Practical maths and Angela Canning
Numerical literacy isn’t just about being able check your change from the cashier. It’s also about whether people are convicted of murder, as Clare Dyer writes today, the day that the disgraced Professor Sir Roy Meadow faces charges of professional misconduct.
There’s much to be vexed about over the Cannings/Meadow case. For me, it’s that a basic error of practical maths wasn’t spotted by the barristers, the judge, any of the twelve members of the jury, or Meadow himself. Let’s do the maths again. Don’t worry, there’s no serious maths involved. Stick with this. You know you really ought to know about this stuff…
First, let’s see what Professor Meadow said. This is how Clare Dyer explained it:
At the time, a government-funded research team, the Confidential Enquiry into Sudden Death in Infancy (Cesdi), was about to publish a new report. The jury was shown a table from the Cesdi report which showed a one-in-8,543 risk of cot death for a baby from a family where the mother was over 26, at least one parent was employed and neither was a smoker - a family, in other words, like the Clarks. To work out the risk of two cot deaths in such a family, Meadow told the jury, you would multiply 8,543 by 8,543 to reach a risk of one in 73 million.
So he’s saying if there’s a one-in-8,543 chance of something happening, then there’s a 8,543 x 8,543 chance of it happening twice. And that figure is near enough 73 million.
Well, the multiplication is fine, but the assumption that got us there is way out.
It’s true that if there’s a one-in-N chance of something happening then there’s a one in N x N chance of it happening twice. But it’s only true if those two events are totally independent. In other words, the calculation assumes that the two occurrences are not connected in any way. That they are truly random.
The multiplication calculation can be applied to things we can safely assume to be random — being hit by lightning, and winning the lottery, say. We can assume such events really do strike their victims/victors at will. The chance of one family member being hit by lightning may be very small, but the chance of two members being hit is vanishingly small.
However, this changes when the events are not truly random. The chance of one person in a family catching measles may be low, but the chance of a two people in the family catching measles is not that much lower– because measles is highly contagious.
So if the two events have some common link then that simple multiplication calculation goes out the window. Clare Dyer continues:
The evidence was clearly wrong. It took no account of any possible genetic predisposition or any unrecognised environmental factors. The risk for a family which has already had a cot death is not the same as for a family which has never had one.
If there was a genetic factor in that family, maybe there was a one-in-100 chance that any of their children might succumb to SDI, and a mere one-in-10,000 chance that two of their children might. Yes, there’s still a one-in-8,543 that any family of that profile might suffer SDI. But once you know the family has suffered it once, surely your world view changes, because you know SDI is not a random event.
The continued story of Angela Canning is told thus:
The statistical error formed the main plank of Clark’s first appeal. But the appeal court refused to allow evidence from statistical experts, arguing that it was “hardly rocket science”.
This isn’t rocket science, but it amazes me how so many educated people can be not only ignorant of some really key practical maths, but also ignorant of just how ignorant they were — they didn’t allow statisticians to come forward because they thought they knew it.
On a side note, I look forward to the return of Radio 4’s very accessible maths programme More or Less. It’s this Thursday, 23 June. It has a natural bias towards statistics, and you can see why that’s so relevant.
Meanwhile, let’s get back to our theme for today. All together now: If there’s a one-in-N chance of something happening then there’s a one in N x N chance of it happening twice, but only if those two events are totally independent.
You have just made yourself a more useful member of society.
