Revision Notes

Why do we have different types of average?

You’ll hear the phrase “on average” used a lot, from politicians talking about the economy to sports analysists to shops talking about their “average customer”. However not all data is numerical (eg the party people voted for in the last election) and even when it is numerical, some of the data may lead to misleading results. This is why we have three types of average.

What do I need to know?

We are sure you wouldn’t have got this far without hearing about mean, median and mode!

1. Mean
This is what is usually meant by “average” – it’s like an ideal world where everybody has the same, everything is shared out equally. It is the TOTAL of all the values DIVIDED by the NUMBER OF VALUES.

eg Find the mean of 4, 6, 7, 9
4 + 6 + 7 + 9 = 26
26 ÷ 4 = 6.5
Mean = 6.5

Problems with the mean occur when there are one or two unusually high (or low) values in the data (outliers) which can make the mean too high (or too low) to reflect any patterns in the data.

2. Median
This is similar to the word medium, which can mean in the middle. So the median is the middle value – but beware, the data has to be arranged into numerical order first.

eg Find the median of 20, 43, 56, 78, 92, 56, 48
In order: 20, 43, 48, 56, 56, 78, 92
To find the median cross out numbers from either end until you meet in the middle (cross them out lightly so you can still read them). This may not be necessary with small lists but is more important when working with lots of data.
20, 43, 48, 56, 56, 78, 92
20, 43, 48, 56, 56, 78, 92
20, 43, 48, 56, 56, 78, 92
Median = 56

We would use the median instead of the mean if we did not want extreme values (outliers) affecting our data.If we have an even number of values we would get two values in the middle. In these cases we take the half way point between these two values. This is usually obvious but if it not we add the two middle values and divide by 2 (this is the same as finding the mean of the middle two values).

eg 20, 43, 46, 48, 56, 56, 78, 92 (as above with an extra 46 in there!)

When crossed out we get
20, 43, 46, 48, 56, 56, 78, 92
So the two middle values are 48 and 56. Halfway is 52 but if you can’t spot that you can work it out …
48 + 56 = 104
104 ÷ 2 =52
Median = 52

3. Mode
Not all data is numerical and that is where the mode comes in.
MOde means the Most Often.
So is often used for things like “favourite …” or “… sold the most” or “… were the most popular”.
Mode is sometimes referred to as modal – so you may see phrases like “modal value” – but they still mean the mode.

eg 12 people were asked about their favourite crisp flavour. The responses are below.
Salt and vinegar   Prawn cocktail     Ready salted
Ready salted         Salt and vinegar  Salt and vinegar
Smokey bacon      Ready salted        Salt and vinegar
Salt and vinegar   Cheese and onion Ready salted

With only a few pieces of data it is quite quick and easy to see here that Salt and vinegar is chosen the most.

With more data it may be wise to create a tally chart or similar to help count the number of each flavour
Mode is Salt and vinegarBe aware that the mode can apply to numerical data as well (from the data used in the example for the median the mode would have been 56).

Sometimes if no value/data occurs more often than others we say there is no mode. If two values occur the most we may say there are two modes (bi-modal). Whether it is appropriate to do this will depend on what the data is about.

Example

Mean, Median & Mode Example 2, IGCSE & GCSE Maths revision notes
Mean, Median & Mode Example 1, IGCSE & GCSE Maths revision notes

Question

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