Posted by: John Colby | Saturday September 27 2008

## My dogs have more paws than average

But before you think there’s something strange about them have a look at their picture.

The Celtic Terrors on the beach at Porth Kidney, near St Ives, Cornwall, August 2008, They have four paws each, quite normal.

In fact they’re quite normal, but that doesn’t stop them having greater than the average number of paws, taking the whole population of dogs into account. In making sense of this you have to consider how many paws dogs could have and still be able to function. Dogs have a maximum of four paws – that’s determined by the species. But a leg may have had to be amputated because of disease or accident, but there are plenty of dogs going round on three legs – they’ve just learned to adapt. Then again, some dogs can survive with just two functional legs, as long as they’re the front pair as they can be fitted with wheeled supports so that they can still lead a decent life. Hence some dogs have two paws.

So not all dogs have four paws.

So how come that my dogs have more paws than the average?

Just consider what an average is. Formally the expression we’re considering is a mean, expressed by:

$\overline x = \dfrac{\sum x}{n}$

The mean is calculated from the sum of all values in the set divided by the number of those values.

So what if we take a sample of 1000 dogs and find that 999 of them have four paws, but the other one has had an accident and has had a leg amputated, and now only has three paws. The average number of paws for this sample of dogs is therefore:

$\overline x = \dfrac{999 \times 4 + 3}{1000}$

$\overline x = \dfrac{3996 + 3}{1000}$

$\overline x = \dfrac{3999}{1000}$

$\overline x = 3.999$

That’s 3.999 paws per dog. That’s the mean, or the average. But 3.999, although close to 4, isn’t 4. If you do the calculations with actual figures instead of the ones that I’ve just made up then the average number of paws per dog is always going to be less than 4 – simply because you don’t get dogs with more than four paws.

In fact, the vast majority of dogs have an above average number of paws. Let’s go further – all dogs with 4 paws have an above average number of paws.

However if you use the data given above (999 dogs with 4 paws and 1 with 3) then two other expressions of an average, the mode and the median, say that my dogs are not unusual. The mode means the most common value, so as dogs with 4 paws are the most common, the modal average is 4. The median is the middle value of the data. You rank the data in order and take the middle value – that’s also going to be 4, and is the median average.

This hasn’t been about unusual dogs, just about interpretations of averages. It’s here because it forms part of my teaching, and has been inspired by Michael Blastland and Andrew Dilnot’s book The Tiger That Isn’t. Andrew Blastland has also written a BBC Magazine series, one of which deals with the question of averages and how they can be misrepresented.

The Tiger That Isn’t was a birthday present to me from my wife. It is on the same plane as the 1954 classic by Darrel Huff, How to Lie with Statistics. I’m going to point our librarians to The Tiger That Isn’t.

The moral of this tale? That when dealing with averages you have to be careful.

## Responses

1. Love this! A great illustration of the danger of using averages without thinking about it!

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