Standard deviation - Definition


The square root of the variance is the standard deviation. It is an easier measure of dispersion as it is expressed in the same units as the observations. The standard deviation of the population is indicated by the Greek symbol sigma.
For a sample only the estimate or sx is available. The formula for a sample of size n is:

The x with the line above it is the arithmetic mean of the x values.

A valid question would be: "why the square of the deviations from the mean?". It would also be possible to take the sum of the absolute differences as a measure of the spread in the data. That is true, but that is a rather awkward tool in further statistical calculations. Another way to avoid that the sum of deviations from the mean does not result in zero is taking squares. Actually the variance is the important measure of "dispersion" of the data. The variance is derived from calculating the moments of the distribution, a process that also leads to the skewness and kurtosis parameters.

The easy way to calculate it, in contrast to the above formula, is shown in the following example:

Porosity, % Squared porosity
SumSum of squares

There are 8 observations, n = 8, and the sum of x and the sum of x2
Now use:

Resulting in sx = 32.034

The uncertainty of a sample standard deviation

With a small sample the theoretical formula as used above gives an unbiased result. The theory also gives the "standard deviation of the standard deviation" as:

The uncertainty of the sample standard deviation can also be estimated by resampling schemes,such as the jackknife and the bootstrap. The the estimate is more efficient, but at the cost of possiby some bias.