Strongest 'average' for a diverse set of numbers?

Question

I have a set of numbers consisting of two general size numbers: size 'a', and size 'b' which are about three times bigger in size than size 'a'. There is some variation and the list might look like this:

{60.2, 180.6, 59.8, 59.9, 61.3, 181.0 ...}

I don't know the order the numbers occur in.

I want to determine what the 'strongest' 'a' is; e.g. '60'. What tool can I use to do this? Fourier transform?

Answer 1

First I would try to guess which numbers were in A vs. B. Order the list and take the elements 3/8 of the way in and 7/8 of the way in and classify elements closer to the first as A and closer to the second as B as a first guess. (The right sense of 'close' will depend on your problem; closer in terms of ratio or in terms of absolute difference or something else.) Then take the median of those elements guessed to be in A as your estimate of the 'strongest'.

Alternately, a bit more work on the initial division might be useful. Split the list at 75%, then compute the median of each set. As long as there is some element on the end of either list which is closer to the median of the other set, move the element over, recalculate the median, and repeat. When you're finally done the median of A is your best guess.

Answer 2

Create a sequence $S$ of all the elements sorted in ascending order.

Then, starting from $n=0$, search for $n$ such that $\dfrac{S_{n+1}}{S_n}>2$.

The value that you are looking for is $S_n$.