Misleading baubles

There's a nice example of how to use graphical representation of statistics to give a misleading comparison in the Asda advertisement in the newspapers this week.

The data refers to the number of products that were 'cheaper' (I presume they mean 'cheapest') in each of four leading supermarket chains on a given date in the previous week, according to independent price checks:

Asda 1206

Morrisons 664

Tesco 546

Sainsbury's 268

To illustrate the comparisons they use pictures of Christmas baubles, which appear on the page as coloured circular shapes, the 'size' of the bauble representing approximately the numbers in the table. But, how have they determined the size of each bauble? Here's the trick: the naughty Asda advertisers have used the diameter of each circle, not the area!

So the ratio of the Asda number to the Morrisons number, for example, (1206 ÷ 664 = 1.82 approximately) is shown by two circles, for which the diameter of one is approximately 1.82 times the diameter of the other. The result of this is that the area of the circle representing the Asda number is about 3.31 times larger than that of the Morrisons figure! (3.31 is about 1.82 × 1.82) So, at a quick glance, because the area is about three and a quarter times larger, the impression is given that the number of cheaper products in Asda is over three times the number in Morrisons.

The comparison between Asda and Sainsbury's looks even more startling! The Asda number is 4.5 times the Sainsbury's number. But the area of the circle representing the Asda number is 20 times larger than the Sainsbury's circle! (4.5 squared = 20.25). So Sainsbury's performance in this comparison looks pathetic!

If you are using a circle to represent data the convention is always to use the area. You would never use the diameter of a circle for this purpose; unless you deliberately aim to give a distorted comparison.

But there's more to this than meets the eye. If they were using actual baubles (three-dimensional spheres) rather than (two-dimensional) pictures, then it would have to be the volumes that represented the data. So, the pictures in the advertisement may be even more misleading – because the volume of the imagined three-dimensional Asda bauble would be about 6 times that of the Morrisons bauble! And the volume of the imagined Asda bauble would be ... wait for it ... over 90 times larger than that of the Sainsbury's one! (4.5 cubed = 91.125).

I feel a letter to the Advertising Standards Agency coming on. Happy Christmas!