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.

goto:
Arithmetic Mean   Median   Mode   Geometric Mean   Harmonic Mean   Weighted Mean

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 may occurr that several values are tied.

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
n_j = size of sample j
= AM of sample j