Known characteristics of the normal curve make it possible to estimate the probability of occurrence of any value of a normally distributed variable. Suppose that the total area under the curve is defined to be 1. You can multiply that number by 100 and say there is a 100 percent chance that any value you can name will be somewhere in the distribution. ( Remember : The distribution extends to infinity in both directions.) Similarly, because half the area of the curve is below the mean and half is above it, you can say that there is a 50 percent chance that a randomly chosen value will be above the mean and the same chance that it will be below it.
It makes sense that the area under the normal curve is equivalent to the probability of randomly drawing a value in that range. The area is greatest in the middle, where the “hump” is, and thins out toward the tails. That is consistent with the fact that there are more values close to the mean in a normal distribution than far from it.
When the area of the standard normal curve is divided into sections by standard deviations above and below the mean, the area in each section is a known quantity (see Figure 1). As explained earlier, the area in each section is the same as the probability of randomly drawing a value in that range.
Figure 1.The normal curve and the area under the curve between σ units.