Measures of Dispersion

Dispersion is the variability around the central tendency.

Absolute dispersion is the amount of variability present without comparison to any reference point or benchmark.

The Range

The range is the difference between the maximum and minimum values in a dataset:

Range = Maximum value – Minimum value

Mean Absolute Deviation (MAD)

where $\bar{X}$ is the sample mean, n is the number of observations in the sample, and the | | indicate the absolute value of what is contained within these bars.

Sample Variance and Sample Standard Deviation

Variance is defined as the average of the squared deviations around the mean.

Standard deviation is the square root of the variance.

Sample variance or standard deviation, $s^2$, is:

where $\bar{X}$ is the sample mean and n is the number of observations in the sample.

Sample Standard Deviation

where $\bar{X}$ is the sample mean and n is the number of observations in the sample.

Downside Deviation and Coefficient of Variation

Downside risk, for example, returns below the mean or below some specified minimum target return.

Downside Deviation

The target downside deviation, also referred to as the target semideviation, is a measure of dispersion of the observations (here, returns) below a target.

The target semideviation, $S_{Target}$, is:

where B is the target and n is the total number of sample observations.

Coefficient of Variation

Relative dispersion is the amount of dispersion relative to a reference value or benchmark.

The coefficient of variation (CV) is the ratio of the standard deviation of a set of observations to their mean value:

where s is the sample standard deviation and $\bar{X}$ is the sample mean.