68–95–99.7 Rule | Hacker News Reader

68–95–99.7 Rule

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131 pointsturrini33 comments

Remember that a lot of data is not actually normally distributed, even though it looks so at first. The horror of "risk management" units of financial institutions, when they get a "7-sigma" event twice in a month.

Can you imagine there was a time when the entire US options market was running on flat volatility vs strike price (implying that financial asset prices are lognormal). Must have been cool to just buy the cheapest, most OTM options and collect big results every now and then.

Also if it’s once in 300 million to happen every day, it could happen to someone every day in the USA (assuming it’s something people do daily).

If the outfall from a 7-sigma is disastrous, you have to try to prevent it, because it WILL happen eventually given enough goes.

That's true but those people were sampling closer to once per day, and still got N-sigma in their models for some high value of N.

The easiest way to get something that's not actually normal in the typical central limit theorem construction is to have the variables be just slightly not independent. Which becomes obvious in retrospect after you start realizing the dependence and shit hits the fan.

Banks failing and CDOs exploding would be two good examples; they’re rare if not nonexistent but if they start they may all flare up.

Also natural distribution assumes you don’t have someone trying to game the extremes to make money. If being seven feet tall resulted in billions we’d have people researching how to grow taller people.

You can see stuff like this in distributions of tax returns - many people will be unexpectedly clustered right below certain cut-off amounts.

Worse is correlation break-down in a sense - e.g., that black swan event happening across an entire asset class because of liquidity squeeze, margin calls, etc.

> ... because it WILL happen eventually given enough goes.

I'm not a statistician, but I don't think that's how that works. Given enough events, the likelihood of something happening goes up, but it never becomes a certainty.

Likewise when the statistical 100 year statistical storm happens, the next one is not 100 years away. It could happen the following week without violating the underlying statistics. (Of course if a 100 year storm keeps happening year after year, then the threshold for what is likely to happen once in a hundred years probably needs to be revised.)

One time for Nassim Nicholas Taleb who's ALL OVER this in a deeply interesting and entertaining way. "The Black Swan" is great at going in on such a weirdly obvious-but-not point.

Unlike other hedge funds, Taleb's/Universa's records, like CAGR, are sketchy. Good records do not exist except what has been 'leaked' to the media by PR. It's not like Bridgewater in this regard.

His barbell OTM put strategy almost certainly did badly in 2022, because of the failure of the Vix index to spike and the decline of the S&P 500 being orderly.

https://greyenlightenment.com/2022/10/08/tail-hedging-strate...

So both parts of the barbell lost money, which is not supposed to happen. This is why you have to be warry of individuals and strategies that are overhyped. If a strategy is getting a lot of positive press, it likely means it's saturated. If a lot of quants are buying the same options for the same hedging purposes, who is going to sell them?

Judging Taleb by a hedge fund's performance, even if it's one of his, is really missing the point entirely.

This may have been possible pre-1987. Since then, out of money put options have a very steep skew, so not possible. Even if OTM put options are very cheap nominally, but by having much higher implied volatility makes them much more expensive compared to how much they would otherwise cost without the skew. This from a Kelly perspective makes them much less lucrative if one was to try to construct a strategy with this. It cuts your ROI big time.

In an efficient market the average ROI on options should be zero.

But it isn’t efficient; banks like overcharge you absurdly especially for OTM options.

I always like the inverse of this, which is that if you calculate that less than 68% is in the first standard deviation, you can guess that the data isn’t normally distributed.

It’s really a heuristic on where to start with analysis, though. Not a result in and of itself.

The gist of the idea is of course correct, but if you actually do this, please use an actual normality test.

https://en.wikipedia.org/wiki/Shapiro–Wilk_test

https://en.wikipedia.org/wiki/Anderson–Darling_test

https://en.wikipedia.org/wiki/Kolmogorov–Smirnov_test

The trouble with normality tests is that you either have too little data (in which case they're useless) or too much (in which case their also useless).

I always think of normality as a nice theory for simple stuff, but you want to be running sensitivity analyses for all of your assumptions, including normality (which I don't think I've ever seen professionally, tbh).

Memorizing a set of numbers doesn't really feel like a shorthand way to remember those numbers. Did you know about the 3.14 rule? It's a way to remember the first three digits of pi are 3.14.

Which is, I don’t know — ironic? — because pi rounded to two decimals is 3.15.

No it’s not- it’s 3.141, which rounds to 3.14

It’s

3.14159

3.1416

3.142

3.14

3.1

Oh, I feel dumb. Some reason I had it memorized without the 1: 3.1459 — for years! Decades!

I wonder what tests I screwed up in collage!

No pi is 3.14159... which is 3.14 when rounded.

The 3.2 rounding of Pi in an Indiana bill[0] is also quite infamous, which confusingly had other values of Pi for use in different contexts. [I'm a Tau touter myself.]

[0] https://www.straightdope.com/21341975/did-a-state-legislatur...

The approximation 355/113 has nice pairs of odd digits.

An more useful thing to remember is Shah's approximation, which estimates the area under the standard normal curve from 0 to a point 𝑧 on the x axis as follows:

  • 𝑧(4.4 – 𝑧)/10 when 0 ≤ 𝑧 ≤ 2.2
  • 0.49 when 2.2 < 𝑧 < 2.6
  • 0.50 when 2.6 ≤ 𝑧

Shah's approximation is accurate to within roughly ±½%.

Examples:

- The area under the standard normal curve within 1 standard deviation of zero is approximately 2(1(4.4 – 1))/10 = 0.68. (The factor of 2 is there to get the area on both sides of zero).

- The area within 2 standard deviations of zero is approximately 2(2(4.4 – 2))/10 = 0.96.

- The area within 3 standard deviations of zero is approximately 1 (because 2.6 ≤ 3).

- The probability of sampling from the standard normal distribution and getting a value less than 1.5 deviations above the mean is approximately 0.5 + 1.5(4.4-1.5) = 0.935. (The 0.5 term is there because we need to include the area to the left of 0, and Shah's approximation only counts the area to the right of zero.)

Reference:

Arvind K. Shah (1985) A Simpler Approximation for Areas under the Standard Normal Curve, The American Statistician, 39:1, 80, DOI: 10.1080/00031305.1985.10479396 (https://twitter.com/jordancurve/status/958026273149915136/ph...)

The table of numerical values is pretty interesting for a layman. I've often heard scientists (such as Brian Cox) talk about x-sigma probabilities, but putting it the context in this table is much more meaningful for someone like me who only has self-study reading as scientific understanding.

Mentioned in the OP, but perhaps buried a bit, I am always partial to Chebyshev's Inequality (https://en.wikipedia.org/wiki/Chebyshev%27s_inequality) since it describes all distributions not just Gaussian. While approximations, they are nice to have when you question the normality of the data and still want to estimate limits - more numbers to remember however, since they are estimates.

This is why also claims of IQs above 180 or so are almost always BS, at least on normed tests. 6sd is about the maximum

Anything outside 70-130 is suspect in practice, as the tests aren't well calibrated beyond ±2σ.

I'm not sure what intellectual performance is even supposed to be implied by any given IQ score, given that the scores commonly used today are just a linear mapping from standard deviations above or below mean.

That said, on the low end at -3σ I would expect a serious deviation from a normal distribution just from the effect of dementia.

Sure, but the point is that 180 wouldn't make sense even if the test were perfect. There aren't enough humans on earth for that many standard deviations to exist in the first place. 180 IQ could as well be 2000 IQ, the sky is the limit when the scale makes no sense.

That's just wrong? 6 sigma would be IQ 100 + 6 * 15 = 190. 6 sigma is 1 in 506,797,346. So there should be 16 people with IQs that extreme if intelligence is normally distributed.

(There's a factor of two somewhere in here for the two tailed nature, but the point stands)

Either I'm misunderstanding the math, or you are. An IQ of 180 is 5 1/3 SD above mean, which is a bit more frequent than 1 in 26,330,254 (which would be 5.5 SD). Half that common because we're only talking about above, and 1 in 50m people on earth is still 160 people. That's not many, but I don't think it's negligible either.

https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rul...

It's also literally impossible for many modern tests. For example the WAIS-IV, a very popular measure, has a maximum possible FSIQ of 160. With a mean of 100 and an SD of 15, that's "only" 4 SDs from the mean (99.997 percentile).

And as you say, modern tests just aren't calibrated in that way. The WAIS-IV is calibrated for the 70-130 range [0]. It's used to provide broad ranges (e.g. 85-100 vs. 115-130), not to provide useful differentiations on the high end like 158 vs. 159.

In fact, the error increases with the score [1]. There is less certainty for a score of 145 than there is for 115. It's just not designed for separating geniuses from super geniuses or however you want to put it.

[0] https://www.quora.com/Whats-the-highest-IQ-score-you-can-get...

(Not a direct source I'm afraid, but the materials are proprietary.)

[1] https://en.wikipedia.org/wiki/IQ_classification#Giftedness

(There are several citations in this section, pertinent discussion below the table.)

I like how they call this a 'rule' instead just a list of three hard to remember numbers in a row.

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