Following the election being called on Tuesday, we were unable to resist the opportunity to delve into some electoral statistics. We consider one of the stranger features of our Electoral System: Labour need fewer votes to win control of Parliament than the Conservatives. Our roughly modelled estimate is that to secure an outright majority, the Conservatives need to win 38% of the vote. And yet Labour could hold on to power with just 34% of the vote. That may seem quite strange enough. But a further observation is that in the presence of equal competition from Labour and the Conservatives, the Liberal Democrats would need a huge lead over both parties to gain control of parliament. According to our modelling they would need the support of 37% of the population, and at the same time Labour and Conservative support levels would both need to fall to around 26%. There are other published models which are even more pessimistic for the Liberal Democrats. Why should this uneven situation arise? To find out, we started by drawing some histograms. The numbers of votes attained by each party across the 646 constituencies in 2005 is shown here.

The histograms and the underlying statistical distributions are obviously very different in character, having different statistical “skewness”. In general, a distribution has negative skewness if more than 50% of the distribution takes values above its mathematical average or mean. This tends to mean having a peak on the right hand side of the distribution. A distribution has positive skewness in the reverse situation (i.e. a peak on the left of hand side). It is surprisingly rare to find skewness at the heart of an issue. However… In Labour’s case, the histogram has a peak towards its right (negative skewness), reflecting the fact that in 381 out of 646 constituencies (59%) the Labour vote was above the overall Labour average of 14800. 14800 is enough to put a party in good contention to win a seat. So Labour support was healthily concentrated in a large number of areas where it held quite high levels of support; and this was a key contributor to winning the election. In the Conservative party’s case, the diagram is more symmetrical, with no clear peak or skewness. This situation was less fortuitous than Labour’s, reflecting – on the left of the diagram – a higher concentration of votes in seats where their number was insufficient for the party to win overall. Also – on the diagram’s right – there are 51 seats with over 25,000 votes. This was bad news for the Conservatives, reflecting some seats where their support was so concentrated that other areas got deprived of it. Why the difference between the Labour and Conservative situations (and skewness)? The simplified summary is as follows: The Boundary Commissions create constituency boundaries with reference to a set of criteria that cover a number of geographical and distributional factors (but which do not cover creating election outcomes that are in proportion to vote numbers). Rural & county seats often have large natural Conservative majorities. Metropolitan constituencies tend to have a mix of Labour & Conservative support, often with natural but small Labour majorities. So Labour win a lot of seats with moderate levels of support and the Conservatives win fewer but with much higher support levels; the overall effect in seat numbers favours Labour. There is currently a trend towards people moving away from cities, which accentuates Labour’s advantage, although in the past the boundaries have also been known to favour the Conservatives. And what of the Liberal Democrats? Their histogram is skewed to the left (positive skewness). Support was concentrated in a few areas where the party performed well above its national average, but the party had an over-large number of areas where it was not in the running. Note in particular the tall bar representing 344 seats where it has 5,000-10,000 votes. If we model a swing towards the Liberal Democrats we find this: They pick up bigger majorities in the seats they already hold and they secure an increased number of second places. But it takes a huge swing before the seats where they have low numbers of votes start to turn into winners. This looks unfair on the Liberal Democrats – but unlike the Labour/Conservative situation it is not connected to boundary issues. Rather it is partly generated by the Liberal Democrat strategy of focussing on some target winnable seats. If over time the Liberal Democrats became a larger party, then their strategy would presumably become focussed more nationally (instead of on 100 target seats). This might allow the party to win an election without needing to push both Labour and Conservative support down to 26%. Does any of this matter? The debate is ignored most of the time, and for good reason. Out of 17 elections since 1945 there have only been two occasions of a party winning the most seats without winning the most votes. The Conservatives won a majority in October 1951 with 47.8% of the vote to Labour’s 49.4%. And Labour formed a short-lived minority government in February 1974 despite trailing the Conservatives by 38.0% to 38.6%. But here’s the interesting thing: Polls are suggesting that the Conservative lead now may be at or around just the critical level for securing a majority of seats. The issue is “live” in 2010: Statistical skewness can have surprising effects.