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.