# Central Tendency

Central tendencies are estimates of the most probable value of a given distribution. There are many different ways to compute the central tendency of a given distribution. The choice of the right central tendency depends mainly on the scale level of the measurements.

## Arithmetic Mean

The arithmetic mean is most commonly used to designate the central tendency of a given distribution.

Requirements:

The individual measures must be at least interval scaled.

Calculation:

in words:
The arithmetic mean is the sum of all observed values divided by the number of observed values.

## Median

The median describes the 50 % point of a given distribution.

Requirements:

The individual measures must be at least ordinal (rank-order) scaled.

Calculation:

All measures are brought into a rank order from the smallest to the largest. If the number of measures N is even:

If the number of measures N is odd:

## Mode

The Mode is the value which has the largest observed frequency. It is possible that there are more than one mode.

Requirements:

No requirements. The mode is also applicable to nominal scaled values.

## Geometric Mean

Geometric mean is used for instance to calculate average growth rate.

Requirements:

All individual measures must be positive and at least ratio scaled.

Calculation:

in words:
The geometric mean ist the n-th root of the product of all individual measures.

## Harmonic Mean

The harmonic mean is used when the individual measures designate sort of indices, for example miles per hour, dollars per week, inhabitants per square mile…

Requirements:

The individual measures must be positive and at least interval scaled

Calculation:

## Weighted Mean

Sometimes it is useful to average the means of one criterion collected from several samples. The weighted mean is the mean of the sample means.

Calculation:

whereas:

k = number of samples
nj = size of sample j
= AM of sample j