1 Averages
What does ‘average’ mean?
Did you notice in the introduction we wrote that Britain drinks 100 million cups of tea per day on average? ‘Average’ is an important word here – otherwise we would be suggesting that the same amount of tea is drunk each day.
The word ‘average’ is used in the English language in different ways. It might refer to something typical, something middling, or something commonplace. In the same way the term ‘average’ can be used in different ways in mathematics and statistics.
Average (noun)
1 the usual or typical amount, extent, quality, number, etc.
2 (in) statistics any number that is representative of a group of numbers or other data, especially the arithmetic mean, which is equal to the sum of a set of n numbers, divided by n.
(Chambers, 2025)
School mathematics typically involves three averages: the median, the mode and the mean.
Activity 1 The different averages
Watch this video about a group of pupils deciding which value to use to represent their height. From the video write down how to calculate the median, mode and mean.
Discussion
The median of a set of numbers is the middle value when the data is written out in order.
The mode is the value or item with the highest frequency in a data set.
The mean of a set of data, is when we add all the values together, and then divide by how many values there are. (Note that it should make sense to add values together in the data set, such as numbers or vectors.)
These are all averages.
In the next activity you will practice calculating median, mode and mean, but do not worry if you find this tricky, you will then go on to look at each of the averages in more detail.
Activity 2 Calculating the different averages
Here are the heights of the original 7 pupils in the video in Activity 1.
| Pupil | Height (in centimetres, cm) |
| Ali | 156 |
| Cal | 156 |
| Ed | 161 |
| Gail | 159 |
| Irena | 156 |
| Jen | 158 |
| Stef | 165 |
- Using the information in the video, calculate the median, the mode and the mean of the pupils’ heights. Use a calculator if needed.
- When the 8th pupil, Vikesh, joins, try to predict (without recalculating) what you think will happen to each of the average values you calculated. For each average think about whether it will stay the same, decrease or increase.
Author can you confirm what the correct answer is for each average. We can then build the multiple choice feature. [Note to Ana]
Please make this second part a multiple choice answer for each of the three averages
i.e.
| Answer options | |||
| Median | Stay the same | Decrease | Increase |
| Mode | Stay the same | Decrease | Increase |
| Median | Stay the same | Decrease | Increase |
Answer
For each answer in part 1, the brackets show the time, in minutes, where to find the information in the video.
The median height of the group is Jen’s height, since she is in the middle when the pupils line up in order: 158 cm (0:31 – 1:12)
The modal height of the group is the height of Ali, Cal and Irena, since they all have the same height: 156 cm (1:13 – 2:01)
The mean height of the group is found using the following calculation:
You may have rounded your answer differently. (2:27 – 3:27)
When Vikesh, who is taller than the other pupils, joins…
…the median will increase a little (158.5 cm) since the middle position of the group will now be between Jen and Gail (a common estimate is halfway between the two values)
…the mode will stay the same (158 cm) since there are still more pupils with this height than any other
…the mean will increase but, without more information about Vikesh’s height, we cannot exactly say what the new value of the mean will be.