Explanation
Fermi estimates generally work because the estimations of the individual terms are often close to correct, and overestimates and underestimates help cancel each other out. That is, if there is no consistent bias, a Fermi calculation that involves the multiplication of several estimated factors (such as the number of piano tuners in Chicago) will probably be more accurate than might be first supposed.
In detail, multiplying estimates corresponds to adding their logarithms; thus one obtains a sort of Wiener process or random walk on the logarithmic scale, which diffuses as (in number of terms n). In discrete terms, the number of overestimates minus underestimates will have a binomial distribution. In continuous terms, if one makes a Fermi estimate of n steps, with standard deviation units on the log scale from the actual value, then the overall estimate will have standard deviation, since the standard deviation of a sum scales as in the number of summands.
For instance, if one makes a 9-step Fermi estimate, at each step overestimating or underestimating the correct number by a factor of 2 (or with a standard deviation 2), then after 9 steps the standard error will have grown by a logarithmic factor of, so . Thus one will expect to be within 1/8 to 8 times the correct value – within an order of magnitude, and much less than the worst case of erring by a factor of (about 2.7 orders of magnitude). If one has a shorter chain or estimates more accurately, the overall estimate will be correspondingly better.
Read more about this topic: Fermi Problem
Famous quotes containing the word explanation:
“Auden, MacNeice, Day Lewis, I have read them all,
Hoping against hope to hear the authentic call . . .
And know the explanation I must pass is this
MYou cannot light a match on a crumbling wall.”
—Hugh MacDiarmid (1892–1978)
“We live between two worlds; we soar in the atmosphere; we creep upon the soil; we have the aspirations of creators and the propensities of quadrupeds. There can be but one explanation of this fact. We are passing from the animal into a higher form, and the drama of this planet is in its second act.”
—W. Winwood Reade (1838–1875)
“Young children constantly invent new explanations to account for complex processes. And since their inventions change from week to week, furnishing the “correct” explanation is not quite so important as conveying a willingness to discuss the subject. Become an “askable parent.””
—Ruth Formanek (20th century)