An average summarises a group of numbers. There are three main types of averages: mean, median and mode. These will be looked at in turn.

Mean

This is the most commonly used average. The mean is calculated by taking the sum of the numbers in a sample and dividing that by sample size. This is the only type of average that takes into account all the numbers in the sample.

Example - Here is a sample of numbers:

1,2,2,2,3,3,4,6

The first thing to do is add the numbers up. In this case, the result is 23.

All that is left to do is divide this number by the sample size. In this case, we are dividing 23 by 8 and we get the result 2.875 .

Potential Problems - Mean averages have two linked problems:

1. If you have a large number of small values with a few very large values in your sample, mean averages get skewed: the mean is nearer to the bigger values even though the small values are more numerous. If you have a few small values and a few large values, the mean average can get skrewed this way too.
2. If you have one, or more, outlying values that do not follow the general trend of the numbers in a sample, the mean average can be affected more dramatically than intended.

Example of the Effect of Outliers - For this example, I will take the other sample and add the number 100 into it:

1,2,2,2,3,3,4,6,100

The sum of these numbers is 123. If we divide this by the sample size - 9, we get 13.6666(recurring) which does not represent the earlier numbers.

Median

This type of average is the middle number in a sample and requires the numbers to be in order.

Example - Here is another sample of numbers:

4,2,2,6,3,3,1,2

To use the median, these numbers need to be in order. So here they are ordered:

1,2,2,2,3,3,4,6

Here the median is 2.5 . It comes out as this because there is a even sample size here. Therefore there is no middle number. To work out the median, you need to take the 2 and the 3 which are the middle numbers and get the mean of them - which is 2.5 .

With an odd sample it is much easier: you just take the middle number as the median.

Potential Problems - One problem with using median is that it requires the numbers to be put in order first. For a large set of numbers, this can be extremely labourous and tiring.

Mode

This type of average is the number that occurs the most times in the sample. Where Mean has problems with representativity, mode focuses on the most common numbers and gives less or no attention to less common numbers.

Example - We'll take another sample of numbers here:

1,2,2,2,3,3,4,6

The mode here is 2 as it appears 3 times. Note: If there are two numbers which are equally common in the sample, then you take both as the mode.

Potential Problems - Mode is less useful when you have a lot of values that are close together but have not been rounded to the nearest whole number. This means an inacurate mode of the numbers will be taken. It would be better in this example to round the numbers first before using mode.

Real-world Example

A class of 15 students took a test that was out of 10. 7 students got 8 marks, 4 got 7 marks, 2 got 6 marks, 1 got 5 marks, and 1 got 4 marks.

Mean - The total number of all the marks the students got is 105. Divide that by the number of students, 15, and you get 7. That is the mean number of the students marks.

Median - Below, the marks the students received are shown, in order from highest to lowest:

4,5,6,6,7,7,7,7,8,8,8,8,8,8,8

The median number is the middle number, so the median of the students marks is 7.

Mode - The marks the students received were:

4,5,6,6,7,7,7,7,8,8,8,8,8,8,8

The mode number is the number that occurs the most times, so the mode number of the students marks is 8.

The average mark the students got depends on which average is used.