Desvio Padrao do Conjunto de Dados

$S = \sum_{i=1}^N(x_i-\bar{x})^2 = \sum_{i=1}^N\Big(x_i-\frac{1}{N}\sum_{j=1}^N x_j\Big)^2$

$S = \sum_{i=1}^N\big[(x_i-\mu)-\frac{1}{N}\sum_{j=1}^N (x_j-\mu)\Big]^2$

$S = \sum_{i=1}^N\big[(x_i-\mu)^2-\frac{2}{N}\sum_{j=1}^N (x_i-\mu)(x_j-\mu) +\frac{1}{N^2}\sum_j\sum_k(x_j-\mu)(x_k-\mu) \Big]$

Calculando o valor médio:

$\langle S \rangle = \sum_{i=1}^N\big[\langle(x_i-\mu)^2\rangle-\f... ..._j-\mu)\rangle +\frac{1}{N^2}\sum_j\sum_k\langle(x_j-\mu)(x_k-\mu)\rangle \Big]$

Assumindo que as amostras são não correlacionadas, $\sigma_{ij} = \sigma_x^2 \partial_{ij}$,

$\langle S \rangle = \sum_{i=1}^N\big[\sigma_x^2-\frac{2}{N}\sigma... ...}\sum_j \sigma_x^2\Big] =\sum_{i=1}^N\big[\sigma_x^2-\frac{1}{N}\sigma_x^2\Big]$

$\langle \sigma^2_{x}\rangle = \frac{1}{N-1}\sigma_x^2$

Volta: Média Mínimos Quadrados Astronomia e Astrofisica