Averages

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There are three averages in common use: the MEAN the MEDIAN and the MODE.

A location tells us where something is. Given a set of data we need to know roughly where we are, roughly what sort of a size we are talking about : the weight of a flea in grams or the price of a house in Coventry in pounds. We need a number and some units to position us.

Measures of location, averages, do just that; they give an indication of the position of the data; they provide a value round which a set of data is located or distributed.

THE MEAN is the sum of the numbers divided by the number of items there are.

It is denoted by   in textbooks and on your calculator.

 

As a formula it rather frighteningly looks like this     
  S is sigma  
a Greek capital S 
for SUM , i.e. ADD UP		 n
the number of items		 xi stands for x1, x2, …..
all the different values of x
we are summing

The mean is the most commonly used average, and is so common that it has become known as THE average, but it is NOT the only one and is often NOT the most helpful. 

The mean uses all the data, but can be distorted by extreme values.
Think about the hourly pay of 6 women:   £5   £5   £5   £5   £5   £25
Is the mean a helpful indicator of their hourly earnings ?

Another problem is that means are often quoted to several decimal places which don't correspond to any actual value of the data, e.g. the typical and mythical 'average family' has 2.45 children !

Calculating the mean for grouped data.

THE MEDIAN is the value of the MIDDLE item (or mean of two middle items) of a set of data arranged IN ORDER.
It has an equal number of data points above it and below it.

Example:
Here are the weekly household incomes of 7 families: £140,  £194,  £ 63, £312,  £253,  £ 97,  £377

To find the median, arrange the incomes in order:

and take the middle value : £194.

If we add an eighth household income of £196 the median would be £195, since it is the mean of £194 and £196 which are the two middle values.

What is the median of the 8 numbers 6, 1, 5, 10, 1, 2 ,4, 1 ?

 

THE MODE is the most commonly occurring value. The mode of the 8 numbers above is 1.
What is the mode of 3, 5, 7, 8, 10, 4, 6, 10, 2, 10, 2?


The mode is the only average we have to describe categorical data or nominal data, e.g. the colour of the 'average' new car.

Back to the Index of Statistics Pages.

Sidney Tyrrell , MIS, Coventry University, January 2001