Averages: mean, median, mode, range

This is a naive collection of mathematical tools for analyzing data and computing averages on it. You have a set of numbers, on which you apply some operations.

Mean

The mean is the average of all the numbers in your set. Add up all the numbers and divide the sum by the size of the set.

§ x̄ = (x_1 + x_2 + ... + x_n) / n §

The line over x is a bar (aka overbar): a horizontal line written above a mathematical symbol to give it some special meaning. In this case it represents the mean of the set x. Another way of writing it:

§ x̄ = 1/n sum_{i=1}^{n} x_i §

For example, I calculate the mean of the following set x:

§ x = {1, 5, 4, 3, 2, 8} §

§ x̄ = (1 + 5 + 4 + 3 + 2 + 8) / 6 = 3.8333... §

Median

The median is the number in the middle of you sorted set. Yes, the set must be sorted first. If the set has a even amount of numbers, you can't pick the central point. In such case the median is the average of the two middle values. Example:

§ x = {1, 2, 3, 4, 5, 8} §

§ "median(x)" = (4 + 3) / 2 = 3.5 §

Mode

The mode is the number that occurs more often in your set. Example:

§ x = {1, 2, 2, 3, 4, 5, 8} §

§ "mode(x)" = 2 §

A set can have more than one mode:

§ x = {1, 2, 2, 3, 4, 5, 8, 10, 10, 10} §

§ "mode(x)" = 2, 10 §

If no number is repeated, then there is no mode for the set.

Range

The range is the difference between the highest and lowest values of a set. Example:

§ x = {1, 2, 3, 4, 5, 8} §

§ "range(x)" = 8 - 1 = 7 §

Sources

Wolfram Mathworld - Bar (link)
Cs.mtu.edu - Computing the Median (link)
Mathisfun.com - How to Find the Median Value (link)
Mathisfun.com - The Range (Statistics) (link)
Purplemath.com - Mean, Median, Mode, and Range (link)