# Standard deviation

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Definition
 Standard Deviation quantifies the spread of a distribution of data points by measuring how far the points are from their mean, $\overline{x}$. is the average (or typical distance) between a data point and the mean, $\overline{x}$. A low standard deviation indicates that the data points are clustered around the mean value, whereas a high standard deviation indicates that the data points are widely spread with points significantly higher and lower than the mean, $\overline{x}$. In most real-life situations, the standard deviation is estimated based on a sample taken from the population. There are many notations for the sample standard deviation: $SD, S, Sd, StDev. \,$ The sample standard deviation is mathematically defined as: $SD = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}\,,$ where $\{x_1,\,x_2,\,\ldots,\,x_n\}$ are the data points in the sample and $\overline{x}$ is the mean of the sample.

## 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:

 152 162 173 188 193 198 203 269

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:

$SD =\sqrt{{1 \over 8-1}\cdot 8\,976} = 35.929$