1. Why does anything differ by many orders of magnitude?Some cities are very big. Some are very small. This fact has probably never bothered you before. But when you look at how cities sizes stack up, it looks somewhat peculiar:
The X-axis is the size of the city, in (natural) logarithmic scale. The Y-axis corresponds to the density (fraction) of cities with that population. The peak is around the mark of 8 on the X-axis, which corresponds to $e^8\approx 3,000$ people.
You can see that the empirical sizes of cities almost perfectly matches a normal ("bell curve") distribution. What's the explanation for this? Is mayoral talent distributed exponentially? When deciding to move to a new city do people first take the log of the new city's size and then roll some normally-distributed dice?
It turns out that this is solely due to dumb luck and mathematical inevitability.
Suppose every city grows by a random amount each year. One year, it will grow 10%, the next 5%, the year after it will shrink by 2%. After these three years, the total change in population is
$$1.10\cdot 1.05\cdot 0.98$$
As in the above graph, we take the log
$$\log\left(1.10\cdot 1.05\cdot 0.98\right)$$
A property of logarithms you may remember is that $\log(a\cdot b)=\log a + \log b$. Rewriting (2) with this property gives
$$\log 1.10+ \log 1.05+\log 0.98$$
The central limit theorem tells us that when you add a bunch of random things together, you'll end up with a normal distribution. We're clearly adding a bunch of random things together here, so we end up with the bell curve we see above.
2. Why charities might differ by many orders of magnitudeSome of Brian's points are about how even if a charity is good in one dimension, it's not necessarily good in others (performance is "independent"). The point of the above is to demonstrate that we don't need dependence to have widely varying impacts. We just need a structure where people's talents are randomly distributed, but critically their talents have a multiplicative effect.
There are some talents which obviously cause a multiplier. A charity's ability to handle logistics ("reduce overhead") will multiply the effectiveness of everything else they do. Their ability to increase the "denominator" of their intervention (number of bednets distributed, number of leaflets handed out, etc.) is another. PR skills, fundraising etc. all plausibly have a multiplicative impact.
More controversially, some proxies for flow-through effects might have a multiplicative impact. Scientific output is probably more valuable in times of peace than in times of war. GDP increases are probably better when there's a fair and just government, instead of the new wealth going to a few plutocrats.
Here's a simulation of charities' effectiveness with 10 dimensions, each uniformly drawn from the range [0,10].
3. Picking winnersSay that impact is the product of measurable, direct impacts and unmeasurable flow-through effects. Algebraically: $I=DF$. By linearity of expectations
So if two charities differ by a factor of say 1,000 in their direct impact then their total impact would (on expectation) differ by 1,000 as well.
This isn't a perfect model. But I do think that it's not always correct to model impacts as a sum of iid variables, and there is a plausible case to be made that not only do charities differ "astronomically" but we can expect those differences even with our limited knowledge.
This post was obviously inspired by Brian, and I talked about it with Gina extensively. The log-normal proof is known as Gibrat's Law and is not due to me.