Estimates, guesstimates, and asstimates
What's in a number…? No, really: where did it come from?
Estimates
An estimate is a value for an unobserved quantity calculated based on available information.
TreatsRUs corporation makes a product called “strawberry jam-filled cookies,” using Red Number 6 to turn soybean oil emulsified with corn syrup into “strawberry jam.”
When a batch of Red Number 6 arrives from a supplier, a sample is given a basic shop-floor test for contamination. The shop-floor test is 99% reliable and symmetric, meaning that the probability of false positives equals that of false negatives.1
TreatsRUs gets its Red Number 6 from two suppliers: Arch-Devilish-Machiavellians, A-D-M, with a certified rate of bad batches of 0.5%, and E. I. (Evil Ignoble) DupeOn, with a certified rate of bad batches of 2.5%.2
The issue facing TreatsRUs is what to do with batches that test positive for contamination. Hence, it needs the probability that a batch is really contaminated (bad) given that it tested positive in that shop-floor test.3
Decision analysis has determined that if that probability is over 60%, it’s cost-effective to refuse the batch and slow down production for a short period while waiting for a rush delivery; if it’s below 60%, it’s worth running a more expensive lab test which determines whether the batch is usable or not (because in more than 40% of cases it should be usable and production delays and rush deliveries are expensive).
To decide what to do when a batch fails a shop-floor test, TreatsRUs needs to estimate the probability that a batch of Red Number 6 from A-D-M or E.I. DupeOn that gets a positive result is actually bad.
Using Bayes’s rule and the numbers above, those probabilities are
Pr(bad batch | positive result; A-D-M) = 33.22%
Pr(bad batch | positive result; E.I. DupeOn) = 71.74%
Therefore if TreatsRUs gets a positive contamination result for a batch from A-D-M it’ll send that batch to the lab for further testing; if TreatsRUs get a positive contamination result for a batch from E.I. DupeOn, it will reject that batch, suspend production, and order a rush delivery.
This is where estimates come from and how they are used to inform decisions.
For the nerdly-inclined: how the reliability of the test and the quality of the suppliers map into the decision space for a variety of values:
Guesstimates
A guesstimate is a calculation of the value for an unknown quantity where the values for some of the parameters of the problem are reasonable guesses.
Guesstimates are used, for example, to answer questions used in quant job interviews to measure fluency of numerical reasoning and level of quantitative sophistication.
Say an interviewer asks “how many dump truck loads of dirt does Werner Ziegler need to move to clear an appropriate space under Lavandería Brillante?”
We start by guessing the size of the Gus Fring’s secret lab (say 15 by 16 by 6 meters, estimated by eye from how big it looks relative to people) and the capacity of a dump truck (say 10 cubic meters4); then the calculations proceed like those in an estimate: volume is 1440 cubic meters, so Werner will need 144 dump trucks.
A more interesting example of guesstimate is this tweet by Prof. Chris Combs
https://twitter.com/DrChrisCombs/status/1673438213057150980
and the one following it, calculating the pressure differential on a Coke can that is heated on a flame then suddenly cooled upside-down in a plate with water. Prof Combs and I guesstimate it at 2:1, rather than the 400:1 of the ill-fated Titan sub.
Many of the replies indicate that people have no idea how Prof. Combs is getting the result from the Ideal Gas Law, PV=nRT, so here’s my version:
The value of 600 °K is based on the paint on the can not running. The red equations are what happens when you solve across the red dotted line preceding them. The black text explains why some variables change and others don’t.
Mechanical engineers make temperature by pressure charts that are very good at explaining these things if you are fluent in the physics, but the above is enough to show what changes in each phase of the experiment.
(There's a chapter in my book DATA to INFORMATION to DECISION elaborating on quantitative reasoning using guesstimates, with suggestions to impress in a quant interview. Get a free sample here; that chapter is not in the sample, though.)
Asstimates
Asstimates are numbers pulled out of a person’s rear end —metaphorically speaking of course— in order to make a word argument seem quantitative. Typically there's some attempt at verisimilitude, but in the service of fooling the audience and not so much in service of accuracy.
For example, instead of saying “you can see very far to the horizon just by being on the deck of the Golden Gate Bridge,” Neil-de-STEM-influencer (generic character) will say “you can see 500 kilometers to the horizon just by being on the deck of the Golden Gate Bridge,” because the number sounds technical and convincing.
(The deck of the Golden Gate Bridge is about 75 meters above mean sea level, so minimal calculation shows that its horizon is around 30 kilometers.)
An asstimate isn’t just a guesstimate with bad guesses for the numbers. Asstimates are props to impress the audience, usually with no more thought than “what would sound credible and convincing here?”
The mother of all asstimates, IMNSHO, was made in 1995 by the then socialist candidate for Prime Minister of Portugal, now UN Secretary-General, António Guterres, who was campaigning on “spending 6% of the GDP on health services.”
When asked by a reporter what that meant in actual money, Guterres —who obviously had no idea what the GDP of the country he wanted to be Prime Minister of was— said something along the lines of “ahem, the GDP is 3 billion thousand-escudos, so times 6 it's 18 so ahem well you can do the math,” which got him roundly mocked for being unable to do the percentage calculation. (He couldn’t figure out how the decimals would work.)
The Portuguese GDP at the time was around 15 billion thousand-escudos. (The thousand-escudo, called “conto,” was the unofficial currency unit of Portugal at the time, an escudo being something like a cent would be to an American.)
Betting on the intelligence of the Portuguese electorate, the opposing social-democratic party made Guterres’s ignorance of the Portuguese economy a central point in their ensuing campaign. Naturally, Guterres won that election.
Portuguese speakers can relive this proud moment in our history here.
There are many examples of asstimates on twitter; but the examples I have are overwhelmingly from accounts that I find good in general (which is why I see their tweets); it would be unfair to present those occasional failures of these accounts as if they were representative.
How to tell if we’re getting an asstimate in three steps
Pay attention. People who use asstimates count on their audience being impressed by the numbers but just coding them as “big” or “small,” not as values. (Ironically, most of these numbers are made up in an attempt to sound technical when the word “big” or “small” would be perfectly correct.)
Source the information. Guterres should have paid attention to the economist behind him heard whispering “fifteen, it’s fifteen” on the video. Often people who asstimate something will do so while linking to a source that contradicts their number, since they count on the audience not being curious.
Calculate. Many asstimates are quickly unmasked by calculation. This is obvious in the Golden Gate Bridge example. Sometimes rescaling in obvious ways (per capita, per day) will show the incongruity of a number, sometimes a little more calculation is needed, but often not much.
We'll have none of that “type I error, type II error” nonsense here: using more words to convey less information (because more error-prone) to fewer people (the few who know which maps into false positive and which to false negative) is counterproductive.
Fictional suppliers; any similarity with real mega-corporations who employ skyscrapers full of lawyers is purely coincidental.
This is a variant on a common exercise in probability classes; also the source of something psychologists call the base rate neglect “bias,” though it’s in fact lack of fluency of technical material that required Euler, two Bernoullis, Gauss, and Bayes to create in the first place.
That's about 30 metric tonnes per truckload given the density of dirt and rock mix at about 3 times that of water so it sounds reasonable. It’s a guess, not a fact.