There is an old joke about the Scotsman who moved to England and raised the average level of intelligence in both countries. The arrival at Westminster of 56 Scottish National Party MPs - most of them political novices - may provide a moment to reflect on the implications.
Averages, and other summary characteristics of a statistical distribution, require careful interpretation when the population the distribution describes is changing.
Last week, Bank of England governor Mark Carney drew attention to the possible role of a shift in the composition of the population in explaining Britain's productivity puzzle.
Like most other issues of central relevance, the productivity puzzle received little attention in the recent election campaign. But it is impossible to understand Britain's economic position and prospects without resolving it. Since the 2008 financial crisis, labour productivity - output per hour worked - has fallen. Mr Carney's staff estimate that if productivity had continued to rise at the trend rate prior to 2008, the British population might on average be 17 per cent better off than it is today.
But average productivity can fall even though no individual worker becomes less productive. This may happen if the composition of the population changes, and those who join the labour force have lower productivity than those who are in it already.
The basic facts behind the productivity puzzle are that, since 2008, employment has risen substantially and working hours have increased but output has barely grown.
And the Scottish joke gets it the wrong way round. Economic migrants tend to be people who have above-average productivity in the countries from which they come; the skilled and enterprising are often those who leave. But they may have below-average productivity in the country in which they arrive, where they may struggle to fulfil their capacities in a foreign environment.
Still, Mr Carney was rightly unwilling to be drawn into blaming Britain's productivity performance on immigrants. The main cause of the rise in employment is not that more foreigners are working in Britain but that more British people are working in Britain.
What have these additional workers been doing? Professor Jonathan Haskel and his colleagues at Imperial College London argue that they have not been employed in low-productivity industries, nor have they been particularly low-skilled.
Yet the Bank of England's analysis is that - especially in the past three years - an increasing proportion of employment growth has been in low-skilled jobs.
Perhaps the abilities of the workforce are underused. Or perhaps easy monetary conditions have enabled low-productivity companies, which disappear in normal recessions, to survive. Any explanation of the productivity puzzle must be found by digging into the detail of how aggregate statistics are built up.
This composition problem is evident elsewhere. Median household income in Britain has recently been static, and in the United States has barely risen since the 1970s.
But it would be wrong to jump to the conclusion that the typical household is no better off. If the population is changing, median income can fall even if everyone is more prosperous.
Additional households - young people leaving home, immigrants or older people increasingly able to live on their own - are mostly in the lower half of the distribution.
The Median Isn't The Message was the title of a very personal essay by science writer Stephen Jay Gould. He was diagnosed with mesothelioma in 1982, and learnt that the median survival time for those with this illness was eight months.
But Gould realised that the underlying distribution necessarily had a long tail, understood compositional effects and determined that his prognosis was better than average. He died in 2002 from an unrelated illness.