Measures of Central Tendency

Statistical measures of central tendency or central location are numerical values that are indicative of the central point or the greatest frequency concerning a set of data. The most common measures of central location are the mean, median and mode.

The statistical mean of a set of observations is the average of the measurements in a set of data. The population mean and sample mean are defined as follows:

Given the set of data values x₁, x₂, .... x_N from a finite
population of size N, the population mean m is calculated as

Given the set of data values x₁, x₂, .... x_n from a sample of
size n, the sample mean is calculated as:

The sample mean is often used as an estimator of the mean of the population from whence the sample was taken. In fact, the sample mean is statistically proven to be a most effective estimator for the population mean.

Median

The median of a set of observations is that value that, when the observations are arranged in an ascending or descending order, satisfies the following condition:

1. If the number of observations is odd, the median is the middle value.
2. If the number of observations is even, the median is the average of the two middle values.

The median is the same as the 50th percentile of a set of data. It is denoted by .

Mode

The mode of a set of observations is the specific value that occurs with the greatest frequency. There may be more than one mode in a set of observations, if there are several values that all occur with the greatest frequency. A mode may also not exist; this is true if all the observations occur with the same frequency.

Another measure of central location that is occasionally used is the midrange. It is computed as the average of the smallest and largest values in a set of data.

 Example of Central Tendency

EX. Given the following set of data

1.2, 1.5, 2.6, 3.8, 2.4, 1.9, 3.5, 2.5, 2.4, 3.0

It can be sorted in ascending order:

1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8

The mean, median and mode are computed as follows:

= (1 / 10) · (1.2 + 1.5 + 2.6 + 3.8 + 2.4 + 1.9 + 3.5 + 2.5 + 2.4 + 3.0)
= 2.48

= (2.4 + 2.5) / 2
= 2.45

The mode is 2.4, since it is the only value that occurs twice.

The midrange is (1.2 + 3.8) / 2 = 2.5.

Note that the mean, median and mode of this set of data are very close to each other. This suggests that the data is very symmetrically distributed.