# Standard deviation

- Calculation from the entire population of the series

- Calculation from a sample of the series

Enter the numbers of the series separated by a space.

## Standard deviation

In statistics, the standard deviation is a measure of the dispersion of a series values relative to the mean. Denoted `sigma` (Greek letter sigma), it is both equal to:

- the square root of the variance

- the quadratic mean deviations from average

Standard deviation is calculated (or estimated) differently depending on whether the available data relates to the entire population or only a sample of the population.

**Calculation of standard deviation from entire population"**

In this case, values are available for the entire population. The calculation of the standard deviation is direct from the above definition:

X is a dataset,

`X = {x_1, x_2, ..., x_n}`

We note `bar x` the average of the X series, then, `bar x = 1/n.sum_{i=1}^{i=n}x_i`

The standard deviation is then defined as follows,

`sigma = sqrt(1/n.sum_{i=1}^{i=n}(x_i-barx)^2)`

Example: `X = {1, 2, 5, 3,8}`

First, we calculate the average,

`bar x = 1/5.(1+2+5+3+8) = 3.8`

Therefore, the standard deviation is,

`sigma = sqrt(1/5( (1-3.8)^2+(2-3.8)^2+(5-3.8)^2+(3-3.8)^2+(8-3.8)^2)) approx 2.48`

**Estimation of standard deviation from a sample**

In this case, observations are not available for the entire population but only for a sample. The standard deviation cannot be calculated directly from the above definition. We use what we call an estimator.

The most commonly used estimator for the standard deviation is as follows:

X is a dataset representing a sample of the entire population,

`X = {x_1, x_2, ..., x_n}`

The sample mean is estimated by `bar x = 1/n.sum_{i=1}^{i=n}x_i`

The standard deviation is estimated as follows,

`sigma = sqrt(1/(n-1).sum_{i=1}^{i=n}(x_i-barx)^2)`

Example: `X = {1, 2, 5, 3,8}`

X represent observations for a randomly drawn sample from a population.

We calculate first the average of the sample,

`bar x = 1/5.(1+2+5+3+8) = 3.8`

An estimator of the standard deviation is calculated as follows,

`sigma = sqrt(1/4( (1-3.8)^2+(2-3.8)^2+(5-3.8)^2+(3-3.8)^2+(8-3.8)^2)) approx 2.77`

## See also

Statistical Variance

Quadratic mean

Arithmetic mean