MathGloss/S/Standard deviation

Examples
Suppose we are interested in the long-jump performance of young adult males. We design an experiment by randomly selecting 100 male students, aged 18-22, to perform the standing long jump. For ease of calculations in this example, we will use the distanced jumped for 8 of the 100 students:

To find the standard deviation of these 8 distances:

1. Calculate the mean of the 8 data points:


 * $$\overline{x}=191$$

2. Calculate the sum of the squared differences of each data point and the mean, $$\sum_{i=1}^8 {(x_i-191)^2} $$:
 * The squared differences for each data point:
 * $$x_1=152\quad\longrightarrow\quad(152-191)^2=1\,521$$
 * $$x_2=162\quad\longrightarrow\quad(162-191)^2=841$$
 * $$x_3=173\quad\longrightarrow\quad(173-191)^2=324$$
 * $$x_4=188\quad\longrightarrow\quad(188-191)^2=9$$
 * $$x_5=193\quad\longrightarrow\quad(193-191)^2=4$$
 * $$x_6=198\quad\longrightarrow\quad(198-191)^2=49$$
 * $$x_7=203\quad\longrightarrow\quad(203-191)^2=144$$
 * $$x_8=269\quad\longrightarrow\quad(269-191)^2=6\,084$$


 * The sum of the squared differences:
 * $$\sum_{i=1}^8 {(x_i-191)^2}=1\,521+841+324+9+4+49+144+6\,084=8\,976$$

3. Divide the resulting sum by $$n-1$$ and take the square root of the result:
 * $$SD =\sqrt{{1 \over 8-1}\cdot 8\,976} = 35.929$$

Attribution