Here is an amazing fact which explains why the normal curve is so important in statistical investigations. If we take many, many random samples from some population of interest and calculate the sample mean in each case, then the distribution of these sample means will be approximately normal in shape provided the sample size is large. Suppose, for example, we selected lots and lots of random samples of size 100,000 from the population of Australian adults and calculated the mean income for each sample. We would then have a big collection of different average incomes, one from each sample. The distribution of these average incomes (means) would be approximately normal, even though the distribution of individual incomes is not normal, as we have seen in Figure 5. Similarly if you tossed a die 100 times, worked out the mean of the numbers that came up,
and then repeated this experiment over and over again, the distribution of these means would be approximately normal.

This surprising result can be mathematically proved. It is a form of a profound and far reaching theorem called the Central Limit Theorem. It explains why many human
characteristics follow the normal curve, as attributes such as height or weight can be thought of as a sort of “average”. If we think of human weight or height as being a “sort of mean” of many factors (such as heredity, diet, race, sex, many others) then the Central
Limit Theorem would lead us to expect that such human characteristics will follow the normal distribution.