Standard deviation

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This tool calculates the standard deviation of a statistical series. There are two options to calculate the 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