Mean

We'll start by calculating the 'average' or 'mean' of a set of numbers.

You can use the terms 'mean' and 'average' interchangeably. Strictly speaking the 'mean' is the correct term. The 'average' has a more general meaning, it can be used to refer to any value that typifies a set of numbers.

You may already know how to calculate the mean, but I want to use the explanation to make a few points. Suppose you want to calculate the mean of five values, say:

1

8

3

6

2

You add them up and divide by the number of values:

I’ve introduced a bit of notation, I’ve represented the mean by the symbol . The ‘bar’ over the 'x' is a standard way of representing a mean.

We say x-bar and that's the way I'll write it in the text from now on.

Now let’s introduce some more notation. We’ll represent the number of values (five in this case) by the variable ‘n’. Although we could use other letters, ‘n’ is the usual first choice when we want to represent the number of values in a sample.

We’ll also substitute the individual values by ‘x_i’, where the subscript ‘i’ takes a number to represent each individual value (‘i’ stands for ‘index’):

This formula can be used to find the mean of any number of values. Just substitute the symbols x₁, x₂ and so on up to x_n with the actual data values.

We can introduce yet more mathematical notation to make it more concise:

Let’s look more closely at the bit on top (the numerator):

The symbol ‘‘ is the Greek symbol capital Sigma, and means ‘summation’ in 'mathspeak':

In full the notation means ‘add all the values of xi from ‘i’ equals 1 to ‘i’ equals ‘n’ (run your mouse over the image to see another view of this). Note that the ‘i = 1’ and ‘i = n’ are sometimes omitted, if they are self-evident.

	The times taken to repair breakdowns of critical equipment, in hours, over the past week were as follows: 5 4 3 2 5 5 What is the mean repair time?