Confirming some observations

In an earlier section we stated the Empirical Rule, which said that if a distribution has a bell shaped curve then

- about 68% of all scores fall within 1 standard deviation of the mean.

- about 95% of all scores fall within 2 standard deviations of the mean.

- about 99.7% of all scores fall within 3 standard deviations of the mean.

We will now show how these figures are obtained using the standard normal distribution.

Example

Find P(-1<z<1)

Solution .

Since the standard deviation of the standard normal is one, P(-1<z<1) will give us the probability of a score falling within 1 standard deviation of the mean. To find P(-1<z<1) we use symmetry and find P(-1<z<1) = 2P(0<z<1) = 2(.3413)= .6826. If you convert this to a percent, you see where the phrase "about 68% of the scores" comes from.

In a similar way we compute P(-2<z<2) and P(-3<z<3).