Not Saussure

November 28, 2006

Mean or median? ‘Average’ household income

Filed under: Blogroll, Economics — notsaussure @ 10:20 pm

As a result of David Cameron’s new interest in relative poverty, as defined as living below 60% of the median income, maybe we need to remind ourselves about the difference between mean and median averages. One person who certainly seems to need reminding is Lord Young of Graffham, one of Mrs Thatcher’s former ministers, who, as Chris Dillow notes in Stumbling and Mumbling, wrote a letter ‘of quite staggering imbecility’ to The Times:

The Conservatives now wish to eradicate “relative” poverty, defined as those living on less than 40 per cent of average incomes. Simple arithmetic shows that, as you increase the incomes for the relatively poor, the average income would rise, promptly putting some back into the relatively poor category.

As Chris points out, one of his imbecilities is that

relative poverty is defined an income below some fraction of median income. And the median doesn’t change just because the incomes of the very poor rise

However, some of the discussion in comments to Chris’ piece suggest Lord Young’s confusion (deliberate sleight of hand?) between ‘average’ in the sense of arithmetical mean and (as it’s normally used in discussions of ‘average’ earnings) median needs clarification.

Briefly, the mean average is what we usually think of when we talk about averages, and it’s the sense in which Lord Young is using it: you take all the numbers in a range, add them together and divide them by the count of those numbers.

Thus, the mean average of 2, 3, 3, 5, 7 and 10 is 30 divided by 6, which is 5.

The median is the middle number of a group of numbers; that is, half the numbers have values that are greater than the median, and half the numbers have values that are less than the median. For example, the median of 2, 3, 3, 5, 7 and 10 is 4.

Another useful ‘average figure’ is the mode, which is the most frequently occurring number in a group of numbers. For example, the mode of 2, 3, 3, 5, 7 and 10 is 3.

(examples lifted from Excel’s Function Help).

And ‘relative poverty’, expressed as less than 60% of the median — the conventional definition in the UK — is anything less than 2.4.

Why’s the median rather than the mean usually used when discussing income? Concept Stew Ltd have a useful (and relevant) explanation:

In early April 2005 there was considerable debate in the media about whether ‘average’ incomes have gone up or down in the UK. The Institute for Fiscal Studies produced a report in which they stated that the mean ‘real household income’ fell by 0.2% over 2003/04 against the previous year. This sounds very authoritative, but it is worth pausing to consider if the mean is really the most appropriate measure.

The mean is calculated by adding together all the values, and then dividing them by the number of values you have. As long as the data is symmetrically distributed (that is, if when you plot them on a frequency chart you get a nice symmetrical shape) this is fine – but it can still be thrown right out by a few extreme values, and if the data is not symmetrical (ie. skewed) it can be downright misleading.

It only takes a moment’s thought to realise that more people earn low salaries than high ones, because a fairly large proportion of the population works part-time – so the mean is not the best ‘average’ to use in this case.

The median, on the other hand, really is the middle value. 50% of values are above it, and 50% below it. So when the data is not symmetrical, this is the form of ‘average’ that gives a better idea of any general tendency in the data. The same report from the IFS states that median real household incomes rose for the same period by 0.5%.

The slightly shocking thing is that where this was reported in the media, some commentators were glorying in this apparent reduction of average incomes as an opportunity to criticise the government. (Gordon Brown, the chancellor, was very frustrated trying to explain that the median is the measure you use for things like income, because the distribution is skewed.)

Thus, average incomes went both up and down simultaneously, depending on which sort of average you meant.

Using my range of figures, 2, 3, 3, 5, 7 and 10, of which the mean is 5 and the median 4, the chap earning 10 can infuriate Polly Toynbee by awarding himself a 28% pay rise, thus paying himself 12.8 while no one else gets an increase.

This actually has no effect on the median — that’s still 4 — or, consequently, ‘relative poverty,’ so I’m not too sure why Polly is so upset about it (it doesn’t make either the number or plight of those in relative poverty any worse). However, if the top earner stays on 10 and the low earner goes up to 3, he’s lifted out of relative poverty, a state in which no one now languishes, contrary to Lord Young’s expectations.

Now that our chap at the bottom of the pile has increased his earnings to 3, we can up the pay of both the chaps earning 7 and 10 to 12 if we want; that doesn’t do anything to the median (and, consequently, to relative poverty). That gives us 3, 3, 3, 5, 12, 12, with a mean of 6.34, a median of 4 and a poverty line of 2.4 still.

The time to worry, though, is if one of the chaps on 3 goes up to 5, giving us 3,3,5,5,12,12. That puts the median up to 5 and, thus, ‘relative poverty’ up to 3 (doesn’t matter about the top pay — we can keep that at 10 or 12, and it doesn’t affect the median). That is, by improving the lot of one of the lower earners, we’ve plunged the other two to the brink of relative poverty.

We can, of course, also abolish relative poverty, as Chris notes in a subsequent post, using a different set of examples.

He writes,

Imagine three people with the following incomes:
A = 50
B = 100
C = 200.
If we follow the convention and define relative poverty as an income less than 60% of the median, A is poor.
Now imagine 30 units are taken from B and given to C, so the new distribution is:
A = 50
B = 70
C = 230.
Relative poverty has been eradicated – no-one’s income is below 60% of the median.
But is this society really better than our original one?
If your only interest is in relative poverty – as Cameron claims – the answer is yes.
But if you care about inequality, the answer’s no.
So, which is it, and why?
Don’t duck the question by claiming that the example is impractical. I’m interested in the theoretical point, of whether we should care more about relative poverty or inequality.

(the relevent Cameron reference in the article is, I think, Cameron now says his only concern about injustice is to bring the bottom nearer to the middle, which Polly Toynbee — for it is she — unaccountably glosses ‘so he never meant what he said about relative poverty’; that’s precisely how relative poverty is defined, though).

I think it’s a tad more complicated than Chris’ example suggests, though. I mean, if we go back to my set of examples where people were were earning 3, 3, 5, 5, 12, 12 and redistribute it to 2, 3, 3, 3, 3, 12, the median is 3 and the poverty line to 1.8. Depends, I think, on how much my lads can buy with their 2 and 3 units each; if this’ll give them a reasonable standard of living, then the main complaint of the chaps who’ve lost out will be that they’ve been expropriated. On the other hand, if we started with most people earning 2 or 3, and enjoying a reasonable standard of living this way, I don’t see that the inequality caused by the one chap earning 12 is much of a problem. I don’t compare my income with that of Sir Richard Branson or the Barclay brothers; I compare it, first, with what people doing broadly similar jobs earn and, second, with — I suppose — what ‘most’ people earn (i.e. the modal income).

Not sure, but I am sure, as one of the people commenting on Chris’ original article suggested, we shouldn’t use the term ‘average’ in this sort of discussion without qualifying it.

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  1. Seems fairly straightforward. “Relative poverty” is a measure of a certain sort of inequality, but it’s a very bad name as Chris’ example shows pretty clearly. If you want to talk about poverty you should probably be thinking about the amount of income you need to purchase certain basic things. If you want to talk about inequality, use better measures than this one of ‘relative poverty’.

    All this of course misses a key point – how come median incomes are so low? 2004 statistics say it’s £16k before tax (compared to mean income of £22k), and that’s the median of tax payers only, so that money has to pay for any dependents too.

    Comment by Dan Goodman — November 28, 2006 @ 10:58 pm

  2. The median is a better measure than a simple mean but it is still far from ideal. The problem is that all it tells you is that 50% of people earn less than that value, without any indication of how much less. Using this measure an increase income of the bottom 20% (for example) would have no effect on the headline figure, even though it represented asignifcant increase in the income of those who most need that increase.

    I would have said that a better measure for most purposes would be a mean that excludes the extremes at either end that can lead to distortion of the results.

    Comment by Andrew — November 29, 2006 @ 10:29 am

  3. I am obliged to you for that excellent explanation. Alas, I was a dunce at school maths but I can assure you that my doodles of Spitfires and footballers were superb!

    The ludicrous notion that any government should concern itself with ‘relative’ poverty is best left for another day whilst I give my brain a rest.

    Comment by David Duff — November 29, 2006 @ 7:55 pm

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