Unfortunately, I cannot answer fully your question. Though I'll give you my partial answer.
First of all, using entire price history of an index (from Yahoo), this is what I got:
Daily Weekly Monthly
DJIA 2.91 2.37 2.33
NASDAQ 100 1.74 1.94 1.91
NYSE Composite 1.61 1.59 1.85
S&P 100 1.75 1.90 1.97
S&P 600 Small Cap 1.59 1.82 1.99
Based on this, I don't think we can claim that the ratio is always 2.
So you agree, that the ratio is not always 2. But still, you want to know why it is equal to 2.91 or 1.59 or whatever.
This is how I would proceed in answering the question. First, the ratio can be expressed as
$$ \frac{E[max(P_1, P_2,...,P_n)-min(P_1, P_2,...,P_n)]} {E[|P_1 - P_n|]}$$
Second, I would start expanding the fraction in order to get a better picture of what exactly influences the ratio. I hope to expand the fraction, and then have some terms cancel each other out and then obtain as an answer 2, $\sigma / \mu$, or something else concise and beautiful. The problem is, it is extremely hard (at least for me) to obtain analytical expression for the expected value of a maximum (or minimum) of a sequence of correlated non-normal variables--the prices. I do not think anyone can give you the analytical expression for that. So this is where it ends as for analytical answer.
You can also use numerical methods, something like Monte Carlo simulation. Assume some model for prices, simulate them, and do some sensitivity analysis in parameters of the model to see how they affect the ratio.
Finally, one thing I do not get is why you are interested in that ratio. Shoundn't you instead be interested in $$ E(\frac{max(P_1, P_2,...,P_n)-min(P_1, P_2,...,P_n)]} {|P_1 - P_n|})$$
For example if the true expected value of the above ratio is equal to 10, and during a trading day the price is such that the ratio is 20, you would start buying the asset because you expect the close price to be higher than the current price. You would then sell the asset for a profit. Using your definition of the ratio, I see no usefulness in it. Care to explain?