I didn't peruse the source code. I just read the linked article in its entirety and it says
> The computation is quite slow. In order to stay as flexible as possible, I'm using the Monte Carlo method. Which means the calculator is running about 250K AST-based computations for every calculation you put forth.
So therefore I conclude Monte Carlo is being used.
Line 19 to 21 should be the Monte-Carlo sampling algorithm. The implementation is maybe a bit unintuitive but apparently he creates a function from the expression in the calculator, calling that function gives a random value from that function.
Wow, this is fantastic! I did not know about squiggle language, and it's basically what I was trying to get to from my unsure calculator through my next project (https://filiph.github.io/napkin/). Squiggle looks and works much better.
I think they should be functions: G(50, 1) for a Gaussian with µ=50, σ=1; N(3) for a negative exponential with λ=3, U(0, 1) for a uniform distribution between 0 and 1, UI(1, 6) for an uniform integer distribution from 1 to 6, etc. Seems much more flexible, and easier to remember.
I actually stumbled upon this a while ago from social media and the web version has a somewhat annoying latency, so I wrote my own version in Python. It uses numpy so it's faster. https://gist.github.com/kccqzy/d3fa7cdb064e03b16acfbefb76645... Thank you filiph for this brilliant idea!
The reason I'm asking: unsure also has a CLI version (which is leaps and bounds faster and in some ways easier to use) but I rarely find myself using it. (Nowadays, I use https://filiph.github.io/napkin/, anyway, but it's still a web app rather than a CLI tool.)
I love this! As a tool for helping folks with a good base in arithmetic develop statistical intuition, I can't think offhand of what I've seen that's better.
i like it and i skimmed the post but i don't understand why the default example 100 / 4~6 has a median of 20? there is no way of knowing why the range is between 4 and 6
mmh ok thanks, i guess i need extra maths training ;)
i didn't mean knowing _that_ the range is between 4 and 6 but _why_, i thought the weighing would be explained by the reasoning, like: "we divide a €100 bill between possibly 4, rather 5 and most probably not 6 persons"
An alternative approach is using fuzzy-numbers. If evaluated with interval arithmetic you can do very long calculations involving uncertain numbers very fast and with strong mathematical guarantees.
It would especially outperform the Monte-Carlo approach drastically.
I'm familiar with fuzzy numbers (e.g. see my https://filiph.net/fuzzy/ toy) but I didn't know there's arithmetic with fuzzy numbers. How is it done? Do you have a link?
There is a book by Hanss on it. It focuses on the sampling approach (he calls it "transformation method") though.
If you want to do arithmetic and not a black box approach you just have to realize that you can perform them on the alpha-cuts with ordinary interval arithmetic. Then you can evaluate arbitrary expressions involving fuzzy numbers, keeping the strengths and weaknesses of interval arithmetic.
The sampling based approach is very similar to Monte-Carlo, but you sample at certain well defined points.
> Range is always a normal distribution, with the lower number being two standard deviations below the mean, and the upper number two standard deviations above. Nothing fancier is possible, in terms of input probability distributions.
It sounds like a gimmick at first, but looks surprisingly useful. I'd surely install it if it was available as an app to use alongside my usual calculator, and while I cannot quite recall a situation when I needed it, it seems very plausible that I'll start finding use cases once I have it bound to some hotkey on my keyboard.
I just threw numbers into there (population x ownership percent x replacement frequency x unit cost) to estimate the annual revenue of the smartphone market and got a few percentage points away from what the internet reports is the true value.
Because there are trig functions, this would also be nice for reverse engineering of complex parts from simple measurements (feature volumes, corner angles, cross-sectional areas)
Multimodal would be nice but not necessary.
Inverse trig functions would be interesting but complicated and not necessary.
In any case, this tool is more convenient than opening Python every time I want to estimate a range of answers.
Would be nice to retransform the output into an interval / gaussian distribution
Note: If you're curious why there is a negative number (-5) in the histogram, that's just an inevitable downside of the simplicity of the Unsure Calculator. Without further knowledge, the calculator cannot know that a negative number is impossible
Drake Equation or equation multiplying probabilities can also be seen in log space, where the uncertainty is on the scale of each probability, and the final probability is the product of exponential of the log probabilities. And we wouldnt have this negative issue
If I am reading this right, a range is expressed as a distance between the minimum and maximum values, and in the Monte Carlo part a number is generated from a uniform distribution within that range[1].
But if I just ask the calculator "1~2" (i.e. just a range without any operators), the histogram shows what looks like a normal distribution centered around 1.5[2].
Shouldn't the histogram be flat if the distribution is uniform?
> Range is always a normal distribution, with the lower number being two standard deviations below the mean, and the upper number two standard deviations above. Nothing fancier is possible, in terms of input probability distributions.
Part of the confusion here is likely that the tool, as seen on the web, probably lags significantly behind the code. I've started using a related but different tool (https://filiph.github.io/napkin/).
The HN mods gave me an opportunity to resubmit the link, so I did. If I had more time, I'd have also upgraded the tool to the latest version and fix the wording. But unfortunately, I didn't find the time to do this.
I'm guessing this is not an error. If you divide 1/normal(0,1), the full distribution would range from -inf to inf, but the 95% output doesn't have to.
I don't quite understand, probably because my math isn't good enough.
If you're treating -1~1 as a normal distribution, then it's centered on 0. If you're working out the answer using a Monte Carlo simulation, then you're going to be testing out different values from that distribution, right? And aren't you going to be more likely to test values closer to 0? So surely the most likely outputs should be far from 0, right?
When I look at the histogram it creates, it varies by run, but the most common output seems generally closest to zero (and sometimes is exactly zero). Wouldn't that mean that it's most frequently picking values closest to -1 or 1 denoninator?
That may be true, but if you look at the distribution it puts out for this, it definitely smells funny. It looks like a very steep normal distribution, centered at 0 (ish). Seems like it should have two peaks? But maybe those are just getting compressed into one because of resolution of buckets?
OK, but do we necessarily just care about the central 95% range of the output? This calculation has the weird property that values in the tails of the input correspond to values in the middle of the output, and vice versa. If you follow the intuition that the range you specify in the input corresponds to the values you expect to see, the corresponding outputs would really include -inf and inf.
Now I'm realizing that this doesn't actually work, and even in more typical calculations the input values that produce the central 95% of the output are not necessarily drawn from the 95% CIs of the inputs. Which is fine and makes sense, but this example makes it very obvious how arbitrary it is to just drop the lowermost and uppermost 2.5%s rather than choosing any other 95/5 partition of the probability mass.
If X is normal and centered around 0, then the average of 1/X does not exist (math speak for "is infinity" in this case). In these cases Monte Carlo simulations are not reliable because they give high variance estimates (math speak for "the histogram varies run by run").
The ASCII art (well technically ANSI art) histogram is neat. Cool hack to get something done quickly. I'd have spent 5x the time trying various chart libraries and giving up.
The android app fits lognormals, and 90% rather than 95% confidence intervals. I think they are a more parsimonious distribution for doing these kinds of estimates. One hint might be that, per the central limit theorem, sums of independent variables will tend to normals, which means that products will tend to be lognormals, and for the decompositions quick estimates are most useful, multiplications are more common
In the grand HN tradition of being triggered by a word in the post and going off on a not-quite-but-basically-totally-tangential rant:
There’s (at least) three areas here that are footguns with these kinds of calculations:
1) 95% is usually a lot wider than people think - people take 95% as “I’m pretty sure it’s this,” whereas it’s really closer to “it’d be really surprising if it were not this” - by and large people keep their mental error bars too close.
2) probability is rarely truly uncorrelated - call this the “Mortgage Derivatives” maxim. In the family example, rent is very likely to be correlated with food costs - so, if rent is high, food costs are also likely to be high. This skews the distribution - modeling with an unweighted uniform distribution will lead to you being surprised at how improbable the actual outcome was.
3) In general normal distributions are rarer than people think - they tend to require some kind of constraining factor on the values to enforce. We see them a bunch in nature because there tends to be negative feedback loops all over the place, but once you leave the relatively tidy garden of Mother Nature for the chaos of human affairs, normal distributions get pretty abnormal.
I like this as a tool, and I like the implementation, I’ve just seen a lot of people pick up statistics for the first time and lose a finger.
I strongly agree with this, and particularly point 1. If you ask people to provide estimated ranges for answers that they are 90% confident in, people on average produce roughly 30% confidence intervals instead. Over 90% of people don't even get to 70% confidence intervals.
I don't think estimation errors regarding things outside of someone's area of familiarity say much.
You could ask a much "easier"" question from the same topic area and still get terrible answers: "What percentage of blue whales are blue?" Or just "Are blue whales blue?"
Estimating something often encountered but uncounted seems like a better test. Like how many cars pass in front of my house every day. I could apply arithmetic, soft logic and intuition to that. But that would be a difficult question to grade, given it has no universal answer.
I have no familiarity with blue whales but I would guess they're 1--5 times the mass of lorries, which I guess weigh like 10--20 cars which I in turn estimate at 1.2--2 tonnes, so primitively 12--200 tonnes for a normal blue whale. This also aligns with it being at least twice as large as an elephant, something I estimate at 5 tonnes.
The question asks for the heaviest, which I think cannot be more than three times the normal weight, and probably no less than 1.3. That lands me at 15--600 tonnes using primitive arithmetic. The calculator in OP suggests 40--320.
The real value is apparently 170, but that doesn't really matter. The process of arriving at an interval that is as wide as necessary but no wider is the point.
Estimation is a skill that can be trained. It is a generic skill that does not rely on domain knowledge beyond some common sense.
I would say general knowledge in many domains may help with this as you can try and approximate to the nearest thing you know from that domain.
How you get good at being a generalist is the tricky part, my best bet is reading and doing a lot of trivia (I found crosswords to be somewhat effective at this, but far from being efficient)
No, that has nothing to do with it. Trivia helps you narrow down an interval. It is not necessary to construct a correct interval, which can be of any width.
So the context of the quiz is software estimation, where I assume it's an intentional parable of estimating something you haven't seen before. It's trying to demonstrate that your "5-7 days" estimate probably represents far more certainty than you intended.
For some of these, your answer could span orders of magnitude. E.g. my answer for the heaviest blue whale would probably be 5-500 tons because I don't have a good concept of things that weigh 500 tons. The important point is that I'm right around 9 times in 10, not that I had a precise estimate.
I don't know, an estimate spanning three orders of magnitude doesn't seem useful.
To continue your example of 5-7 days, it would turn into an estimate of 5-700 days. So somewhere between a week or two years. And fair enough, whatever you're estimating will land somewhere in between. But how do I proceed from there with actual planning or budget?
I mean it's no less useful than a more precise, but less certain estimate. It means you either need to do some work to improve your certainty (e.g. in the case of this quiz, allow spending more than 10 minutes or allow research) or prepare for the possibility that it's 700 days.
Edit: And by the way given a large enough view, estimates like this can still be valuable, because when you add these estimates together the resulting probability distribution narrows considerably. e.g. at just 10 tasks of this size, you get a 95% CI of 245~460 per task. At 20, 225~430 per task.
Note that this is obviously reductive as there's no way an estimate of 5-700 would imply a normal distribution centred at 352.5, it would be more like a logarithmic distribution where the mean is around 10 days. And additionally, this treats each task as independent...i.e. one estimate being at the high end wouldn't mean another one would be as well.
> But how do I proceed from there with actual planning or budget?
You make up the number you wanted to hear in the first place that ostensibly works with the rest of the schedule. That’s why engineering estimates are so useless - it’s not that they’re inaccurate or unrealistic - it’s that if we insisted on giving them realistic estimates we’d get fired and replaced by someone else who is willing to appease management and just kick the can down the road a few more weeks.
Your question is akin to asking ‘how do I make the tail to wag the dog?’
Your budget should be allocated for say 80% confidence (which the tool helpfully provides behind a switch) and your stakeholders must be on board with this. It shouldn’t be too hard to do since everyone has some experience with missed engineering deadlines. (Bezos would probably say 70% or even less.)
It shouldn't matter how familiar you are with the question. If you're pretty familiar, give a narrow 90% credence interval. If you're unfamiliar, give a wide interval.
I did a project with non-technical stakeholders modeling likely completion dates for a big GANTT chart. Business stakeholders wanted probabilistic task completion times because some of the tasks were new and impractical to quantify with fixed times.
Stakeholders really liked specifying work times as t_i ~ PERT(min, mode, max) because it mimics their thinking and handles typical real-world asymmetrical distributions.
[Background: PERT is just a re-parameterized beta distribution that's more user-friendly and intuitive https://rpubs.com/Kraj86186/985700]
This looks like a much more sophisticated version of PERT than I have seen used. When people around me have claimed to use PERT, they have just added together all the small numbers, all the middle numbers, and all the big numbers. That results in a distribution that is too extreme in both lower and upper bound.
This jives with my general reaction to the post, which was that the added complexity and difficulty of reasoning about the ranges actually made me feel less confident in the result of their example calculation. I liked the $50 result, you can tack on a plus or minus range but generally feel like you're about breakeven. On the other hand, "95% sure the real balance will fall into the -$60 to +$220 range" feels like it's creating a false sense of having more concrete information when you've really just added compounding uncertainties at every step (if we don't know that each one is definitely 95%, or the true min/max, we're just adding more guesses to be potentially wrong about). That's why I don't like the Drake equation, every step is just compounding wild-ass guesses, is it really producing a useful number?
It is producing a useful number. As more truly independent terms are added, error grows with the square root while the point estimation grows linearly. In the aggregate, the error makes up less of the point estimation.
This is the reason Fermi estimation works. You can test people on it, and almost universally they get more accurate with this method.
If you got less certain of the result in the example, that's probably a good thing. People are default overconfident with their estimated error bars.
They are meaning the same thing. The original comment pointed out that people’s qualitative description and mental model of the 95% interval means they are overconfident… they think 95 means ‘pretty sure I’m right’ rather than ‘it would be surprising to be wrong’
Read a bit on Fermi estimation, I'm not quite sure exactly what the "method" is in contrast to a less accurate method, it's basically just getting people to think in terms of dimensional analysis? This passage from the Wikipedia is interesting:
By contrast, precise calculations can be extremely complex but with the expectation that the answer they produce is correct. The far larger number of factors and operations involved can obscure a very significant error, either in mathematical process or in the assumptions the equation is based on, but the result may still be assumed to be right because it has been derived from a precise formula that is expected to yield good results.
So the strength of it is in keeping it simple and not trying to get too fancy, with the understanding that it's just a ballpark/sanity check. I still feel like the Drake equation in particular has too many terms for which we don't have enough sample data to produce a reasonable guess. But I think this is generally understood and it's seen as more of a thought experiment.
I think the point is to create uncertainty, though, or to at least capture it. You mention tacking a plus/minus range to $50, but my suspicion is that people's expected plus/minus would be narrower than the actual - I think the primary value of the example is that it makes it clear there's a very real possibility of the outcome being negative, which I don't think most people would acknowledge when they got the initial positive result. The increased uncertainty and the decreased confidence in the result is a feature, not a bug.
I think to do all that you’d need a full on DSL rather than something pocket calculator like. I think adding a triangular distribution would be good though.
Normal distributions are the maximum entropy distributions for a given mean and variance. Therefore, in accordance with the principle of maximum entropy, unless you have some reason to not pick a normal distribution (e.g. you know your values must be non-negative), you should be using a normal distribution.
Without having fully digested how the Unsure Calculator computes, it seems to me you could perhaps "weight" the ranges you pass to the calculator. Rather than a standard bell curve the Calculator could apply a more tightly focused — or perhaps skewed curve for that term.
If you think your salary will be in the range of 10 to 20, but more likely closer to 10 you could:
10<~20 (not to be confused with less-than)
or: 10!~20 (not to be confused with factorial)
or even: 10~12~20 to indicate a range of 10 to 20 ... leaning toward 12.
The correlation in this case isn't about the distribution for the individual event, it's about the interactions between them - so, for instance, Rent could be anywhere between 1200 and 1800, and Food could be anywhere between 100 and 150, but if Rent is 1200, it means Food is more likely to be 100, and if Food is 150, it means Rent is more likely to be 1800. Basically, there's a shared factor that's influencing both (local cost of living) that's the actual thing you need to model.
So, a realistic modeling isn't 1200~1500 + 100~150, it's (1~1.5)*(1200 + 150) - the "cost of living" distribution applies to both factors.
Great points. I think the idea of this calculator could just be simply extended to specific use cases to make the statistical calculation simple and take into account additional variables. Moving being one example.
>rent is very likely to be correlated with food costs - so, if rent is high, food costs are also likely to be high
Not sure I agree with this. It's reasonable to have a model where the mean rent may be correlated with the mean food cost, but given those two parameters we can model the fluctuations about the mean as uncorrelated. In any case at the point when you want to consider something like this you need to do proper Bayesian statistics anyways.
>In general normal distributions are rarer than people think - they tend to require some kind of constraining factor on the values to enforce.
I don't know where you're getting this from. One needs uncorrelated errors, but this isn't a "constraint" or "negative feedback".
The family example is a pat example, but take something like project planning - two tasks, each one takes between 2 and 4 weeks - except that they’re both reliant on Jim, and if Jim takes the “over” on task 1, what’s the odds he takes the “under” on task 2?
This is why I joked about it as the mortgage derivatives maxim - what happened in 2008 (mathematically, at least - the parts of the crisis that aren’t covered by the famous Upton Sinclair quote) was that the mortgage backed derivatives were modeled as an aggregate of a thousand uncorrelated outcomes (a mortgage going bust), without taking into account that at least a subset of the conditions leading to one mortgage going bust would also lead to a separate unrelated mortgage going bust - the results were not uncorrelated, and treating them as such meant the “1 in a million” outcome was substantially more likely in reality than the model allowed.
Re: negative feedback - that’s a separate point from the uncorrelated errors problem above, and a critique of using the normal distribution at all for modeling many different scenarios. Normal distributions rely on some kind of, well, normal scattering of the outcomes, which means there’s some reason why they’d tend to clump around a central value. We see it in natural systems because there’s some constraints on things like height and weight of an organism, etc, but without some form of constraint, you can’t rely on a normal distribution - the classic examples being wealth, income, sales, etc, where the outliers tend to be so much larger than average that they’re effectively precluded by a normal distribution, and yet there they are.
To be clear, I’m not saying there are not statistical methods for handling all of the above, I’m noting that the naive approach of modeling several different uncorrelated normally distributed outcomes, which is what the posted tool is doing, has severe flaws which are likely to lead to it underestimating the probability of outlier outcomes.
There's an amazing scene in "This is Spinal Tap" where Nigel Tufnel had been brainstorming a scene where Stonehenge would be lowered from above onto the stage during their performance, and he does some back of the envelope calculations which he gives to the set designer. Unfortunately, he mixes the symbol for feet with the symbol for inches. Leading to the following:
Cool! Some random requests to consider: Could the range x~y be uniform instead of 2 std dev normal (95.4%ile)? Sometimes the range of quantities is known. 95%ile is probably fine as a default though.
Also, could a symbolic JS package be used instead of Monte-Carlo? This would improve speed and precision, especially for many variables (high dimensions).
Could the result be shown in a line plot instead of ASCII bar chart?
This is neat! If you enjoy the write up, you might be interested in the paper “Dissolving the Fermi Paradox” which goes even more on-depth into actually multiplying the probability density functions instead of the common point estimates. It has the somewhat surprising result that we may just be alone.
Here (https://uncertainty.nist.gov/) is another similar Monte Carlo-style calculator designed by the statisticians at NIST. It is intended for propagating uncertainties in measurements and can handle various different assumed input distributions.
I think I was looking at this and several other similar calculators when creating the linked tool. This is what I mean when I say "you'll want to use something more sophisticated".
The problem with similar tools is that of the very high barrier to entry. This is what my project was trying to address, though imperfectly (the user still needs to understand, at the very least, the concept of probability distributions).
169 comments
[ 3.2 ms ] story [ 250 ms ] threadMeans "100 divided by some number between 4 and 6"
https://github.com/filiph/unsure/blob/master/lib/src/calcula...
I assume this is a montecarlo approach? (Not to start a flamewar, at least for us data scientists :) ).
> The computation is quite slow. In order to stay as flexible as possible, I'm using the Monte Carlo method. Which means the calculator is running about 250K AST-based computations for every calculation you put forth.
So therefore I conclude Monte Carlo is being used.
Someone also turned it into the https://github.com/rethinkpriorities/squigglepy python library
Thanks for the link!
Want to use it every 3 months or so to pretend that we know what we can squeeze in the roadmap for the quarter.
> 0.4.0
> BRAKING: x~y (read: range from x to y) now means "flat distribution from x to y". Every value between x and y is as likely to be emitted.
> For normal distribution, you can now use x+-d, which puts the mean at x, and the 95% (2 sigma) bounds at distance d from x.
https://github.com/filiph/unsure/blob/master/CHANGELOG.md#04...
The reason I'm asking: unsure also has a CLI version (which is leaps and bounds faster and in some ways easier to use) but I rarely find myself using it. (Nowadays, I use https://filiph.github.io/napkin/, anyway, but it's still a web app rather than a CLI tool.)
>there is no way of knowing why the range is between 4 and 6
??? There is. It is the ~ symbol.
i didn't mean knowing _that_ the range is between 4 and 6 but _why_, i thought the weighing would be explained by the reasoning, like: "we divide a €100 bill between possibly 4, rather 5 and most probably not 6 persons"
Because that is how it is defined by the author of the article.
It would especially outperform the Monte-Carlo approach drastically.
Although fuzzy-number can be used to model many different kinds of uncertainties.
If you want to do arithmetic and not a black box approach you just have to realize that you can perform them on the alpha-cuts with ordinary interval arithmetic. Then you can evaluate arbitrary expressions involving fuzzy numbers, keeping the strengths and weaknesses of interval arithmetic.
The sampling based approach is very similar to Monte-Carlo, but you sample at certain well defined points.
> Range is always a normal distribution, with the lower number being two standard deviations below the mean, and the upper number two standard deviations above. Nothing fancier is possible, in terms of input probability distributions.
Consider https://f-droid.org/en/packages/com.nunosempere.distribution...
Because there are trig functions, this would also be nice for reverse engineering of complex parts from simple measurements (feature volumes, corner angles, cross-sectional areas)
Multimodal would be nice but not necessary.
Inverse trig functions would be interesting but complicated and not necessary.
In any case, this tool is more convenient than opening Python every time I want to estimate a range of answers.
But if I just ask the calculator "1~2" (i.e. just a range without any operators), the histogram shows what looks like a normal distribution centered around 1.5[2].
Shouldn't the histogram be flat if the distribution is uniform?
[1] https://github.com/filiph/unsure/blob/123712482b7053974cbef9...
[2] https://filiph.github.io/unsure/#f=1~2
> Range is always a normal distribution, with the lower number being two standard deviations below the mean, and the upper number two standard deviations above. Nothing fancier is possible, in terms of input probability distributions.
The HN mods gave me an opportunity to resubmit the link, so I did. If I had more time, I'd have also upgraded the tool to the latest version and fix the wording. But unfortunately, I didn't find the time to do this.
Apologies for the confusion!
I really don't known how good it is.
If you're treating -1~1 as a normal distribution, then it's centered on 0. If you're working out the answer using a Monte Carlo simulation, then you're going to be testing out different values from that distribution, right? And aren't you going to be more likely to test values closer to 0? So surely the most likely outputs should be far from 0, right?
When I look at the histogram it creates, it varies by run, but the most common output seems generally closest to zero (and sometimes is exactly zero). Wouldn't that mean that it's most frequently picking values closest to -1 or 1 denoninator?
For normal it is higher but maybe not much more so.
As the mean of this distribution does not exist (it's "infinite") the Monte Carlo estimates aren't reliable
Now I'm realizing that this doesn't actually work, and even in more typical calculations the input values that produce the central 95% of the output are not necessarily drawn from the 95% CIs of the inputs. Which is fine and makes sense, but this example makes it very obvious how arbitrary it is to just drop the lowermost and uppermost 2.5%s rather than choosing any other 95/5 partition of the probability mass.
The actual distribution of 1/X is fairly interesting, see https://en.m.wikipedia.org/wiki/Inverse_distribution#Recipro...
[1] https://github.com/stefanhengl/histogram
- for command line, fermi: https://git.nunosempere.com/NunoSempere/fermi
- for android, a distribution calculator: https://f-droid.org/en/packages/com.nunosempere.distribution...
People might also be interested in https://www.squiggle-language.com/, which is a more complex version (or possibly <https://git.nunosempere.com/personal/squiggle.c>, which is a faster but much more verbose version in C)
```
5M 12M # number of people living in Chicago
beta 1 200 # fraction of people that have a piano
30 180 # minutes it takes to tune a piano, including travel time
/ 48 52 # weeks a year that piano tuners work for
/ 5 6 # days a week in which piano tuners work
/ 6 8 # hours a day in which piano tuners work
/ 60 # minutes to an hour
```
multiplication is implied as the default operation, fits are lognormal.
900K 1.5M # tonnes of rice per year NK gets from Russia
* 1K # kg in a tone
* 1.2K 1.4K # calories per kg of rice
/ 1.9K 2.5K # daily caloric intake
/ 25M 28M # population of NK
/ 365 # years of food this buys
/ 1% # as a percentage
I actually quite like it. Really clean, easy to see all the important elements. Lovely clear legible monospace serif font.
In the grand HN tradition of being triggered by a word in the post and going off on a not-quite-but-basically-totally-tangential rant:
There’s (at least) three areas here that are footguns with these kinds of calculations:
1) 95% is usually a lot wider than people think - people take 95% as “I’m pretty sure it’s this,” whereas it’s really closer to “it’d be really surprising if it were not this” - by and large people keep their mental error bars too close.
2) probability is rarely truly uncorrelated - call this the “Mortgage Derivatives” maxim. In the family example, rent is very likely to be correlated with food costs - so, if rent is high, food costs are also likely to be high. This skews the distribution - modeling with an unweighted uniform distribution will lead to you being surprised at how improbable the actual outcome was.
3) In general normal distributions are rarer than people think - they tend to require some kind of constraining factor on the values to enforce. We see them a bunch in nature because there tends to be negative feedback loops all over the place, but once you leave the relatively tidy garden of Mother Nature for the chaos of human affairs, normal distributions get pretty abnormal.
I like this as a tool, and I like the implementation, I’ve just seen a lot of people pick up statistics for the first time and lose a finger.
I love this. I've never though of statistics like a power tool or firearm, but the analogy fits really well.
You can test yourself at https://blog.codinghorror.com/how-good-an-estimator-are-you/.
> Heaviest blue whale ever recorded
I don't think estimation errors regarding things outside of someone's area of familiarity say much.
You could ask a much "easier"" question from the same topic area and still get terrible answers: "What percentage of blue whales are blue?" Or just "Are blue whales blue?"
Estimating something often encountered but uncounted seems like a better test. Like how many cars pass in front of my house every day. I could apply arithmetic, soft logic and intuition to that. But that would be a difficult question to grade, given it has no universal answer.
The question asks for the heaviest, which I think cannot be more than three times the normal weight, and probably no less than 1.3. That lands me at 15--600 tonnes using primitive arithmetic. The calculator in OP suggests 40--320.
The real value is apparently 170, but that doesn't really matter. The process of arriving at an interval that is as wide as necessary but no wider is the point.
Estimation is a skill that can be trained. It is a generic skill that does not rely on domain knowledge beyond some common sense.
How you get good at being a generalist is the tricky part, my best bet is reading and doing a lot of trivia (I found crosswords to be somewhat effective at this, but far from being efficient)
For some of these, your answer could span orders of magnitude. E.g. my answer for the heaviest blue whale would probably be 5-500 tons because I don't have a good concept of things that weigh 500 tons. The important point is that I'm right around 9 times in 10, not that I had a precise estimate.
To continue your example of 5-7 days, it would turn into an estimate of 5-700 days. So somewhere between a week or two years. And fair enough, whatever you're estimating will land somewhere in between. But how do I proceed from there with actual planning or budget?
Edit: And by the way given a large enough view, estimates like this can still be valuable, because when you add these estimates together the resulting probability distribution narrows considerably. e.g. at just 10 tasks of this size, you get a 95% CI of 245~460 per task. At 20, 225~430 per task.
Note that this is obviously reductive as there's no way an estimate of 5-700 would imply a normal distribution centred at 352.5, it would be more like a logarithmic distribution where the mean is around 10 days. And additionally, this treats each task as independent...i.e. one estimate being at the high end wouldn't mean another one would be as well.
You make up the number you wanted to hear in the first place that ostensibly works with the rest of the schedule. That’s why engineering estimates are so useless - it’s not that they’re inaccurate or unrealistic - it’s that if we insisted on giving them realistic estimates we’d get fired and replaced by someone else who is willing to appease management and just kick the can down the road a few more weeks.
Your budget should be allocated for say 80% confidence (which the tool helpfully provides behind a switch) and your stakeholders must be on board with this. It shouldn’t be too hard to do since everyone has some experience with missed engineering deadlines. (Bezos would probably say 70% or even less.)
Stakeholders really liked specifying work times as t_i ~ PERT(min, mode, max) because it mimics their thinking and handles typical real-world asymmetrical distributions.
[Background: PERT is just a re-parameterized beta distribution that's more user-friendly and intuitive https://rpubs.com/Kraj86186/985700]
This is the reason Fermi estimation works. You can test people on it, and almost universally they get more accurate with this method.
If you got less certain of the result in the example, that's probably a good thing. People are default overconfident with their estimated error bars.
You say this but yet roughly in a top level comment mentions people keep their error bars too close.
Being overconfident with error bars means placing them too close to the point estimation, i.e. the error bars are too narrow.
By contrast, precise calculations can be extremely complex but with the expectation that the answer they produce is correct. The far larger number of factors and operations involved can obscure a very significant error, either in mathematical process or in the assumptions the equation is based on, but the result may still be assumed to be right because it has been derived from a precise formula that is expected to yield good results.
So the strength of it is in keeping it simple and not trying to get too fancy, with the understanding that it's just a ballpark/sanity check. I still feel like the Drake equation in particular has too many terms for which we don't have enough sample data to produce a reasonable guess. But I think this is generally understood and it's seen as more of a thought experiment.
...if the only things you know about an uncertain value are its expectation and variance, yes.
Often you know other things. Often you don't know expectation and variance with any certainty.
Without having fully digested how the Unsure Calculator computes, it seems to me you could perhaps "weight" the ranges you pass to the calculator. Rather than a standard bell curve the Calculator could apply a more tightly focused — or perhaps skewed curve for that term.
If you think your salary will be in the range of 10 to 20, but more likely closer to 10 you could:
10<~20 (not to be confused with less-than)
or: 10!~20 (not to be confused with factorial)
or even: 10~12~20 to indicate a range of 10 to 20 ... leaning toward 12.
So, a realistic modeling isn't 1200~1500 + 100~150, it's (1~1.5)*(1200 + 150) - the "cost of living" distribution applies to both factors.
Not sure I agree with this. It's reasonable to have a model where the mean rent may be correlated with the mean food cost, but given those two parameters we can model the fluctuations about the mean as uncorrelated. In any case at the point when you want to consider something like this you need to do proper Bayesian statistics anyways.
>In general normal distributions are rarer than people think - they tend to require some kind of constraining factor on the values to enforce.
I don't know where you're getting this from. One needs uncorrelated errors, but this isn't a "constraint" or "negative feedback".
This is why I joked about it as the mortgage derivatives maxim - what happened in 2008 (mathematically, at least - the parts of the crisis that aren’t covered by the famous Upton Sinclair quote) was that the mortgage backed derivatives were modeled as an aggregate of a thousand uncorrelated outcomes (a mortgage going bust), without taking into account that at least a subset of the conditions leading to one mortgage going bust would also lead to a separate unrelated mortgage going bust - the results were not uncorrelated, and treating them as such meant the “1 in a million” outcome was substantially more likely in reality than the model allowed.
Re: negative feedback - that’s a separate point from the uncorrelated errors problem above, and a critique of using the normal distribution at all for modeling many different scenarios. Normal distributions rely on some kind of, well, normal scattering of the outcomes, which means there’s some reason why they’d tend to clump around a central value. We see it in natural systems because there’s some constraints on things like height and weight of an organism, etc, but without some form of constraint, you can’t rely on a normal distribution - the classic examples being wealth, income, sales, etc, where the outliers tend to be so much larger than average that they’re effectively precluded by a normal distribution, and yet there they are.
To be clear, I’m not saying there are not statistical methods for handling all of the above, I’m noting that the naive approach of modeling several different uncorrelated normally distributed outcomes, which is what the posted tool is doing, has severe flaws which are likely to lead to it underestimating the probability of outlier outcomes.
https://www.youtube.com/watch?v=Pyh1Va_mYWI
https://arxiv.org/abs/1806.02404
The problem with similar tools is that of the very high barrier to entry. This is what my project was trying to address, though imperfectly (the user still needs to understand, at the very least, the concept of probability distributions).
Any reason why we kept it 250k and now a lower number like 10k