[gruez]

>unbiased = all companies, or unbiased random sampling of companies)

Most indexes are market-cap weighted for a reason. Suppose we weigh each company equally. If you have 90 shitty companies and 10 good companies, but the 10 good companies combined are 10x as big as the 90 shitty companies combined, is it fair to conclude that "companies are shitty"? Suppose the 10 good companies spun themselves out into 100 separate companies, did the quality of companies improve by 5x (10% good to 53% good)?

[Nevermark]

I think you are mixing two different points.

1) Of course indexes are biased. Lots of good reasons for that as you point out.

2) But that makes indexes unrepresentative of average returns. Average returns include all caps, or a good estimate could be had from a random selection, weighted by their individual caps (as you correctly point out).

(That is what it means to have an unbiased estimate of total returns on total cap of all companies. An accurate estimate cannot have biases: not a bias toward smaller companies simply because there are more of them, nor biased toward large companies, as most indexes do.)

[gruez]

>2) But that makes indexes unrepresentative of average returns

But average returns should be market cap weighted to be meaningful. If you made investments of $10 and $1, which were later valued $12 and $2 respectively, what was your "average return"? The average of the two investments, (12/10 + 2/1)/2 = 1.60? or the weighted average (12+2)/(10+1) = 1.27?

[Nevermark]

You are right.

Actual average returns are simply total cap at one time divided by total cap at a previous time: TC(t2)/TC(t1)

That probably is easy enough to do.

But an estimate of that would be an average of returns for randomly chosen companies, but with either:

1) straight random selection of companies, with the individual returns weighted by their company size, or ...

2) ... random selection of companies, with probabilities of each company weighted by size, then the unweighted average of those returns.

So you are right, one way or another a subsample of companies needs to weight companies by size to account for the greater levels of investment in large companies.