### Nutty clusters

'But surely,' said my friend, 'you would expect something like leukemia cases to be spread evenly across the country. Where there's a cluster of cases, there must be a cause. Like a phone mast.' No, no, no.

First you've got to realize that clusters are natural in something that's random. You wouldn't expect a coin toss to go heads, tails, heads, tails... repeatedly or the results on a roulette wheel to flip between red and black, time after time, all nice and evenly spaced out. You get clusters. Similarly if you drop a whole tray full of ball bearings on the floor, it would be really strange if they all lined themselves up nice and evenly spaced. You will get clusters and gaps. That's just the nature of something random. Yes, on average, across a huge sample, it will kind of even out, but in any particular place there's likely to be either a cluster or a gap.

So we can't make the leap in logic from 'there's a cluster' to 'it has a cause'. Many clusters don't have causes.

We also have to be careful when saying about something common like phone masts 'Look, lots of clusters are near phone masts, so phone masts must cause them.' It would be strange if lots of clusters weren't near phone masts. They'll also be near lots of other things that occur pretty evenly across the country. There will be lots near pubs and churches too. But those who seek to blame phone masts for the ills of the world don't think it through, they just jump in.

I'm not saying that no cluster has a cause. The cluster of asbestosis cases near the Turner & Newall asbestos factory in Rochdale was no coincidence, for example. But most clusters are simple statistical artefacts, arising from the nature of random numbers. There are good tests to see if a cluster is likely to be random or causal - but these are rarely employed by those who panic about clusters.

So next time there's a cluster of cases of something locally, don't fall for the same mistake as the medieval types who decided this must be down to their version of a phone mast, the local witch. Make sure you understand the numbers before using them in anger.

(The picture (from Wikipedia), by the way, is a star cluster. And, no, it wasn't caused by a phone mast.)

### Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5.  However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor

### Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's recent gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some ex