# Can percentages be averaged?

I've been sent two sets of timings for a task as carried out by several subjects: first using Method A, and the other using Method B.

The experimenter has taken the mean timing for each method, worked out the percentage improvement (e.g. Subject 1 took 10 seconds on average using Method A and 5 seconds using Method B, showing a 50% improvement while Subject 2 took showed a 25% improvement), and then averaged those percentages (i.e. (50% + 25%) / 2 = Average 37.5% improvement using Method B).

Is this valid?

Specific advice is difficult here. We see no data and have only a broad idea of what is being done. What are the precise questions being answered? Averaging percents is often going to be a bad idea, but the question needs to be made broader to allow good recommendations.

For example, if reducing time spent is of primary interest, then the savings as measured in seconds are key. Often the fast can't get much faster, but the slower have a lot of scope to be slower still if there is some physical or mental skill that is essential for rapid completion, but difficult for some. Working with percents could then obscure the key issues.

So the first thing to get clear is whether percent improvement is a good scale at all, on which some general advice is possible. Working with percents can make sense if changes are generally multiplicative. So talking about price or income changes as percents can make sense because that is, to a good approximation, the way that many bodies do change prices or incomes. Is there something similar here?

1. Are often best analysed as they come, as there is scientific, and especially practical, interest in time as it would be spent.

2. Sometimes are best analysed on logarithmic scale, as they are often highly skewed (imagine times to run 1 km, even among those who can run). Working on logarithmic scale and percent change are basically the same idea.

3. Sometimes are best analysed on reciprocal scale, as that gives a speed or rate of completion. (People who never finish can be regarded as having zero speed, which is unflattering in the abstract, but makes them easier to plot and average.)

Suppose person X changes on A from 10 s to 20 s and on B from 20 s to 10 s. That is a 50% improvement in one case and a -100% improvement in the other. What would be an appropriate summary? It is easy to imagine cases in which different kinds of changes would be averaged to the same average percent, which will be at best not helpful and at worst highly confusing. But do they occur in the dataset?

1. The raw data should always be accessible to anyone asked to judge on this so any reduction can be checked and revised.

2. If the two sets of improvement percents are very close, then that is the best basis for averaging, but even if that is true, it is often better just to present both sets of results any way. That should not take more space as you can use the same graphs and tables.

3. Your examples may be plucked out of the air, but if there are substantial differences between percent change on the methods, you need to be focusing on the differences, not taking the average.

I quite agree with Nick Cox's response, to give an example where one has to be wary, take compliance with a particular standard by an organisation with just two cost centres where the sample size in the first case is 10 and in the second it is 100. In the first case 8 out of 10 may be compliant, while in the second only 40 out of 100 may be compliant. If one wants to know what the average among cost centres is then the answer is 60%, while if one wants to know what the average is across the organisation as a whole, then it is better to aggregate the underlying numbers, i.e. 48/110 making 44%.

If Subject 1 went from 10 sec to 5 sec (50% improvement) while Subject 2 went from 10 sec to 7.5 sec (25% improvement) then it is valid to average the percentages in this case and say that the average improvement is about 37.5%.