Standard Deviation

Let $\{x_1,~x_2,~\ldots,x_n\}$ be a set of data.

1. The mean
   \begin{eqnarray*}
   \overline{x} & = & \sum_{i=1}^n \frac{x_i}{n} \\
   & = & \frac{x_1+x_2+\cdots+x_n}{n}
   \end{eqnarray*}

2. The variance
   \begin{eqnarray*}
   \sigma^2 & = & \sum_{i=1}^n \frac{(x_i-\overline{x})^2}{n} \\
   & = & \frac{(x_1-\overline{x})^2+(x_2-\overline{x})^2+\cdots+(x_n-\overline{x})^2}{n}
   \end{eqnarray*}

3. The standard deviation
   \begin{eqnarray*}
   \sigma & = & \sqrt{\sum_{i=1}^n \frac{(x_i-\overline{x})^2}{n}} \\
   & = & \sqrt{\frac{(x_1-\overline{x})^2+(x_2-\overline{x})^2+\cdots+(x_n-\overline{x})^2}{n}}
   \end{eqnarray*}

4. The standard score
   \begin{eqnarray*}
   z & = & \dfrac{x_i -\bar{x}}{\sigma}\ \text{, for $i=1,2, \ldots, n$.}
   \end{eqnarray*}

Same Topic:

Expectation Inter-quartile Range Permutations Range