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This question came to mind when I was discussing a contest with a friend in which the challenge was to estimate the number of particles in a glass tube. The amount of particles (close to $5\cdot 10^4$) was large enough to have people give wrong estimates by more than 1 order of magnitude (i.e. $5\cdot 10^3$ and $5\cdot 10^5$).

My question is: does the wisdom of the crowd still apply to this situation with orders of magnitude estimation errors? What I could imagine is that if there are equally many people off by an order of magnitude in the 'up' and 'down' directions you would rather need to take something like a logarithmic average.

A second question is related to the way many people actually made the estimate. Instead of a random guess they made an estimative calculation saying: well, the length of the particle bed in the tube is $h_t\approx0.5$ m, the tube diameter is $d_t\approx 3 $ cm, the particles are approximately spherical with a diameter of $d_p\approx3$ mm and the void fraction will be about $\epsilon=40\%$. Then they came up with a number based on the calculation: $$N_p=\frac{V}{V_p}=\frac{6(1-\epsilon)h_t d_t^2}{d_p^3} $$

My second question is: does the wisdom of the crowd still apply for this estimative calculation? Or should we rather start working with averages per variable (particle diameter, void fraction etc) and use those averages for the final calculation? Or will the analysis be flawed anyway if the guesses are calculated.

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The first question is very interesting but psychological and empirical, not mathematical. You'd need to either understand how humans arrive at estimates or analyze large samples of estimates to determine the distribution of estimates. I suspect that the result would be different in different scenarios. Once you know how estimates are distributed, it becomes a mathematical question to find a suitable estimator. The second question is more mathematical, but you should specify a distribution for the estimates of the individual quantities. Are they normally distributed? – joriki Apr 29 '13 at 6:46
@joriki yes, I think it is safe to assume that the estimates for the individual quantities are normally distributed – Michiel Apr 29 '13 at 6:50

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