Misunderstandings of averages

There are many colloquial words that are used for the idea of average, such as 'typical' or 'common' or even the word 'average' itself. The mean, median and mode are the three averages, or measures of centre, that the curriculum focusses upon.

For data sets that are symmetric, the median and the mean will be the same (or nearly so) but for skewed distributions they are different. Activities where students collect their own data and compare the mean and median (and sometimes the mode) can be useful in building intuitions for distinguishing the two measures of centre.

Using the median meaningfully is perhaps the most difficult. The two main difficulties are:

• the median is only used for numerical data that can be ordered
• the data must actually be ordered.

Students can believe that it is possible to find the median of categorical data sets. Activities that challenge students' thinking about mean, median, mode, and the difference between numerical and categorical data are needed.

When focussing on the mean, students at times forget outliers, which can affect the summary statistic or a claim, especially if there is no accompanying graph to attract attention. Activities where outliers are included can be useful.

Sometimes the typical part of a data set (or its middle) is better represented by a range of values rather than a unique number. Students need to be exposed to this idea.