Just before the Trinity test, Enrico Fermi decided he wanted a rough estimate of the blast's power before the diagnostic data came in. So he dropped some pieces of paper from his hand as the blast wave passed him, and used this to estimate that the blast was equivalent to 10 kilotons of TNT. His guess was remarkably accurate for having so little data: the true answer turned out to be 20 kilotons of TNT.

Fermi had a knack for making roughly-accurate estimates with very little data, and therefore such an estimate is known today as a Fermi estimate.

Why bother with Fermi estimates, if your estimates are likely to be off by a factor of 2 or even 10? Often, getting an estimate within a factor of 10 or 20 is enough to make a decision. So Fermi estimates can save you a lot of time, especially as you gain more practice at making them.

 

Estimation tips

These first two sections are adapted from Guestimation 2.0.

Dare to be imprecise. Round things off enough to do the calculations in your head. I call this the spherical cow principle, after a joke about how physicists oversimplify things to make calculations feasible:

Milk production at a dairy farm was low, so the farmer asked a local university for help. A multidisciplinary team of professors was assembled, headed by a theoretical physicist. After two weeks of observation and analysis, the physicist told the farmer, "I have the solution, but it only works in the case of spherical cows in a vacuum."

By the spherical cow principle, there are 300 days in a year, people are six feet (or 2 meters) tall, the circumference of the Earth is 20,000 mi (or 40,000 km), and cows are spheres of meat and bone 4 feet (or 1 meter) in diameter.

Decompose the problem. Sometimes you can give an estimate in one step, within a factor of 10. (How much does a new compact car cost? $20,000.) But in most cases, you'll need to break the problem into several pieces, estimate each of them, and then recombine them. I'll give several examples below.

Estimate by bounding. Sometimes it is easier to give lower and upper bounds than to give a point estimate. How much time per day does the average 15-year-old watch TV? I don't spend any time with 15-year-olds, so I haven't a clue. It could be 30 minutes, or 3 hours, or 5 hours, but I'm pretty confident it's more than 2 minutes and less than 7 hours (400 minutes, by the spherical cow principle).

Can we convert those bounds into an estimate? You bet. But we don't do it by taking the average. That would give us (2 mins + 400 mins)/2 = 201 mins, which is within a factor of 2 from our upper bound, but a factor 100 greater than our lower bound. Since our goal is to estimate the answer within a factor of 10, we'll probably be way off.

Instead, we take the geometric mean — the square root of the product of our upper and lower bounds. But square roots often require a calculator, so instead we'll take the approximate geometric mean (AGM). To do that, we average the coefficients and exponents of our upper and lower bounds.

So what is the AGM of 2 and 400? Well, 2 is 2×10⁰, and 400 is 4×10². The average of the coefficients (2 and 4) is 3; the average of the exponents (0 and 2) is 1. So, the AGM of 2 and 400 is 3×10¹, or 30. The precise geometric mean of 2 and 400 turns out to be 28.28. Not bad.

What if the sum of the exponents is an odd number? Then we round the resulting exponent down, and multiply the final answer by three. So suppose my lower and upper bounds for how much TV the average 15-year-old watches had been 20 mins and 400 mins. Now we calculate the AGM like this: 20 is 2×10¹, and 400 is still 4×10². The average of the coefficients (2 and 4) is 3; the average of the exponents (1 and 2) is 1.5. So we round the exponent down to 1, and we multiple the final result by three: 3(3×10¹) = 90 mins. The precise geometric mean of 20 and 400 is 89.44. Again, not bad.

Sanity-check your answer. You should always sanity-check your final estimate by comparing it to some reasonable analogue. You'll see examples of this below.

Use Google as needed. You can often quickly find the exact quantity you're trying to estimate on Google, or at least some piece of the problem. In those cases, it's probably not worth trying to estimate it without Google.

Standard Intro

The following section will be at the top of all posts in the LW Women series.

About two months ago, I put out a call for anonymous submissions by the women on LW, with the idea that I would compile them into some kind of post.  There is a LOT of material, so I am breaking them down into more manageable-sized themed posts. 

Seven women submitted, totaling about 18 pages. 

Crocker's Warning- Submitters were told to not hold back for politeness. You are allowed to disagree, but these are candid comments; if you consider candidness impolite, I suggest you not read this post

To the submittrs- If you would like to respond anonymously to a comment (for example if there is a comment questioning something in your post, and you want to clarify), you can PM your message and I will post it for you. If this happens a lot, I might create a LW_Women sockpuppet account for the submitters to share.

Standard Disclaimer- Women have many different viewpoints, and just because I am acting as an intermediary to allow for anonymous communication does NOT mean that I agree with everything that will be posted in this series. (It would be rather impossible to, since there are some posts arguing opposite sides!)

Please do NOT break anonymity, because it lowers the anonymity of the rest of the submitters.

Suppose a general-population survey shows that people who exercise less, weigh more. You don't have any known direction of time in the data - you don't know which came first, the increased weight or the diminished exercise. And you didn't randomly assign half the population to exercise less; you just surveyed an existing population.

The statisticians who discovered causality were trying to find a way to distinguish, within survey data, the direction of cause and effect - whether, as common sense would have it, more obese people exercise less because they find physical activity less rewarding; or whether, as in the virtue theory of metabolism, lack of exercise actually causes weight gain due to divine punishment for the sin of sloth.

vs.

The usual way to resolve this sort of question is by randomized intervention. If you randomly assign half your experimental subjects to exercise more, and afterward the increased-exercise group doesn't lose any weight compared to the control group [1], you could rule out causality from exercise to weight, and conclude that the correlation between weight and exercise is probably due to physical activity being less fun when you're overweight [3]. The question is whether you can get causal data without interventions.

For a long time, the conventional wisdom in philosophy was that this was impossible unless you knew the direction of time and knew which event had happened first. Among some philosophers of science, there was a belief that the "direction of causality" was a meaningless question, and that in the universe itself there were only correlations - that "cause and effect" was something unobservable and undefinable, that only unsophisticated non-statisticians believed in due to their lack of formal training:

"The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm." -- Bertrand Russell (he later changed his mind)

"Beyond such discarded fundamentals as 'matter' and 'force' lies still another fetish among the inscrutable arcana of modern science, namely, the category of cause and effect." -- Karl Pearson

The famous statistician Fisher, who was also a smoker, testified before Congress that the correlation between smoking and lung cancer couldn't prove that the former caused the latter.  We have remnants of this type of reasoning in old-school "Correlation does not imply causation", without the now-standard appendix, "But it sure is a hint".

This skepticism was overturned by a surprisingly simple mathematical observation.

I think people who are not made happier by having things either have the wrong things, or have them incorrectly.  Here is how I get the most out of my stuff.

Money doesn't buy happiness.  If you want to try throwing money at the problem anyway, you should buy experiences like vacations or services, rather than purchasing objects.  If you have to buy objects, they should be absolute and not positional goods; positional goods just put you on a treadmill and you're never going to catch up.

Supposedly.

I think getting value out of spending money, owning objects, and having positional goods are all three of them skills, that people often don't have naturally but can develop.  I'm going to focus mostly on the middle skill: how to have things correctly¹.

(This is the first post of a new Sequence, Highly Advanced Epistemology 101 for Beginners, setting up the Sequence Open Problems in Friendly AI.  For experienced readers, this first post may seem somewhat elementary; but it serves as a basis for what follows.  And though it may be conventional in standard philosophy, the world at large does not know it, and it is useful to know a compact explanation.  Kudos to Alex Altair for helping in the production and editing of this post and Sequence!)

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I remember this paper I wrote on existentialism. My teacher gave it back with an F. She’d underlined true and truth wherever it appeared in the essay, probably about twenty times, with a question mark beside each. She wanted to know what I meant by truth.
-- Danielle Egan

I understand what it means for a hypothesis to be elegant, or falsifiable, or compatible with the evidence. It sounds to me like calling a belief ‘true’ or ‘real’ or ‘actual’ is merely the difference between saying you believe something, and saying you really really believe something.
-- Dale Carrico

What then is truth? A movable host of metaphors, metonymies, and; anthropomorphisms: in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished, and which, after long usage, seem to a people to be fixed, canonical, and binding.
-- Friedrich Nietzche

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The Sally-Anne False-Belief task is an experiment used to tell whether a child understands the difference between belief and reality. It goes as follows:

1. The child sees Sally hide a marble inside a covered basket, as Anne looks on.

2. Sally leaves the room, and Anne takes the marble out of the basket and hides it inside a lidded box.

3. Anne leaves the room, and Sally returns.

4. The experimenter asks the child where Sally will look for her marble.

Children under the age of four say that Sally will look for her marble inside the box. Children over the age of four say that Sally will look for her marble inside the basket.