The mythical median voter

Most people have an above average number of legs, and what that means for our political imagination

There were local elections in the UK last week, so many of us have been viscerally confronted with the fact that other people think and vote differently than we do. It’s one of the salutary shocks that democracy regularly delivers, the epistemic equivalent of a cold shower or vigorous morning exercises.

Pre- and post-election there has been lots of discussion of what the voters want. I was struck by this discussion of centrism from Ian Leslie, in which he complains about the incoherence of the views of “the median voter”

According to this data, the median voter distrusts politicians; wants change; backs deportations; wants large reduction in legal migration; favours a wealth tax and wage ratios, is anti big-business. But also dislikes inflammatory rhetoric; worries about an unstable world; wants to see solutions over political point-scoring.

Let me get this straight: the median voter distrusts conventional politicians and most non-conventional politicians. They want lower taxes on their household and higher taxes on the rich. They want to cut public spending but only on migrants and scroungers. They want higher living standards, tougher regulation on big business, and much lower immigration. Oh and they’re also a big fan of pragmatic solutions. Sure.

Here's the piece Ian was riffing off, from Scarlett Maguire, research director at pollsters JL Partners, in the Telegraph:

“Indeed, the problem with the traditional concept of the centre ground is that it now looks like a very small patch of land indeed…Voters who adhere to these [centrist] beliefs make up roughly 5 per cent of the electorate…chasing an almost purely imagined centre is clearly not a winning strategy”.

Ian has good things to say about the meaning of centrism, including about the common misconceptions of many voters, and concludes that the idea of centrism needs a lot of updating. I think I can offer some encouragement here, in that centrism isn’t in some new kind of trouble (or at least that Maguire’s data don’t show this).

If you did want to update centrism, the incoherent rationality of the median voter is the wrong starting point. Instead, I think the whole idea of the median voter needs exploding. Explaining why shows us something important about what we mean when we think about “the general public”, as well as about the systematic bias in perception that polling data introduces into politics. And there’s some fun with the non-intuitive quirks of statistics. We’ll also find out why expecting the centre ground to be well populated in the first place is completely wrong, but first, let’s start with the median voter.

The average is not typical

There are two big mistakes here, and the first is assuming that the average voter should be the typical voter. There can be cases where the average is typical and the typical is average, but this doesn't have to be the case, and when you are polling beliefs I'd argue we shouldn't expect it to be.

A clear case for this is when you have two distinct populations. If I take the average height of ten ducks and ten giraffes then I get an average which doesn’t reflect ducks or giraffes. Similarly if I have a polarised opinion. If I poll people who love the EU and those who hate it, that doesn’t mean that an attitude of neutrality represents any individual well.

A fun thought that illustrates this is the line “most people have an above average number of legs”. Because nobody has three legs and some have zero or one, the average must be less than two (admittedly probably 1.99 or similar), so the vast majority have a number above the average. This makes clear that the average is an idea which, as in this case, doesn’t have to reflect well any individual.

This problem of the non-representativeness of averages is compounded when we measure multiple attributes of individuals, as in the attitudes across multiple separate issues that opinion pollsters measure. When these attitudes are individually averaged and the averages “put back together again” in the idea of a median voter, there is a strong risk of creating a chimera which doesn’t reflect any existing individual.

Here’s a toy example, which I think speaks to Ian’s complaint about the incoherence of the median voter’s views. Imagine we only ask two questions, for which we expect to get coherent answers; if you’re YES on one you should be YES on the other. You can imagine them as something like “Do you want higher taxes?” and “Do you want more public spending?”. Imagine that we have many people who are double YES, and many people who are double NO.

Lining them up you get this¹:

Question 1, responses: YYYNNN

Question 2, responses: YYYNNN

So far, so coherent.

Now imagine that we find a single person who, for whatever reason, is not typical. Either they are incoherent in their views, or they feel ambiguous and respond strongly to the way the question is worded (flipping YES on one question and NO on the other, or vice versa), or they have some idiosyncratic interpretation of the meaning which means they endorse what look like contradictory views in good faith, because those answers are consistent in a way you’re not imagining. It doesn’t matter why, really, all that matters is you concede that a) this is possible b) such a person will by definition affect the middle of the rankings.

Adding them to the line up we get:

Question 1, responses: YYYYNNN

Question 2, responses: YYYNNNN

Like before, most people either say YES to both Question 1 and Question 2 or NO to both, but now the median voter is YES on Question 1 and NO on Question 2. It would be entirely wrong to reason about this median voter as if their middle-ness means they are a good representation of the rest of the population. They are in the middle precisely because they are unrepresentative.

It would be interesting to know more about this person. To know why they are different from everyone else, and to know why do they hold views that are conventionally interpreted as contradictory. But they are interesting precisely because they are not representative. We shouldn’t use them to give up on the rest of the population’s powers of consistent reasoning.

The centre will be underpopulated, by definition

This toy example illustrates the point that we shouldn’t assume the average is typical. When we start polling voters about more and more issues, as is the case these days, there’s an argument that the average isn’t even possible.

Most issues are not polarised, meaning that non-extreme attitudes are more common. You’d be forgiven for thinking that in this case the centre should be more densely populated than the edges. Surprisingly, although this is true for the one-issue case, it is less true the more issues you consider simultaneously. This is the mistake of the “the traditional concept of the centre ground”, as invoked by Maguire and Leslie. The error is not in the claim that the centre is sparsely populated, but in the assumption that this is surprising. Far from it, it is in fact inevitable and understanding why can help us abandon an implausible expectation and adopt a more reasonable definition of centrism.

(I tried this explanation on my friend, M., and he said that what follows is trivially obvious to anyone who is familiar with the geometry of n-dimensional hyperspheres. But I think there is something here worth explaining, so on the off chance that your hypersphere geometry is a little rusty, let’s go through this step by step).

For most attitudes or beliefs there is a centre ground which is more densely populated than the extremes. If we ask people the extent of their agreement on a seven point scale, as a polling company might, we’ll tend to get a cluster around some central tendency. There are some polarised, marmite-like, issues where this isn’t the case, but it is a true for most issues that attitudes cluster around a central value. Indeed there are deep reasons measured attitudes will tend to take the form of what we know as the Gaussian, or normal, distribution:

Because this distribution is so common, for everything from the distribution of people’s heights to their exam scores, it tends to dominate our imagination. Lots of people in the centre, fewer at the edges.

Like the one-dimensional case, the two-dimensional case is easy to visualise, and anchors our intuitions. The two-dimensional case is also the framing for things like the political compass, which lets you answer questions to position yourself in a space with an economically left-right axis and perpendicular socially left-right axis.

More generally, if you measure two attitudes and plot the answers you’ll get a two-dimensional space, and it looks like the intuition still holds - a dense centre, with a sparse periphery.

But wait, as you increase into the (harder to visualise) dimensions of 3 or more our intuition starts to mislead. As you increase the number of dimensions, the centre, strictly defined, tends to be less and less populated.

To see this, let’s take a definition of the centre as being within 1 standard deviation of the middle value, for one dimension this looks like this and captures an average of 68% of the population:

A 1 standard deviation (sd) centre shown in red

So, yes, the centre is more densely populated than the periphery, for one dimension.

For two dimensions the same definition of the centre looks like this, a circle. That circle captures an average of 39% of the population :

1 centre of 1 sd radius shown in red.

At this point, note that already more people are outside centre than are in it.

For three dimensions we’d be drawing a sphere to define the centre. For four we’d have to draw a 4-dimensional sphere — a hypersphere, which we couldn’t easily visualise — and so on for as many dimensions as we think are relevant. For politics this would be the number of different issues which we think are relevant to the definition of centrism (Leslie’s opening complaint which I quoted mentioned ten issues).

For 3 dimensions the 1 standard deviation sphere captures 20% of cases, for 4 dimensions it captures 9%, and so on towards 0%.

A way of understanding this is that every added dimension gives each individual a chance to ‘escape’ the centre. Your beliefs about Europe, and tax, and immigration might be completely standard and in the middle range, but if your belief about a fourth issue, say national service, is out of the normal range then, by this definition, you aren’t in the centre any more.

If we consider enough dimensions then the centre will tend to complete emptiness — every single individual is outside it on at least one issue².

What this means for centrism

We’ve seen that our intuitions can be misleading. The average isn’t typical, and once we add more dimensions to our account of which political issues matter, simple maths means that the centre can’t be more populous than the peripheries.

In one-dimensional space, like that of the traditional left-right divide, the centre is densely populated and closer to each of the extremes than the extremes are to each other. In a high-dimensional space our intuitions break down. Nobody is truly in the centre and every individual is ‘next to’ every other individual on some dimension. The logic of high-dimensional attitude spaces is horseshoe theory on steroids.

This doesn’t mean centrism is impossible, only that you need to think of it differently. Conceived of as a space, the centre is empty, the median voter doesn’t exist and wouldn’t be representative anyway. But seen as a network, you can imagine the people in a polity, holding their diverse - often maddening, sometimes contradictory - beliefs over multiple issues, with everyone connected to everyone else by some shared views (and everyone distant to even their closest allies, by a difference on some non-shared view). Within this network, many possible coalitions can be found.

Manoeuvring in this higher-dimensional space we call politics. Understanding the impossible-to-visualise possibilities well enough to be able to reconfigure them is the stuff that makes political geniuses.

Centrism, and the idea of a centre, still has value. I am not talking down the importance of consensus (or of recognising reality) in politics, but finding the point in the complex space of beliefs where consensus can be built takes work and imagination, and neither of those are assisted by bafflement at a median voter who is more of a product of polling methods than a reflection of actual voters.

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See below other things I’ve been thinking about.

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Other stuff, but sticking with politics…

Sheffield City Council Elections

Results for Sheffield were nicely summarised by local quality newspaper The Sheffield Tribute (‘Labour suffers worst Sheffield election results in living memory’), who republished this map of the results, by Sheffield City Council

The results for Sheffield City Council election. Source: Sheffield City Council.

I immediately thought of the US election results maps of 2016, where larger, less densely populated countries tended to vote Republican, and smaller, dense, counties tended to vote Democrat. This meant a map looked like most of the US voted Republican — but land doesn’t vote, people do. I realised I could make the visualisation of the results easier to grok if each ward was the same size.

And voila:

These are just the new elections, which define approx 1/3rd of Sheffield councillors. These newly elected councillors will join existing councillors to make up a council for which Labour is still the largest party, but must work in coalition to have a majority. Here’s what this looks like:

… And finally

From warandpeas

Comments? Feedback? Hypersphere tennis? I am tom@idiolect.org.uk and on Mastodon at @tomstafford@mastodon.online

AI declaration: I write all the words and think all the thoughts myself. I asked Gemini to check for spelling and grammar. This time I also asked Gemini to draw the plots, from a fairly detailed specification that I provided.

1. The example also works if you imagine groups of people responding, or if you ask people to respond on a scale, it is just easier to think about in the simplified version.

2. This is true regardless of which distribution of values people take on each dimension, it doesn’t just apply to the normal distribution case.