Originally Posted by
Mathman
Here is a "population" of five numbers: 9, 10, 10 10, 11. The average (arithmetic mean) is 10. So far so good.
Here is another population: 0, 5, 10, 15, 20. The average of this population is 10, too.
So how are the two populations different, if they both have the same average? Well, in the first population, all the numbers are close together. They are all very close to the mean. In the second population they are spread apart. Some of them are a long way from the average. The variance measures, "how far from the mean are these numbers?"
So for each number, we look and see how far from average it is. The numbers 9, 10, 10, 10, 11 are, respectively, -1 unit, 0 units, 0 units, 0 units, and 1 unit away from the average of 10. Because we don't really care about the + or - sign, we square these differences, then take the average. That's the variance:
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