# 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 = x_{max} - x_{min}

**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:**

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

The same holds for the estimated standard deviation of a population

**Standard error of the mean**

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

if is estimated from the sample:

**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:

**Calculation:**

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

Sample skewness:

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

**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:

The estimation of the population kurtosis is as follows: