## Saturday, 20 September 2014

### Scotland referendum results

The reporting of the victory of the Nosers over the Yessers in the referendum about Scottish independence provides another example of the potential confusion when people compare percentages.

The result was reported as 55% for No and 45% for Yes. So was the Nosers vote 10% higher than the Yessers, as I heard someone say? Well, no! There are two errors here. Let me explain.

When you compare two quantities you can do this either by using the difference between them or the ratio of one to the other. For example, let's say that I earn £45 an hour and you earn £55 an hour. Using difference I could say that you earn £10 an hour more than me. But using ratio, I could say that your hourly rate is 22.2% higher than mine (to one decimal place). This is because that extra £10 you earn is 22.2% of what I earn (22.2% of £45). This is similar to saying that when a price goes up from £45 to £55 that is a 22.2% increase.

This gets tricky when you are comparing percentages using percentages!

Let's look at the actual data, reported in The Times this morning:

Calculate the No vote as a percentage of the Total vote:   2001926 ÷ 3619915 × 100 = 55.30%
Calculate the Yes vote as a percentage of the Total vote:  1617989 ÷ 3619915 × 100 = 44.70%
(percentages given to 2 decimal places).

The difference between these two percentages is 10.6%, Note that this is closer to 11% than the 10% that was reported. So, that's the first error: a classic rounding error!

But it would still be confusing to report this by saying that the No vote was 10.6% higher than the Yes vote. What we can say, correctly, is:

'The No vote was 10.6 percentage points greater than the Yes vote.'

This is understood to mean that the comparison being used is the difference between the two percentages.

If we compare the actual figures, using ratio, by what percentage is 2001926 greater than 1617989? (Compare my example above for comparing two hourly rates). The No vote is 383937 more than the 1617989 votes for Yes. As a percentage that is 23.73% higher than the Yes vote (383937 ÷ 1617989 × 100). So a correct reporting would be:

'The number of people who voted No was 23.73% greater than than the number who voted Yes.'

That makes the extent of the victory clearer!