Unbiased estimate of Population Parameters

Unbiased Estimate of Population Mean,

\(\mu =\overline{x}=\frac{\sum{x}}{n}=\frac{\sum{\left( x-a \right)}}{n}+a\) where \(a\) is a constant.

Unbiased Estimate of Population variance,

\({{s}^{2}}=\frac{1}{n-1}{{\sum{\left( x-\overline{x} \right)}}^{2}}=\frac{1}{n-1}\left[ \sum{{{x}^{2}}}-\frac{{{\left( \sum{x} \right)}^{2}}}{n} \right]\) [in MF 26]

\(=\frac{n}{n-1}\times \)(sample variance) or \(\frac{n}{n-1}{{\sigma }_{x}}^{2}\)where \({{\sigma }_{x}}^{2}\) is the sample variance.

\(=\frac{1}{n-1}\left[ \sum{{{(x-a)}^{2}}}-\frac{{{\left( \sum{(x-a)} \right)}^{2}}}{n} \right]\) where \(a\) is a constant.