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Biased sample variance, unbiased sample variance
2022-07-05 23:39:00 【Uncertainty!!】
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 x ˉ = 17.6 \bar{x}=17.6 xˉ=17.6, Biased sample variance σ X 1 2 = 81.4 \sigma_{X_1}^2=81.4 σX12=81.4
The assumed overall mean μ = 20 \mu=20 μ=20, Total variance σ 2 = 87.2 \sigma^2=87.2 σ2=87.2
σ X 1 2 < σ 2 \sigma_{X_1}^2\lt\sigma^2 σX12<σ2
example 2: Another set of sample data
Sample mean x ˉ = 22.6 \bar{x}=22.6 xˉ=22.6, Biased sample variance σ X 2 2 = 105 \sigma_{X_2}^2=105 σX22=105
The assumed overall mean μ = 20 \mu=20 μ=20, Total variance σ 2 = 112 \sigma^2=112 σ2=112
σ X 2 2 < σ 2 \sigma_{X_2}^2\lt\sigma^2 σX22<σ2
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
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