Dice World by Clegg Brian

Author:Clegg, Brian
Language: eng, eng
Format: epub
Tags: statistics, randomness, probability, science
Publisher: Icon Books Ltd
Published: 2013-03-04T16:00:00+00:00

Male height distribution (in centimetres)

Using data for US men, you might be surprised to learn that the average height is just 5ft 6in (167cm). This is because, rather than being a symmetrical distribution, the range of heights has a much greater spread at the short end than at the tall end. Around 99 per cent of all men are no more than 11in (28cm) taller than the average height. But to get to that 99 per cent figure you have to include men as short as 2ft 7in (79cm) shorter than the average height. If you look at a graph of the distribution, the tail-off to the left – the short men – is much longer than that to the right. This also shows in the way that the median height, the middle value, is 5ft 8in (172cm), significantly taller than the average. Most men are above average height.

As we have seen, we also have to be careful with applying these distributions where there is a chaotic element to the randomness. If we look at net worth or book sales, a few individuals can hugely distort the whole picture, making a traditional distribution almost worthless. This isn’t the case with the height distribution. There are clear limits. You will never find a man with twice the average height, whereas someone with twice the average earnings is fairly commonplace. And even though height does not fit a normal distribution, the distribution of heights can be dealt with well in determining a significance level.

For the purposes of understanding significance levels, let’s assume that we are looking at something that does have a normal distribution – a nice, symmetrical bell-shaped curve. If we just take the tall middle section and cut off the two shallow tails we will capture most of the values. The 95 per cent confidence level is the central chunk of the distribution where 95 per cent of the values are expected to fall. So should we get a result that falls in one of the two tails that are left behind when we take out that middle chunk, it will only happen by chance 5 per cent of the time.

The measures of confidence in a result are often expressed in terms of ‘sigma’ levels. Sigma is the symbol for ‘standard deviation’, which is a measure of how quickly a distribution spreads out into its tails. The higher the number of sigmas, the less likely it is that the result would be obtained purely randomly. Our 95 per cent confidence level is a 2 sigma level. As we have seen, when there were indications that fitted with the Higgs boson from the Large Hadron Collider at CERN in 2012, this was reported to be to a ‘5 sigma level’. This was the equivalent of saying there is just a 1 in 3.5 million probability of these results happening anyway without something that looked like a Higgs boson.

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