Proof: Why Sample Variance uses n-1 (Unbiased Estimator Derivation) #mathematics #varianceanalysis

Why do we divide by n-1 instead of n in the sample variance formula? This high-speed mathematical derivation proves that the sample variance S^2 is an unbiased estimator of the population variance sigma^2.

We cover:

Expanding the sum of squares.
Using the linearity of expectation.
Applying the variance identity E[Z^2] = Var(Z) + E[Z]^2.
Final algebraic simplification to reach sigma^2.
Plain-text Timestamps
0:00 The Goal: Show E[S^2] equals sigma^2
0:09 Defining the Expectation of S^2
0:18 Expanding the Sum of Squares
0:27 Simplifying the Sum using the Sample Mean
0:36 Applying Linearity of Expectation
0:45 Expected Value of individual Sample Points
0:54 Expected Value of the Sample Mean Squared
1:03 Substituting components into the Equation
1:12 Final Algebraic Simplification
1:21 Conclusion: Unbiased Estimator Proof
#variance #invariance #statistical inference

Видео Proof: Why Sample Variance uses n-1 (Unbiased Estimator Derivation) #mathematics #varianceanalysis канала nicefa