller than z=−2 (that is, two standard deviations below the mean) in a normal distribution.

The hard-copy tables work with z-scores to two decimal places, so consider the z-score as z=−2.00. On the tables, find −2.0 in the left margin of the table, and find the second decimal place (in this case, 0) in the top margin of the table (Fig. 17.6): where these intersect is the area (or probability) less than the z-score. So the probability of finding a z-score less than z=−2 is 0.0228, or about 2.28%. (The online tables work differently.)

Using the hard-copy tables to compute the probability that $z$ is less than $-2$

FIGURE 17.6: Using the hard-copy tables to compute the probability that z is less than −2

The tables give the area to the left of the z-score that is looked up.

17.6.2 Using the online tables

The online tables work differently to the hard-copy tables. Consider the same example again: the percentage of observations smaller than z=−2.

Like the hard-copy tables, the online tables (Appendix B.2) work with two decimal places, so consider the z-score as z=−2.00. In the tables, enter the value -2 in the search region just under the column labelled z.score (see the animation below). After pressing Enter, the answer is shown in the column headed Area.to.left: the probability of finding a z-score less than −2 is 0.0228, or about 2.28%.

Using either the hard-copy or online tables gives an answer of about 2.28%. Using the 68–95–99.7 rule, the answer we obtained was 2.5%. Recall that the 68–95–99.7 rule is an approximation only.