Con-Lab Gap Analysis

This page first posted 3 September 2006

Even after the Boundary Commisson changes, there seems to be a difference in the electoral geography for the two main parties. This is typified by the fact that if Labour and the Conservatives have equal support, then Labour still has a majority in Westminster, but the Conservatives need about a 10% lead to get a majority themselves. Some people call this a "bias", but we will use the more neutral term of "gap". Whatever name we use, the question remains as to what causes this gap, is it a rational feature of a fair electoral system, and will it persist to the next election.

The article on this page examines the causes of the gap by consideration of the likely electoral geography at the next general election under the new boundaries.


  1. Observing the Gap
  2. Constituency sizes
  3. Differential Turnout
  4. Support Distributions
  5. Tactical Voting
  6. Conclusions

1. Observing the Gap

For our analysis, we will work with an adjusted version of the parties' support as at 1-Sep-2006, using polls from 7-Jul-2006 to 20-Aug-2006. Those support figures are Con 37.18%, Lab 32.59%, LibDem 20.86%. We adjust the Con and Lab figures to be equal at their average value of 34.88%. This gives Labour a predicted majority of 10 in the House of Commons, with 90 seats more than the Conservatives.

The following table shows clearly the existence and extent of the electoral gap between Labour and the Conservatives. The table varies the Con and Lab support figures around that. central case, keeping the LibDem support constant.

Con-Lab %Con %Lab %Con

The columns of the table are:

The rows of the table are coloured red or blue according to the largest party. If the party has an absolute majority (326 seats or more), the colour has a darker hue.


Useful rule-of-thumb

From the table's data, we can also get a useful approximation rule. Each 1% change in the Con-Lab% difference is worth about 17 seats in the Con-Lab seat difference.

We will use this gearing of 17 later.

We can confirm some key facts.

  1. Labour has a majority up to a Con-Lab% difference of zero.
  2. Conservative majority starts from a Con-Lab% difference of nine.
  3. The equal-seats point is around a Con-Lab% difference of five (precisely 4.9%)
  4. Around 4% on either side of the equal-seats point is a hung parliament

The motivation for the three parties in the next election is clear. Labour need to be in the darker-red area, which means doing at least as well as the Conservatives. The Conservatives need the darker-blue area, which needs about a 9% lead in the opinion polls. The Liberal Democrats would like a hung parliament, and would like the Conservatives to lead Labour by around 5% or so.

In one sense the electoral landscape is symmetric. There is a 4% buffer zone on either side of the equal-seats point. Beyond that zone, each major party can enjoy a parliamentary majority. But the equal-seats point is not symmetric. We would expect it to be at 0%, but instead it is at 4.9%. This is the "gap" that we have observed, and now have to explain.

2. Constituency sizes

Despite the efforts of the Boundary Commissions, the seats are not all the same size, in terms of numbers of electors. There are three reasons for this. Firstly there are 59 seats in Scotland and 40 in Wales. This forces the average Scottish seat to have 65,300 electors and for Wales only 55,762. The average for England is higher at 69,735. This over-represents Scotland and Wales which are predominantly Labour supporting. To be equal, Scotland should have 55 seats and Wales should have 32 seats.

The number of seats in Scotland is now controlled by the Boundary Commission for Scotland, but the Scotland Act 1998 requires them to make seats with the same average electorate as England. In its fifth report in 2004, it calculated the quota number of Scottish seats at 57, but nevertheless produced 59 in practice. The small seats of Orkney and Shetland, and Na h-Eileanan An Iar (Western Isles) contributed to the discrepancy.

Although the number of seats in Wales is set by the Parliamentary Constituencies Act 1986 to be at least 35 (which is now slightly too high), the rules also require the Boundary Commission for Wales to keep the number of seats approximately unchanged. In its fifth report (2005), the Boundary Commission for Wales kept the number of seats unchanged at 40.

Secondly, the Boundary Commission for England has to work with the electorates as at 2000. In the years since then, we would expect continued poplulation drift away from poorer (Labour) areas towards richer (Conservative) areas. This also over-represents Labour voters slightly.

Thirdly, it is not possible to have seats exactly equal, given that seats are made up of entire local election wards, and that some regard should be given to county, historic and natural boundaries.

We can estimate how much this over-representation is worth. Working with our base case of Con and Lab equal at 34.88%, we can weight each seat by its own electorate and count how many weighted seats each party wins. This alters the Con-Lab seat difference from -90 to -78, implying that uneven constituency sizes are worth 12 seats to Labour. Using our gearing of 17, we can think of this as about 0.7% of the total gap of 4.9%.

3. Differential Turnout

It has long been observed that Labour supporters are less likely to actually vote than supporters of the other parties. Indeed there is a strong negative correlation between the turnout of a constituency and the Labour support there. (The correlation is -0.69.) We can estimate that a Labour supporter is only 65% as likely to vote as Conservative and Lib Dem supporters.

This is a major effect, but it has already been allowed for by the pollsters. They try to restrict their sampling to those most likely to vote. So non-voting Labour supporters will be ignored both by the opinion polls and (automatically) by the general election. So we should be comparing like-with-like and differential turnout will not matter. But the Boundary Commissions, rightly, do not ignore non-voters and they draw the seat boundaries to ensure equal numbers of electors in each seat, not equal numbers of actual voters. This means, rather elegantly, that Labour non-voters are represented as long as they live in a seat with a lot of Labour supporters anyway. However, it does also mean that safe Labour seats will contribute less than they should to the measured national support figures, and thus cause some "gap".

We can estimate how much gap is caused by this effect. We do this by estimating how many votes would be cast for each party if the turnout in each seat was identical. Our formula for this is

V(i) = U(i) x Scale,

where U(i) is the original number of votes cast for party i in the seat, Scale is the seat's scale-factor and V(i) is the adjusted number of votes cast for party i. We define the seat's scale-factor to be

Scale = 80% x Electorate / (U(1) + ... + U(n) ),

to ensure an equal turnout of 80% in each seat. We can then calculate the national support figures by adding up the adjusted votes V(i) over all the seats. Starting with the base case of Con 34.88% and Lab 34.88%, this gives adjusted support figures of Con 34.02% and Lab 35.86%. So this has caused a support gap of 1.84%.

This effect is legitimate. Labour is strongly handicapped by the failure of many of its supporters to vote, and this effect is a much smaller move in the other direction. This part of the gap is a direct consequence of the principle of having seats with equal electorates rather than equal turnouts. It is hard to argue with that principle, though it does result in safe Labour MPs being elected with fewer votes than equally safe Conservative MPs.

Compared to the total gap of 4.9%, we see that this effect (1.8% gap) plus the uneven constituency sizes (0.7% gap) accounts for about half the total gap. We can check this calculation intuitively by taking a simple average of each party's percentage support over all the seats (instead of adding up all the votes cast for the party). This gives a Labour lead of 2.46%, which is again about half the gap.

4. Support Distributions

It is important to remember the support of a party is not spread uniformly across all seats. If it were, the most popular party would win every seat. A useful way of analysing this is to look at the distribution across seats of party support. For the, say, Conservatives, look at how many seats they have a support of 0%-5%, how many 5%-10%, and so on. The average over the seats must (arithmetically) match the national average, but the shape of the distribution can be important.

Here are two extreme examples for a party which has 35% support nationally. There are 632 seats (excluding Northern Ireland).

1. Strong case Suppose the party has 45% support in 492 seats, and 0% support in 140 seats. This has the correct average of ( 45% x 492 + 0% x 140 ) / 632 = 35%. Usually 45% support in any seat is sufficient to win, so the party will win about 492 seats and have a landslide majority of 334.

2. Weak case Instead, suppose the party has 30% support in 587 seats and 100% support in 45 seats. This also has the correct national average of ( 30% x 587 + 100% x 45 ) / 632 = 35%. In a three-party system, 30% support in any seat will always lose, so the party will win just 45 seats.

For those familiar with second moments of distributions, it is interesting to note that the standard deviations of these two distributions are almost identical. So the two distributions are equal in both first and second moment, but have completely different electoral outcomes.

We now work with a base case which has been adjusted for the effects of uneven constituency size and for differential turnout. We do this by applying the explained gap of 2.46%, to give national support figures of Con 36.11% and Lab 33.65%. This has the effect of ensuring that the average percentage support in each constituency is 34.9% for both parties. The seat gap is now 49 which, as expected, is about half the original seat gap of 90.

Frequency distribution of seats by Labour and Conservative support

The histogram shows the distribution across seats of support for both the Conservatives and Labour at this adjusted base case. For instance, the peak of 110 for Labour at 45% means that there are 110 seats in which Labour's support is between 40% and 45%. Of those, all but 8 were seats which Labour won.

Thinking about the two extreme cases above, let us consider whether the parties have a relatively "weak" or "strong" distribution. The Conservatives have more seats than Labour in the weak area of 25%-35%. These seats are bad for the Conservatives because they "use up" many votes in seats which they do not win. Conversely Labour has more seats than the Conservatives in the strong area of 40%-45%. These seats are good for Labour because they can win the seat without spending too many votes. Both of these effects are good for Labour and bad for the Conservatives.

On the other hand, the Conservatives have more seats than Labour in the slightly strong areas of 50%-55%. Labour also "wastes" some votes in very safe seats with 60%-70% support. These effects are bad for Labour and good for the Conservatives.

We need to quantify these effects so that we can determine which are more important. We can look at the means and standard deviations of the distributions, but we do not expect this to reveal very much. Another simple measure is just to count how many seats have support above, say, 40% (marginal seat), 45% (winnable seat), or 50% (safe seat).

PartyMeanStd devNum seats
above 40%
Num seats
above 45%
Num seats
above 50%

The table shows the relevant statistics for the two parties, as well as the difference. The surprising result is that the distributions are very similar. The means and standard deviations are very close, but even the number of marginal, winnable and safe seats are similar compared with the remaining gap of 49 seats. Labour lead by 10 seats over winnable or safer seats, so they have a slightly more favourable support distribution.

The main difference appears, not in the distribution, but in the number of seats won. Labour has been much more successful at winning the borderline seats in the 40% bucket, where it has won nearly half the possible seats. The Conservatives only win about one sixth of their possible seats.

5. Tactical Voting

Our analysis of the support distributions has not explained very much more of the gap, but it has shown us where to look next. Let us now consider the set of seats in the 40% support bucket (which includes seats where support lies between 35% and 40%). There are 80 of those seats for the Conservatives, who won 14, and 88 of those seats for Labour, who won 42. What was different for the two major parties in those seats?

The following two charts show all the marginal Conservative and Labour seats respectively. The result of each seat is shown as a column of 100% of all votes divided up into Con, Lab, LibDem and the rest (Nat, Min and Oth). The winning party is highlighted in a bolder hue.

Conservative marginal seats: vote breakdown

And for the 88 Labour marginals:

Labour marginal seats: vote breakdown

We can see clearly that the Conservatives do not win many of their marginal seats, but Labour win quite a few of theirs. At first glance, it is hard to see any particular reason why this is. Looking more closely we see that Labour often win when the LibDems have good support, and the non-Labour vote is split. In Conservative seats, the non-Conservative vote is less often split, so the Conservatives usually lose. This suggests that tactical voting against the Conservatives may be taking place.

An informative statistic is that in the 18 Conservative marginal seats, where Labour was third, the Labour vote averaged just 12.04% and the Conservatives only won 2 of those seats. But in the 16 Labour marginal seats, where the Conservatives were third, the Conservative vote averaged 17.81% and Labour won 13 of the seats. It appears as if Labour supporters are quite ready, if Labour is trailing, to vote tactically to support the Lib Dems against the Conservatives. But Conservatives are less inclined to vote Lib Dem to stop Labour. So tactical voting by Labour supporters could be worth about 10 seats of the gap. Testing with a tactical unwind of 5% for third-placed Labour supporters in the election simulator confirms a change of 9 seats in the gap.

It is hard to estimate the amount of Lib Dem tactical voting against the Conservatives, if any. Speculatively, we attach a 1% tactical vote by third-placed Lib Dems in favour of Labour. This is partly a matter of judgement, though there is some weak numerial evidence. (The average support for third-placed Lib Dem candidates in Conservative marginals is about 1% less than that in Labour marginals.) The tactical voting feature of the election simulator values this gap effect at 16 seats.

6. Conclusions

We have observed a gap of 90 seats (equivalent to 4.9% support) between Labour and the Conservatives when the parties have equal support nationally. We have investigated some possible causes of this gap and tried to quantify them. The results of the analysis are shown below:

CauseSeat gap Support 
gap %
Uneven seat sizes120.7%Scotland and Wales are over-represented
Differential turnout331.8%Labour supporters don't vote in (safer) seats
Support distribution100.5%Differences in clustering of supporters
Tactical voting by
Labour supporters
100.5%Labour supporters vote LibDem over Con
Tactical voting by
Lib Dem supporters
161.0%LibDem supporters vote Lab over Con

So we see that most of the gap has been explained. There is no single cause, but the two biggest effects are differential Labour turnout and tactical voting. In terms of the fairness or legitimacy of each of these effects, the following comments could be made:

To summarise, the gap is mostly due to the technical factor of differential Labour turnout (which hurts Labour more than it helps it) plus tactical voting against the Conservatives. It is not due, as far as is measurable, to the way in which the Boundary Commission for England has drawn the new seat boundaries. However, the total number of seats for Scotland and Wales are too high, making those areas over-represented.

Looking forward to the next election, we can confidently assume that the uneven seat sizes and differential turnout will continue to operate. The support distribution might change, but it is a small effect anyway. But tactical voting could alter significantly between the 2005 election and the next one. If public sentiment switches from wanting to "keep the Tories out" to either neutrality or even a desire to punish Labour, then perhaps a third of the 90 seat gap could disappear.

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