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Given $X_1, \dots , X_n$ i.i.d. and the two sample moments $M_1 = \frac{1}{n} \sum_{i = 1}^{n} X_i = \bar{X}$ and $ M_2 = \frac{1}{n} \sum_{i = 1}^{n} X_i^2$ how can I compute: $ S^2 = \frac{1}{n} \sum_{i = 1}^{n} (X_i - \bar{X})^2$ such as: $S^2 = f(M_1, M_2)$

Thank you.

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    This has to do with statistics in that the derived formula is commonly used for computing the variance estiamte.2012-06-13

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$S^2 = \frac{1}{n}\sum_{i=1}^n (X_i - M_1)^2 = \frac{1}{n}\sum_{i=1}^n (X_i^2 - 2 X_i M_1 + M_1^2) =$

$= \frac{1}{n}\sum_{i=1}^n X_i^2 - 2 M_1 \frac{1}{n}\sum_{i=1}^n X_i + \frac{1}{n}\sum_{i=1}^n M_1^2 =$

$= M_2 - 2M_1^2 + M_1^2 = M_2 - M_1^2$