Title: The Mean
Course: UGQuants (SSPS) > Section: Univariate Analysis

The Mean

The mean (or more properly the ‘arithmetic mean’) is often the number to which people are referring in everyday life when they use the term ‘average’. For example, ‘the average number of goals scored in the football world cup was 2.5’, or ‘the average height of children in the class was 1.6 metres’. To calculate the mean, we add together all the values for avariable and divide by the number of cases. In doing this, we usually want to exclude those cases which, for whatever reason, did not provide data for that variable (for a further discussion see section x.xx on missing data). So, for example, in the 2010 European Social Survey for the UK, 2,413 respondents stated their age in years, and the total of all their ages (in statistical terms, usually described as the ‘sum’) was 120,639. Therefore:

Mean age in years = 120639 / 2413 = 50 (to the nearest year)

Note that since all respondents to this survey were adults aged 18 years or over, the mean does not give a good indication of the likely mean age of the population of the UK at the time of the survey. Note also that the actual value for mean age was not exactly 50, but 49.995 (rounded to three decimal places). This highlights a significant feature of the mean: in trying to calculate a figure that most closely summarises the age of all respondents, we often end up with a figure that does notprecisely represent any one case, since no cases have a value for age of 49.995.

In fact in this example it is possible that one of the cases does actually have an age very close to 49.995, since it is quite legitimate to measure years in fractions of a year. But imagine we were expressing the mean number of children in households in Britain: it is likely that this would not be a whole number, but a decimal such as 1.6, even though it is not possible to have .6 of a child. This is worth bearing in mind when presenting even the simplest data such as a mean value: should you round the final value so that it does represent a value that could be meaningful for any individual case, or in doing that would you lose so much precision that it would result in a quite misleading picture? It might be impossible to have 1.6 children, but rounding the figure to 2 would give a very misleading representation of the number of children per household.

Some further examples of mean values for ratio level variables are as follows:

Even these basic figures give us useful immediate evidence – a ‘model’ – of the amount of time people people in Europe typically spend on paid work each week; the extent to which ‘teenage’ motherhood might be prevalent across the world; and the degree to which women of the world are represented in democratic institutions.

Why is it not legitimate to calculate the mean for nominal or ordinal variables? For nominal variables, this should be obvious: because the values for these variables have no numerical significance other than being a way of distinguishing one category from another, it does not make any sense, e.g. to say that the ‘mean’ sex is 1.1, or that the ‘mean’ party voted for at an election was 3.2. While this is never done (at least not by anyone who has reached the most basic level of understanding), the situation with regard to ordinal variables is a little more complex. For example, with an ordinal variable that expresses level of interest in politics on a range from 1 = very interested to 4 = not at all interested, we might argue that it is legitimate to express a mean value for this variable of, e.g. 2.5, which would suggest that ‘mean’ interest in politics is somewhere between ‘quite interested’ and ‘not very interested’. But this would rely on the (questionable) assumption of an equal difference between any pair of adjacent values of the variable. For other ordinal variables, calculating the mean seems even more questionable. For example, for highest educational qualification, calculating that, e.g., the ‘mean’ qualification lies somewhere between post-16 and pre-16 school qualifications does not seem a very informative way of representing the data.