Averages and Ranges

There are three types of average:

Mean

To find the mean we add all the numbers together and divide by how many numbers we have.

Example: If we have the numbers 3, 4, 4 and 9 (so we have 4 numbers). So the mean would be

^(3+4+4+9)/₄ = ²⁰/₄ = 5

Median

To find the median we first put the numbers in size order and then the median is then the middle number.

Example: If we have the numbers 7, 9, 2, 3, 3 first we put them in size order: 2, 3, 3, 7, 9. Then the median is the middle number:

2, 3, 3, 7, 9

Mode

This is just the most common number (note: we can have two modes, or three modes or no modes if there’s the same number of each).

Example: We have the numbers 2, 2, 4, 5, 6, 9, 9, 9. There are more nines than any other number so

mode = 9

Next we consider ranges, here there are only two: the range and the interquartile range.

Range

Simply the difference between the biggest and smallest number.

Example: We have the numbers 2, 2, 3, 3, 5, 9, 90. The range is simply

90 – 2 = 88

Interquartile Range

This is the difference between the middle half of the data (i.e. between a quarter way of the through the data, the lower quartile, and three quarters way through the data, the upper quartile ). This is a far more reliable test of the spread of data because it means one or two outliers can’t skew the result like it can with the range.

Example: Consider the same data as above 2, 2, 3, 3, 5, 9, 90. The range was 88 which suggests the data was really spread out when in fact all the data is quite close together except one which is much bigger than all the others. If we consider the interquartile range:

2, 2, 3, 3, 5, 9, 90

3 is the middle number (i.e. the median) now what we do to find the upper quartile is effectively find the median of all the numbers greater than the median and for the lower quartile we find the median of all the numbers below the median.

So the upper quartile is 9 and the lower quartile is 2 so,

Interquartile range = 9 – 2 = 7

We can see this far more accurately reflects the small spread of the data barring the outliers.