For a normal distribution, the percentage of values falling within any interval can be calculated exactly. For example, in a normal distribution with a mean of 100 and a standard deviation of 15 (as in Figure 4.1), 68% of all values fall between 85 (one standard deviation less than the mean) and 115 (one standard deviation more than the mean). And 95% of all values fall in the range 70 to 130, within two standard deviations from the mean.
A normal distribution can have any mean and standard deviation. However, the percentage of cases falling within a particular number of standard deviations from the mean is always the same. The shape of a normal distribution does not change. Most of the observations are near the mean, and a mathematical function describes how many observations are at any given distance (measured in standard deviations) from the mean. Means and standard deviations differ from variable to variable. But the percentage of cases within specific intervals is always the same in a true normal distribution. We are going to use this principle in Volumes IV, V and VI.
It turns out that many variables you can measure have a distribution close to the mathematical ideal of a normal distribution. We say these variables are "normally distributed," even though their distributions are not exactly normal. Usually when we say this, we mean that the histograms look like Figure 4.1. For example, Figure 4.2 shows that an actual distribution that is indeed pretty close to normal.