Friday, April 16, 2010

Mean, Mode and Median

I loathed Statistics at school. It put me in mind of the tedious double Geography lessons that I would always try to skive out of with a judiciously placed piano lesson. For me, Statistics represented the dirty side of mathematics, where the elegance edifice of pure mathematics met with murky reality. It's only since I've taught the subject that I realise how beautiful it can be.

Yet I've always been troubled by the notion of "average" - i.e. trying to find a single number which best represents a statistic of a population. School statistics gets you as far as learning about the mean, median and mode.

To refresh the memory of my readers who may not have touched upon this for some time, the mean is what we tend to call THE average - add up all the data and divide by the number of pieces of data you have. This is where you'll get your 2.5 children. The mode is the most frequently occurring value of the data, and the median is the middle number that you would find if you arranged all the data in numerical order.

What you may not have been taught in school is that two of these averages are biased!

By biased, I mean that the expected value of the average depends very significantly on the sample of the population that you choose, i.e. if you use a biased average you can't get an accurate value of the average height of the population by recording the average height of 200 people.

In fact, the median is biased and so is the mode. The mean is not. We can reasonably expect that value of the mean of any sample will be the mean of the population. Not so the median and mode.

But let me remind you about the mean: add up all the values and divide by the number of values you have. If you have 10 people and the total amount of money that they have in their pockets is £453.20, then the mean amount that one person has is £45.32. The mean is a rather Procrustean quantity: it levels the playing field by cutting off the bits that are too high and gluing them to the bits that are too low. The mean is a truly communist average.

And yet it is the mean that we think of most when we think of "average".

How willing are we to describe ourselves as average? Does that mean that we fit a Procrustean bed of qualia? If we mean "mean" when we say "average", then to describe ourselves or anyone else as average we are fitting them into a mould within our minds to "aid" understanding. Except we lose information and make our average humans less than human. The mean human being is not a human being.

Human beings are biased.

That's not something to be ashamed of. In fact it is something to rejoice in, provided that we recognise our bias when we make objective judgments. Our bias is the result of our upbringing/heritage/nature/interactions and without our bias we would cease to be the persons we are.

We can have a bias for pure mathematics, but pure mathematics can naturally lead us into contact with the filthiest mathematics going - statistics and theoretical physics. But it is knowing this bias that allows us to engage with the mathematics at its most necessary as a language of pure reason to make reasonable judgments about the universe. However, we should not focus on the mean values but try to see how our reality possesses bias away from what we expect and imprint upon our understanding.

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