Hidden side to satisfaction

The contention that smaller institutions seem to do better than larger ones for student satisfaction scores in the National Student Survey (“Non-Russell Group institutions lead in satisfaction stakes”, News, 14 August) falls into the trap of what the statistician Howard Wainer calls “the most dangerous equation”. This is Abraham de Moivre’s equation, which describes how the averages of small samples vary more than those of large samples. When you are ordering averages from different sized groups, all other things being equal, you would expect to see smaller groups disproportionately represented at both the top and the bottom.

Wainer gives a nice example of how this misunderstanding can cost large amounts of money: having noticed that the top of school league tables was often dominated by smaller schools in the US, billions of dollars were spent on encouraging smaller schools, until it was noticed that smaller schools also dominated the bottom of the tables.

The same happens here: a simple plot of satisfaction score against number of responses in the most recent NSS results shows that smaller institutions dominate the bottom as well as the top. Indeed, the plot also suggests a small but positive relationship between size and satisfaction score.

There may be many very good reasons for preferring smaller institutions, but De Moivre’s equation isn’t one of them.

Adrian Simpson
Josephine Butler College, Durham University