«Bariantza»: berrikuspenen arteko aldeak

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:<math>s_X = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2\,}=\sqrt{\frac{\sum_{i=1}^n {x_i^2}}{n}-\overline{x}^2}\,</math>
 
{{kaxa zabalkorra
|Formula laburtuaren dedukzioa
|ta2=center
|
 
<math>s_X = \frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2\,=\frac{1}{N} \sum_{i=1}^N (x_i^2 + \overline{x}^2-2\overline{x}x_i)\,=\frac{\sum_{i=1}^N x_i^2}{N} + \frac{\sum_{i=1}^N \overline{x}^2}{N} -2 \overline{x}\frac{\sum_{i=1}^N x_i}{N}\,=</math>
 
 
<math>\frac{\sum_{i=1}^N x_i^2}{N} + \frac{N \overline{x}^2}{N} -2\overline{x}\overline{x}\,=\frac{\sum_{i=1}^N x_i^2}{N} + \overline{x}^2 -2 \overline{x}^2\,=\frac{\sum_{i=1}^N x_i^2}{N} - \overline{x}^2\,</math>
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=== Adibidea datu bakanduetarako ===
:<math>\operatorname{var}[X]=\sigma^2_X=E[(X-\mu)^2]=E[X^2]-\mu^2=E[X^2]-E[X]^2</math>
 
{{kaxa zabalkorra
|Formula laburtuaren dedukzioa
|ta2=center
|
 
<math>var[X] = E[(X-\mu)^2]=E[(X^2-2\mu X+\mu^2)]=E[X^2]-2\mu E[X]+E[\mu^2]=</math>
 
 
<math>=E[X^2]-2\mu^2+\mu^2=E[X^2]-\mu^2</math>
}}
 
=== Banakuntza diskretu baterako adibidea ===
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