Dispersion

Dispersions are estimates of the variability of measurements in a given distribution.


Range

The range defines the distribution's width from the minimum to the maximum and is defined as follows:

Range = xmax - xmin


Percentiles

The n-th percentile is the value which cuts n% of the distribution.


Variance and standard deviation

Variance and standard deviation are most commonly used to describe the variability of a given distribution.

Requirements:

The individual measures must be at least interval scaled.

Calculation:


variance

std. dev.

For the estimation of a population variance the variance of a sample distribution must be corrected as follows:


est. pop. variance

The same holds for the estimated standard deviation of a population


est.pop. std. dev.

Standard error of the mean

This is a measure for the accuracy of the estimated population mean.


standard error of mean


if sigma pop. is estimated from the sample:


SEM

Skewness

Skewness is a measure for the symmetry of a given distribution.
A negative skew means that the left tail of the distribution is elongated (AM > Median > Mode)
A positive skew means that the right tail of the distribution is elongated (AM < Median < Mode)

Examples:


positive skewness


negative skewness

Calculation:

Skewness is the third standardized moment of the distribution and is computed as follows:

Sample skewness:


skewness

The estimation of the population skewness from a given sample is:


pop.skewness

Kurtosis

Kurtosis is a measure of the “peakedness” of a given distribution. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations.

Calculation:

Sample kurtosis:


kurtosis

The estimation of the population kurtosis is as follows:


pop. kurtosis

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