1.1.5 Mean, Median and Mode

Mean

• The mean of a numerical distribution is found by summing up the values of all individual data points, then dividing by the number of data points.
• It is represented by either a capital letter with a bar drawn above it, or the Greek symbol mu (µ):

\bar{X}=\frac{\sum_{i=1}^{N} x_{i}}{N}

Where N is the total number of data points, and represents the i’th datapoint.

Note: the symbol \Sigma is short for “sum of”, so \sum_{i=1}^{N} x_{i} represents the sum of all individual data points (from datapoint 1, to datapoint N)

• The mean is an appropriate measure of the “centre” of a distribution when that distribution is approximately symmetrical.

Example

For the dataset: 2 4 6

The average is: \bar{X}=\frac{2+4+6}{4}=3

Median

• The median is the value of the “middle point” of a distribution when the data points are placed in ascending or descending order.
• If there is an even number of data points, the median is the average of the two middle points.

Example

For the dataset: 1 3 4 5 8

The median is 4, as it is the middle number

For the dataset: 1 8 10 14

The median is the average of the two middle numbers, 8 and 10: \frac{8+10}{2}=9

Mode

• The mode of a categorical distribution is the number or range which occurs the most

Example

For the dataset: 1 3 3 5 7 7 7

The mode is 7, as it occurs the most.