Difference between N and n-1 when variance is obtained

There are two types of variance. One is the sample variance and the other is the population variance. When the sample variance is obtained, the denominator is n-1; when the population variance is obtained, the denominator is N. In mathematical statistics, the common request is the sample variance, which requires the construction of a statistical sample variance s ^ 2 (note that this is a random variable ), the expected and population variance of the constructed statistic must be equal to that of the population, so that the statistical measuring tool is biased. In elementary school mathematics, it may be that junior high school mathematics can only encounter the situation of overall variance. For example, 50 people in a class, each person's mathematical score knows, let you calculate the average and variance. In this case, the variance is the population variance. There is no sampling problem here, so there is no random variable or expectation.

Why do we need the sample variance? For example, if there are 0.1 billion data records in a large population, I want to know the total variance. It is very difficult to count the 0.1 billion data. Therefore, we hope to extract 10 thousand samples from the population and estimate the population variance by calculating the sample variance. This requires us to construct an appropriate statistic s ^ 2 to estimate the population variance.

Next we will explain why we need to divide the sample variance by n-1. First, we may ask how to construct the statistic s ^ 2? In fact, it can be constructed at will, but the statistic you construct may not be able to properly estimate the unknown parameters in the population. There are three criteria for judging the quality of the constructed statistic: non-biased, effective, and consistent (Here we only talk about non-biased ). Unbiased, refers to the statistic constructed for unknown parameters. The expected statistic is equal to the parameter to be estimated.

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