4 Making sense of data

4.8 Mean, median and mode

Most of us are familiar with the word ‘average’. We regularly encounter statements like ‘the average temperature in May was 4 °C below normal’ or ‘underground water reserves are currently above average’. The term average is used to convey the idea of an amount, which is standard; typical of the values involved. When we are faced with a set of values, the average should help us to get a quick understanding of the general size of the values in the set. The mean, median and mode are all types of average, which are typical of the data they represent. Each of them has advantages and disadvantages, and can be used in different situations. Here we provide brief definitions and some idea of how each can be used.

4.8.1 Mean

The mean is found by adding up all the values in a set of numbers and dividing by the total number of values in the set. This is what is usually meant by the word ‘average’.

For example, if a company tests a sample of the batteries it manufactures to determine the lifetime of each battery, the mean result would be appropriate as a measure of the possible lifetime any of the batteries and could be used to promote the product.

Activity 12

A sample of 15 batteries was tested. The batteries lasted, in hours:

23, 28, 25, 30, 18, 29, 23, 19, 30, 24, 25, 21, 26, 25, 23

  1. Calculate the mean life of a battery, correct to one decimal place (i.e. with one figure after the decimal point).
  2. How many hours of life will the company claim?

Discussion

  1. Adding up all the values gives 369. Dividing 369 by 15 gives 24.6.
  2. The company is likely to claim an average battery life of 25 hours (24.6 rounded up to the nearest hour).

However, the mean is not always a good representation of the data. To illustrate this, consider the annual salaries of the people employed by a small company. The salaries, in euros, are:

22 000, 22 000, 25 000, 28 000, 35 000, 65 000, 92 000

The mean salary = 289 000 ÷ 7 = 41 286 euros (approximately). Here, however, the mean is very misleading because the two highest salaries raise the mean considerably. You might like to confirm this by calculating the mean of just the five lowest salaries.

4.8.2 Median

The median is the middle value of a set of numbers arranged in ascending (or descending) order. If the set has an even number of values then the median is the mean of the two middle numbers. For example:

1, 1, 2, 5, 8, 10, 12, 15, 24 This set of nine values is arranged in ascending order and the median is 8.
32, 25, 20, 16, 14, 11, 9, 9, 4, 0 This set of 10 values is arranged in descending order and here the median is (14 + 11) + 2, which is 12.5.

The salaries considered in the discussion above are in ascending order and the median is 28 000 euros. This value is more representative of the state of affairs in the company than the mean (41 286 euros) as half the employees earn less than the median and half earn more.

Activity 13

Consider the scores of one batsman during the cricket season:

0, 0, 0, 4, 7, 10, 12, 15, 24, 30, 42, 48

Calculate the median and the mean of this set of scores.

Discussion

The median = (10 + 12) + 2 = 11 The mean = 192 ÷ 12 = 16

The median is not appropriate here as the number of runs may vary greatly during the season and will probably include a number of zeros. The average actually used to rank players is the mean.

4.8.3 Mode

The mode, or modal value, is the most popular value in a set of numbers, the one that occurs most often. However, it is not always possible to give the mode as some sets of values do not have a single value that occurs more than each of the others. Like the median, the mode can help us to get a better feel for the set of values. Retailers find the mode useful when they want to know which item to restock first.

Activity 14

The following list is the number of goals scored by a football team in a series of 18 games:

0, 0, 0, 3, 0, 1, 1, 2, 0, 2, 0, 0, 4, 0, 1, 2, 0, 0

Calculate the mode, median and mean for this list.

Discussion

Arranging the values in ascending order gives:

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4

The mode (the value that appears most frequently in the list) = 0

The median (here the mean of the two middle numbers) = (0 + 1) ÷ 2 = 0.5

The mean (total of all the goals scored divided by the number of games) = 18 ÷ 18 = 1

This football team would probably prefer us not to quote the mode as their average score during a season as it is zero!