Unweighted Mean of Percentages

Question

I have come across an interesting problem, which I'm sure there must be a standard solution to, but lots of Googling hasn't yielded much (which may mean I'm well off track...).

I am trying to calculate the unweighted mean of several percentages which represent both positive and negative change.

I realise that in most situations, the unweighted mean of percentage increase is fairly meaningless, unless the initial values for all of the percentages are the same; however in this situation I feel that it is the best way to go.

I am trying to calculate the average effect of an event on how much an individual gives, which can increase or decrease. If someone doubles their giving, it is just as significant if it goes from 100 to 200 as from 1 to 2, hence unweighted mean. It is also just as significant if someone doubles their giving as if someone halves their giving etc.

My issue however, is that taking a standard arithmetic mean of the percentages yields a value that is unrepresentatively high as an increase counts for more than a decrease. Using the standard formula for percentage change:

q = (b-a)/a,

Yields answers of q = 0 to ∞ for an increase, and q = 0 to -1 for a decrease.

I want a representative figure, such that if the event causes someone to double their giving, this is as significant as if it causes someone to halve their giving. For example:

before (a), after (b), percentage change (q):

100, 200, 1

2, 1, -0.5

Arithmetic mean of percentage change = 0.25; whereas I would like it to equal 0.

I can arrive at something closer to what I am hoping for if I use:

average = (10^mean(q's)) - 1 where q = log(b/a)

or average = (sum(a+b)/sum(a))*mean(q's) where q = (b-a)/(b+a)

but neither of these are quite satisfactory and certainly not standard methods, and so I can't really use them for what I'm working on.

I feel like there must be other situations where this kind of analysis is useful, but is there any standard method for doing this?

Thanks in advance for anyone who can post their thoughts on this.

Answer 1

Indeed an unweighted average is pretty meaningless 99% of the time. However there can be some rare cases (in my opinion) where unweighted is the way to go.

I will describe bellow the case I personally decided to go with unweighted.

I got a database with some values. All those values represent a single Indicator (lets say forest area in KM2 per Country per year).

So we got one indicator record for 100 Countries for 50 last years. Total 5000 records.

However there are some years where some Countries have no record at all (empty record instead). That happens though not because there was no forest.. but because there had been no survey (for example some less developed countries could have last 50 years only 4 or 5 year records..).

Keep also in mind that values from that Indicator are pretty hard to change much.

So... (Lets simplify above example with 2 Countries and not 50)

We have a small, developed Country with every year records (total 50) and all of them are between [100km2 - 120km2]

And a big, less developed Country with records (1960, 1990, 2015) between [2000km2 - 1500km2]

and I want to show diachronically some representative average (forest area) on those 2 Countries. What would the weightend average show me? A too low price.

What about unweightend average? Well that would show me a better description of what I am searching for. The mean across countries.

And why should I just not take the last year record or the first or the median or one other? Because as I said, I want to include the diachronical sence. Also there are some cases with Indicators, where a single record could include some errors and an average would hide them.

Anyway. There you go.