Biased sample variance, unbiased sample variance

1. Biased sample variance 、 Unbiased sample variance

Note source ：Why Dividing By N Underestimates the Variance

1.1 Why is the sample variance always smaller than the population variance ？

Because the total quantity is too large , We have to spend a lot of human, material and financial resources to understand the overall situation... Or never , So we can only use sample data to estimate the overall situation , For the convenience of explanation, we assume a population mean

Sample mean $\bar{x}=17.6$, Biased sample variance ${\sigma }_{X_1}^2=81.4$
The assumed overall mean $\mu =20$, Total variance ${\sigma }^2=87.2$
$${\sigma }_{X_1}^2<{\sigma }^2$$

example 2： Another set of sample data
Sample mean $\bar{x}=22.6$, Biased sample variance ${\sigma }_{X_2}^2=105$
The assumed overall mean $\mu =20$, Total variance ${\sigma }^2=112$
$${\sigma }_{X_2}^2<{\sigma }^2$$

1.2 Why can't we use absolute value for variance ？

We find the minimum variance by deriving the formula of the opposite difference

When we replace the square in the variance formula with the absolute value , We cannot find the minimum variance , Because it is not differentiable at the minimum