2.1.1 Averages – mean and median

This section covers two different types of averages – the mean and the median.

An average value is often used to provide a ‘typical’ value for a set of data. There are several ways of expressing a typical value, but the most common are the mean and the median.

In this study note you will look at two different types of averages – the mean and the median.

Mean values

The mean value (commonly called the ‘average’, although it is just one type of average) is obtained by adding up all the values of a set of data and dividing by the number of items in the set. For example, if three individuals have carbon footprints of 13, 9 and 17 tonnes CO₂e per year, their total footprint is 13 + 9 + 17 = 39 tonnes CO₂e per year. Their mean footprint is the total divided by the number of individuals, i.e. 39/3 = 13 tonnes CO₂e per person per year.

To calculate the mean carbon footprint of a UK inhabitant (in 2019) using the information in Figure 4, divide the total UK emissions of 774 million tonnes CO₂e by the 2019 UK population of 66.4 million. The result is 11.67 tonnes CO₂e per person per year rounded to one decimal place. (See Rounding numbers, decimal places and significant figures later in this section.)

Median values

Although the example above gave a value for the amount of CO₂e emissions that could be attributed to each person on an equal basis, this may not represent a ‘typical’ person. This is because there may be a group of heavy emitters that have a larger effect on the total (and hence the mean) than the majority of more modest emitters. In this case the median is a better indication of a typical individual’s footprint.

So, if the carbon footprints of a sample of 11 individuals were 21, 6, 15, 16, 10, 9, 13, 25, 12, 11, 27 tonnes CO₂e per year, the mean footprint is 165 tonnes CO₂e per year / 11 = 15 tonnes CO₂e per year.

The median footprint is obtained by arranging the data into numerical order, as follows:

6, 9, 10, 11, 12, 13, 15, 16, 21, 25, 27 tonnes CO₂e per year the median footprint is the middle value in the set = 13 tonnes CO₂e per year, i.e. the sixth value in the set, with five below and five above it.

If there are an even number in the sample, say 10 individuals with footprints of

9, 10, 11, 12, 13, 15, 16, 21, 25, 27 tonnes CO₂e per year

then the median is halfway between the two middle numbers (13 and 15), i.e. 14 tonnes CO₂e per year, again with five values below and five above.

Consider another example, the UK’s income distribution. Some people have very high incomes and some have low incomes, with the majority spread between the extremes. Figure 6 shows the official distribution of the UK population by household incomes (after tax and before deduction of mortgages and rents) in 2021–22. The mean household income in 2021–22 is £688 per week , calculated by adding up all the incomes and dividing by the number of individuals living in households whose incomes were measured.

The median income is the amount which divides the income distribution into two equal groups, half having an income above that amount, and half having an income below that amount. The median household income in 2014–15 is £473 per week (£24 596 per year), which is less than the mean and more representative of an ‘average’ or ‘typical’ household.

Maximise

Figure 6 Distribution of the UK population by household incomes in 2021–22. Group 1 is the distribution of household incomes for the 10% of individuals living in the poorest households. Group 9 is the household incomes for the 10% of individuals living in the richest households (excluding 8.8 million individuals with a household income above £1000 per week). Equivalised means the incomes are adjusted to take account of the numbers of adults and children in a household (adapted from DWP, 2023).

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This is a bar chart showing the official distribution of the UK population by net household incomes in 2021–22. The height of each bar shows the total number of individuals in the UK who live in a household that has a particular weekly income, from the lowest group (more than five hundred thousand individuals with a net household income of 0 to £10 per week) to the highest (nearly two hundred thousand individuals with a net household income of £990 to £1000 per week). The 100 bars on the chart are divided into nine alternating purple and orange groups, numbered 1 to 9 and each representing 10 per cent of the population. The caption notes that the richest 10 per cent is not included, but consists of 8.8 million UK individuals living in households with incomes of over £1000 per week. Labels on the graph note that the mean net income is £688 per week (which is just under halfway through group 7) and the median net income is £565 per week (which is nearly at the end of group 5). Between £20 and £200 a week, the number of individuals increases gradually from about 50 thousand to just over 200 thousand. From £200 a week to £500 a week, the number of individuals increases more rapidly to about 1.1 million. The number of individuals in each income band then gradually decreases to just under 200 thousand in the highest income group shown (£1000 per week).

Figure 6 Distribution of the UK population by household incomes in 2021–22. Group 1 is the distribution of household incomes for the 10% ...