## Measures of Central Tendency

Central tendency can be measured in several ways, means, median, mode, and midpoint. We will consider each of these separately.

### Means

• Arithmetic Mean - The arithmetic mean is often referred to as the simple mean or average. This is the average that you are familiar with. It is expressed as follows:

$$\bar{x} = \frac{ \displaystyle\sum\limits_{i=1}^{n} x_{i}}{n}$$

where Xi is the ith variable of interest, n is the number of individuals in the sample.

In Excel® the buildin function:

=AVERAGE(range of data)

Example:
=AVERAGE( A1:A25 )
Which will produce:
Arthimetric mean of the data range

• Geometric mean - The geometric mean is obtained by multiplying all the individuals and then taking the nth root.

$$\bar{x}_{g} = \sqrt[n]{\frac{ \displaystyle\prod\limits_{i=1}^{n} x_{i}}{n}}$$

This can be computed easily by:
$$\bar{x}_{g} = exp \left(\frac{ \displaystyle\sum\limits_{i=1}^{n} ln x_{i}}{n} \right)$$

where exp is the exponential of the result within the brackets and ln is the natural logarithm of Xi. This mean is useful when examining rates.

In Excel® the buildin function:

=GEOMEAN(range of data)

Example:
=GEOMEAN( A1:A25 )
Which will produce:
Geometric mean of the data range

• Quadratic Mean - Quadratic means is quite useful when used with diameters. It can be thought of as the diameter of the individual with average area.
$$\bar{x}_{q} = \sqrt{\frac{ \displaystyle\sum\limits_{i=1}^{n} x^{2}_{i}}{n}}$$

In Excel® the there is no quadratic mean buildin function, See the programming section of this course.

### Median

Median is the middle value in a dataset. If the data set is ordered from smallest to largest value the value that is the (n+1)/2th value in the ordered list is the median.  For even numbered datasets there are two middle numbers. The median is the midpoint between these two middle numbers.

In Excel® the buildin function:

=MEDIAN(range of data)

Example:
=MEDIAN( A1:A25 )
Which will produce:
Median of the data range

### Mode

The mode is the most frequent value in the data set. A data set can have many modes as they are just points of frequent values in the data set.

In Excel® the buildin function:

=MODE(range of data)

Example:
=MODE( A1:A25 )
Which will produce:
Mode of the data range

### Midrange

Midrange is the value half way between the largest value and the smallest value in the dataset. The problem with this measure is that only the largest and smallest values effect the result.

Also See:

Chapter 3 - Measures of Central Tendency pages 25-34 in:

Phillips, J. L. 2000. How to think about statistics. W. H. Freeman and Co. New York. 202 pp. ISBN 0-7167-3654-3

Chapter 3 - Measures of Central Tendency pages 18-23 in:

Zar, J. H. 2007. Biostatistical Analysis. Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 718 pp.