The "standard" definition of a weighted average of n variables P_{1}, P_{2} ... P_{n} is:
[A]
Weighted Average = (W_{1} P_{1} + W_{2} P_{2} + ... + W_{n} P_{n}) /
(W_{1} + W_{2} + ... + W_{n})
= Σ W_{j} P_{j} / Σ W_{j}
where the Ws are the weights assigned to each variable.

>So?
So, I've always felt that the typical Stock Market Indexes are not weighted at all.
Indeed, I wrote this when I was trying to understand why the adjective "weighted'.
>So?
I'd like to try again.
>Be my guest.
The DOW Index is proportional to the sum of stock prices. It's usually called a "Price Weighted Index":
DOW(1) = k (P_{1} + P_{2} + ... + P_{n}) = k Σ P_{j}
where k is a constant and the Ps are the stock prices.

Suppose the gain factors for each stock were g_{1}, g_{2} ... g_{n}
(where, by "gain factor", we mean: if a $1 investment changes to $1.23, the gain factor is 1.23).
Then, applying such gains to each stock price, the Index would become
DOW(2) = k (g_{1} P_{1} + g_{2} P_{2} + ... + g_{n} P_{n}) = k Σ g_{j} P_{j}
where the gs are the gain factors and P_{j} are the stock prices.

The gain factor for the Index is then:
[B]
DOW Gain Factor = DOW(2) / DOW(1) = Σ g_{j} P_{j} / Σ P_{j}

Indexes like the S&P500 are socalled "Market Cap Weighted". They're proportional to the sum of market caps of all the stocks in the Index:
Mkt Cap Wgt'd Index:
S&P(1) = k (N_{1} P_{1} + N_{2} P_{2} + ... + N_{n} P_{n}) = k Σ N_{j} P_{j} = k Σ M_{j}
where k is a constant, the Ns are the number of outstanding shares and the Ps are the prices of the stocks and M_{j} = N_{j} P_{j} is a market Capitalization.

Applying gain factors to each stock price, the Index would become
S&P(2) =k (g_{1} M_{1} + g_{2} M_{2} + ... + g_{n} M_{n}) = k Σ g_{j} M_{j}
where the gs are the gain factors and the M_{j}s are the Market Caps.

The gain factor for the Index is then:
[C]
S&P Gain Factor = S&P(2) / S&P(1) = Σ g_{j} M_{j} / Σ M_{j}
where M_{j} = N_{j} P_{j} are the Market Caps for the individual stocks.

Note:
We may argue about whether the DOW or the S&P Indexes are "really" weighted averages, BUT
The Gain Factor for each Index is a real, live weighted average of the individual gain factors.
[B] says the DOW Gain Factor has weights equal to the individual Prices.
[C] says the S&P Gain Factor has weights equal to the individual Market Caps.
So, what prescription makes more sense?
There seems to be little to choose between 'em, eh?
In fact:
If you bought one share of each of the 30 DOW stocks, your gains would be the same as the DOW.
If you invested $M_{j} in each of the 500 S&P stocks, your gains would be the same as the S&P.
>Huh? M_{j}?
Yes ... an amount equal to the Mkt Cap.
 
>And if I invested $A in each of the stocks? What then?
Your gains would be the Average Gain of the stocks in the Index.
>Huh?
Your total investment would be $nA ... that's $A for each of the n stocks in the Index.
When the stocks increase by factors g_{j}, your investment would be worth:
Σ gain * investment = Σ g_{j}A = A Σ g_{j}
Starting with $nA, that'd be a Gain Factor of A Σ g_{j} / (nA):
[D]
Gain Factor = (1/n)Σ g_{j} = the Average (or Mean) of n Stock Gain Factors
assuming you invest equal amounts in each of n stocks.

>Are you happy now?
Very!
Note:
If we compare the growth (decay?) of the DOW and S&P (over the past 10 years) and the gIndex, then ...
>Huh? gIndex?
Well ... uh, that's what I'm calling the Index where you just average the gains (beginning at some convenient point in time).
>Like 10 years ago?
Yes. Anyway, you'd get this
See? The gIndex done good, eh?
There's a spreadsheet to play with: click here.
 
>Why would anyone use that gIndex instead of a Priceweighted or MktCapweighted Index?
I dunno. Suppose you wanted to invest in the DOW stocks and didn't know which would be the best performer.
How much would you invest in each stock? Would you be guided by the prices or the mkt caps?
>Knowing nothing about their future behaviour? I think I'd invest equal amounts in each stock.
Exactly ... and that'd be the gIndex.
Okay, so I ask my brotherinlaw (Gerry) how he'd buy the 30 DOW stocks.
Not knowing which stocks should be overweighted, surely he'd invest equal amounts in each, right?
No. He figures: the smaller the mkt cap, the more he'd invest.
So I interpret that to mean the amount invested is inversely proportional to the mkt cap.
So I whip out a ...
>So you whip out a spreadsheet ...
Exactly ... and, using the DOW stocks over the past year, I get this
>It's better than your gIndex !!
Uh ... yes.
 
>And the spreadsheet?
It looks like this: (Click on the picture to download the spreadsheet).
>But don't you need the market caps, too?
Yes. There's a sheet for that:
Note that, for the past year, the largest weights are assigned to Alcoa (AA), DuPont (DD) and Travelers (TRV).
Surprise! They were all up about 20% over the past year.
On the other hand, some of the big guys, like Exxon (XOM) and Walmart (WMT), had negative returns.
P.S.
Gerry wrote to say that he thinks allocations inversely proportional to the Mkt Cap is too drastic.
The spreadsheet now has an allocation proportional to MktCap^{P} and you get to pick P.
You can also click a button and get the spreadsheet to search for the best P in a range.
Turns out that P = 1.24 is "best". That is, allocation should be proportional to 1 / MktCap^{1.24}.
