Editing Variance (section)

===Population variance===
In general, the '''''population variance''''' of a ''finite'' [[statistical population|population]] of size {{mvar|N}} with values {{math|''x''<sub>''i''</sub>}} is given by
<math display="block">\begin{align}
  \sigma^2 &= \frac{1}{N} \sum_{i=1}^N {\left(x_i - \mu\right)}^2
            = \frac{1}{N} \sum_{i=1}^N  \left(x_i^2 - 2 \mu x_i + \mu^2 \right) \\[5pt]
           &= \left(\frac{1}{N} \sum_{i=1}^N x_i^2\right) - 2\mu \left(\frac{1}{N} \sum_{i=1}^N x_i\right) + \mu^2 \\[5pt]
           &= \operatorname{E}[x_i^2] - \mu^2 
\end{align}</math>

where the population mean is <math display="inline">\mu = \operatorname{E}[x_i] = \frac 1N \sum_{i=1}^N x_i </math> and <math display="inline">\operatorname{E}[x_i^2] = \left(\frac{1}{N} \sum_{i=1}^N x_i^2\right)  </math>, where <math display="inline">\operatorname{E} </math> is the [[Expected value|expectation value]] operator.

The population variance can also be computed using<ref>{{cite conference|author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |date=June 2012|title=Some new deformation formulas about variance and covariance|conference=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012)|pages=987–992}}</ref>

<math display="block">\sigma^2 = \frac {1} {N^2}\sum_{i<j}\left( x_i-x_j \right)^2 = \frac{1}{2N^2} \sum_{i, j=1}^N\left( x_i-x_j \right)^2.</math>

(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because
<math display="block">\begin{align}
      &\frac{1}{2N^2} \sum_{i, j=1}^N {\left( x_i - x_j \right)}^2 \\[5pt]
  ={} &\frac{1}{2N^2} \sum_{i, j=1}^N \left( x_i^2 - 2x_i x_j  + x_j^2 \right) \\[5pt]
  ={} &\frac{1}{2N} \sum_{j=1}^N \left(\frac{1}{N} \sum_{i=1}^N x_i^2\right) - \left(\frac{1}{N} \sum_{i=1}^N x_i\right) \left(\frac{1}{N} \sum_{j=1}^N x_j\right) + \frac{1}{2N} \sum_{i=1}^N \left(\frac{1}{N} \sum_{j=1}^N x_j^2\right) \\[5pt]
  ={} &\frac{1}{2} \left( \sigma^2 + \mu^2 \right) - \mu^2 + \frac{1}{2} \left( \sigma^2 + \mu^2 \right) \\[5pt]
  ={} &\sigma^2.
\end{align}</math>

The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.