Is there an example of an *exact* normal distribution in nature?

Factual Questions
1

The normal (or Gauss) distribution is a terrific approximation in a wide variety of settings. And it’s backed by the central limit theorem, which I admittedly do not understand.

But is it exactly observed in nature, for samples with an arbitrarily large number of observations? By exact, I mean that large samples consistently reject the hypothesis of non-normality.

In finance, rates of return tend to be leptokurtotic - they have long tails. There’s even some (negative?) skew. That doesn’t stop analysts from using the Gaussian distribution as a rough and ready model (sometimes to the chagrin of their investors, but that’s another matter).

What about other settings? Biology? Demography? Chemistry? Atmospheric science? Engineering?

I’m guessing that a reliably perfect Gaussian process is empirically rare, since there’s typically some extreme event that kicks in periodically. A large and important part of reality may consist of sums of very small errors, but I suspect that another aspect involves sporadic big honking errors. For every few thousand leaky faucets, we get a bust water main.
While I’m at it, why do the Shapiro-Wilk and Shapiro-Francia tests for normality cap out with sample sizes of 2000 and 5000?

I posed this question at the now defunct website, teemings.org in 2008. The short answer (by the esteemable ultrafilter) was, “no, nothing’s going to be exactly normally distributed. On the other hand, particularly in cases where the central limit theorem applies, the difference between what you see and what you’d expect to see is negligible.”

A tighter version of my question follows:

  1. Has anybody stumbled upon an unsimulated and naturally occurring dataset with, say, more than 100 observations that looks exactly like a textbook normal curve?

  2. Does any natural process consistently spin off Gaussian distributions, with p values consistent with normality virtually all the time? (Presumably this would be produced by something other than the central limit theorem (CLT) alone.) Ultrafilter says, “No”, if I understand him correctly.

  3. Does any natural process consistently spin off large datasets (thousands of observations each) where normality is not rejected - at least 95% of the time (at the 5% level of confidence)? If the CLT is the only thing in play, there should be natural processes like this. But I suspect that black swans are pretty much ubiquitous.

Then again, it should be possible to isolate a process that hews to a conventional Gaussian.
Bonus question: Do any of the tests for normality evaluate moments higher than 4?

As always wiki is helpful: Normal distribution - Wikipedia , but I’m not sure whether I should trust their claims of exactness.

Finally, if anybody has an empirical dataset whose process is plausibly Gauss, sample is huge and is in a reasonably accessible computer format, feel free to link to it if you’re curious. I’ll run some tests at some point using the statistical package Stata. Datasets are admittedly “All around the internet”, but extracting lots of fairly large (2000+) samples typically requires some work.

2

Your question is a little goofy. If I generate 20 random samples from a standard normal distribution, I would expect the hypothesis of normality to be rejected for one of them at alpha = .05. That doesn’t mean that the points it produces aren’t normally distributed, just that strange things can happen on a small sample.

In fact, I just ran this experiment, and the seventh sample didn’t look normal to the robust Jarque-Bera test (p = .014).

3

Exact normal distributions can be found in physics. They are not rare. Wikipedia has some examples.

4

Question #2 is a little goofy (provided the CLT is the only driver of normality, a fair assumption). Question #3 covers your concern though. (You addressed question #1 in 2008: to wit, a Gauss random number generator will typically produce distributions that don’t look perfectly normal, but are statistically indistinguishable from a perfect normal distribution. So Q1 is a hunt for an anomaly of sorts.).

iamnotbatman: Ok. It’s just that Wikipedia’s image of “The ground state of a quantum harmonic oscillator,” doesn’t look remotely Gauss, as it lacks a pair of inflection points. Velocities of ideal gases are not empirical observations (are they?).

5

Sorry, my question is actually a lotta goofy. “Gauss is an approximation MfM! Who cares what happens at the 12th moment?”

I reply that models based upon rough approximations made by highly paid analysts got us into trouble during the dawn of this Little Depression. And in a more general context, the late and great Peter Kennedy wrote:

6

Wiki’s picture for the quantum harmonic oscillator is a bit confusing. You see how it has those pixellated-looking horizontal bands across it? Each of those is one energy level. So the bottom-most band represents the ground state (with intensity as a function of position instead of height as a function of position, like on a graph). The ground state of a harmonic oscillator is, in fact, a perfect Gaussian.

But that doesn’t really answer the question, either-- It just moves it to the question of whether there are really any perfect harmonic oscillators in nature. The harmonic oscillator, like the Gaussian itself, is something that doesn’t actually show up exactly, very often, but which is a very good and simple approximation for a lot of things that do show up. The best I can come up with is a charged particle in a magnetic field with no other influences on it, but that “no other influences” is a killer. Especially since the particle would also have to have no magnetic dipole moment.

But I’m a bit unclear about what the OP means by a “natural” process. For instance, would rolling a whole bunch of dice and adding them up count as “natural”? Because it’s really easy to get an arbitrarily-good Gaussian that way.

7

Nice point. Roll a reasonably fair die and you have a uniform random number generator. Add a bunch of them up, and the CLT kicks in. So that’s one example.

But the spirit of the OP seeks:
a) a one million observation empirical Gaussian dataset (give or take a magnitude) on the internet that I can evaluate in Stata and/or
b) verification of the hypothesis that in practice no natural process is wholly governed by the CLT: black swans are ubiquitous for example. And that’s just the 4th moment.

Hypothesis b) is falsified depending upon whether you consider rolled dice a natural process. Are there even better examples?

My knowledge of physics is pretty sparse so I can’t tell which phenomena are a step or two removed from direct observation. I am not contesting that a regression of a near-Gaussian process could strip away the non-Gaussian bits, leaving behind systematically Gaussian errors. That’s what I meant by Gaussian processes, whose existence I accept. Solid examples of such processes would be welcome as well. But I’m really asking about pure Gaussian empirical data.

Anyway, Wikipedia gives examples of exact Gaussian processes. I can’t evaluate them: I can’t tell which are theoretical artifacts and which correspond to actual datasets. Would anyone like to take a crack? Normal distribution - Wikipedia

8

Wikipedia gives three examples of exact normality. In truth, none of them are exact, although in practice I believe that the exactness in all three cases could (and I’m quite sure it has, though I’m too lazy to find a cite) been confirmed over thousands of trials to many decimal places.

The first example is the velocities of the molecules in the ideal gas. One reason this isn’t truly exact (in the sense of wanting a true continuous distribution) is because the number of molecules is finite. But in practice, for a macroscopic ensemble like a liter of gas, the normality of the distribution would be impossible to differentiate from non-normality with 21st century technology.

The second example is the ground state of the quantum harmonic oscillator. As Chronos pointed out, there may not be any truly perfect quantum harmonic oscillators in nature. But any potential (smooth, continuous) with local minima will have ground states that to a very good approximation have normal distributions. Nature provides such minima in abundance (in magnetic or electric field configurations, vibrations of diatomic molecules…). The problem is that if you include the finite size of the universe, or the effect of tiny perturbations due to interactions with the quantum vacuum, or even the tiny contribution of gravity waves from the stars in the sky, you are bound to imperceptibly distort your potentials so that they are not exactly gaussian.

The third example is the position of a particle in quantum mechanics. If you measure the position exactly, then wait some time, and measure again, the distributions of measured positions is exactly Gaussian. Of course, this is assuming an idealized particle, and no other potentials in the vicinity, which, as I mentioned above, is not ever going to be achieved perfectly in practice. Similarly you are not going to be able to perfectly measure the position of the particle in the first place.

9

Radioactive decay.

10

That’s not a normal distribution.

11

iamnotbatman:
I have difficulty working through these science examples as I lack a background in either college physics or chemistry. Apologies for my inaccuracy and imprecision: I’m really winging it here.

Ok. In nature there are no ideal gases, no perfect harmonic oscillators and no perfect measures of any given particle’s position. These models approximate the world quite well though. This isn’t exactly what I’m getting at.

Is there a dataset with ~10K observations of the velocity of the molecules of a liter of nitrogen? [1] If so, I’m not asking whether the distribution produced is perfectly Gauss. I’m asking whether it is consistently indistinguishable from Gauss at the 95% level of confidence. Sufficiently small perturbations only matter with sufficiently large sample sizes. (My call for huge datasets was so that I could have a set n=2000 subsamples, and work out the share of them that reject normality at 5%).

In practice though, I might wonder whether measurement errors make a difference. Ok, now I’m wandering into the goofy again.

[1] Seriously, what are we measuring here? Is it the average velocity of molecules in a liter of gas? I’m guessing that the data would be measuring temperature and pressure, which is something different. The Gauss would be used to transform empirical observation into a postulated velocity: it would address some processes and set others aside. This is something different than “A Gaussian dataset”. I’m calling it “A Gaussian process”. So what I’m gathering from this thread is that “There are lots of examples of the exact Gauss in physics”, though I don’t have a clear idea of any particular Gauss dataset in physics. Could we specify the experimental setup in greater detail?

12

It might help to know what you’re after. For example, why can’t you produce your own dataset using a software package such as Root, R, or Mathematica (or C++)? In terms of the physical world, surely we have many indirect ways of testing normality. But you seem to be after a direct measurement of normality, which, from my perspective, is odd, or without motivation that makes sense to me. For example, we know that the normality of the velocity distribution is true, because of the macroscopic properties of the gas that we can measure with great precision. Usually we would not resort to looking at the literal distribution of individual velocities. Unfortunately I don’t know of a paper published regarding your question in which they include their actual dataset for you to peruse. For the gas velocity distribution (we would be measuring the distribution of velocities of the gas molecules), an example of an experimental setup would be to have a chamber of gas at some temperature and pressure, and have a very tiny opening through which the gas molecules can escape more or less one at a time at a very low rate. You then a collimator that makes sure the ‘beam’ of molecules is straight, followed by a detector that records the velocities (perhaps ensuring the particles are ionized and measuring how much they are deflected in a given electric field). This was in fact done, I think, in 1926 by Stern, in a series of experiments, perhaps with input form Sterlach and Einstein, which showed more or less directly that the distribution of the velocities are normally distributed. I do not think they published their dataset though.

13

iamnotbatman:
Thanks for spelling out the ideal gas example.

I concede that a random number generator can produce Gauss random variables; I've even fooled around with that a little.  My original motivation was linked to the tendency in the social sciences to assume Gauss errors without credible or even explicit justification.  Now that's not necessarily a bad thing: it depends upon the problem in question.  And in physics I now understand that there are very good reasons for assuming that certain types of distributions are indeed Gauss.  But it is bogus and dangerous to blithely assume Gauss for financial market returns in the face of strong evidence to the contrary, at least without applying robustness checks and general due diligence.  And yet that is what was done routinely some years back: this sort of mismanagement formed one of the necessary conditions for the financial crisis and subsequent Little Depression.

Construct the Casual Fact
So much for high-level motivation. For this thread I’m working on a more general level. I’d like to say something along the lines of, “The Gauss distribution doesn’t exist empirically in nature: what we have are distribution mixtures.” But I don’t think that’s quite correct. I’m trying to work out the proper rough characterization about the prevalence of observed exact Gaussians.

Again, in many applications this doesn't matter.  If you're conducting an hypothesis test and the underlying distribution is even Laplace, applying the student's t-statistic probably won't steer you that far wrong.  Type I and II errors will be less than optimal, but arguably acceptable. Or so I speculate: I haven't read the relevant Monte Carlo study.  But if you are forecasting central tendency and dispersion, that's another matter entirely.
14

That’s so far down the list of causes that it barely even rates a mention.

15

I disagree, but that’s a matter for another thread. The failure of due diligence and ignoring once-every-eight-year events was a big part of the story. Admittedly over-application of the Gaussian distribution is only a proximate cause: it’s not like the financial engineers lacked sufficient training or understanding.

Note to future readers (if any): interesting discussion of the Gaussian distribution occurs here: Why half-life and 1 standard devation = 68%? - Factual Questions - Straight Dope Message Board

16

I read the Taleb book a couple years back and found it very compelling, so I know where you are coming from. But it is important to separate out two separate things, because generally what causes the ‘fat tail’ is not the underlying thing you are trying to measure, for which the theoretical gaussian distribution is well understood and thoroughly correct, but some external factor. In physics we would call this external factor the ‘systematic error’ in the measurement, and it is considered of great importance to account for it correctly. Consider an example:

If you toss a coin N times, the number of times the coin shows ‘heads’ (N_h) is a random variable that is binomially distributed, but for large N is normally distributed. (Btw if you are a masochist you can use this method to produce your own dataset, but you should really just use a software method). Now, one source of systematic error is whether or not the coin is ‘fair’. But let’s ignore that, because even if it wasn’t fair, the distribution shouldn’t have a fat tail. Now suppose you needed a coin tossed a trillion times, so you farmed out the work to some company that used a robot to flip coins and an employed image recognition software to determine which side of the coin landed up. The question is, do you trust the company to do this without error? Perhaps the robot can flip with such precision that if it produces the same ‘flipping’ force the coin will always land heads-up, and the company’s software was never tested beyond a few million flips, and since some of their variables were 32-bit and reset after 2^32 flips, after a few billion flips the robot gets into a pattern where it keeps throwing heads over and over again. If the company was incompetent (which is extremely common in the real world), they may not notice the bug, and hand you the dataset claiming that the systematic error is zero. But in reality you would get a very fat tail – not because the underlying process was non-gaussian, but because of external factors which were not correctly accounted for. I think a common problem in the financial world may be an arrogance regarding the evaluation of systematic errors combined with a lot of top-down pressure and under-regulated competitive pressure (tragedy of the commons, etc) – not honestly accounting for systematic error. For example, if you have companies competing to build coin-flipping machines in the market place, they are going to make competitive shortcuts and unrealistic promises regarding low systematic errors. Cheaper and flimsier coin-flipping machines may be built and trusted because it is necessary to compete against others making the same mistakes. Needless to say, in such a market, I would not ‘bet on’ normally distributed data – humans make mistakes, and it is unrealistic to expect that the probability for making such a mistake is as small as a normal distribution says it can be.

I think your focus on normal distributions could be expanded to any distribution that claims probabilities that can be vanishingly small (disregarding boundary conditions). In the real world your systematic error is generally large enough that when you add it to the statistical prediction, you always expect some fatness to your tails, to some extent. Most people know this intuitively. For example we know that in quantum mechanics it is possible for your stapler to tunnel through your desk and onto the floor. The probability distribution is a tail very much like that of the tail of the normal curve, and is unimaginably small. But if you are doing a home experiment, you have to account for the possibility that someone took your stapler while you weren’t looking, and someone else dropped a stapler near your desk and it ended up below yours. That is in fact analogous to one of the systematic errors that must be controlled for when doing some of these actual quantum mechanical experiments.

17

Strictly speaking, an error which causes fat tails on both sides wouldn’t be a systematic error, since systematic errors by definition will bias your data in one direction.

18

Surely it would be a systematic error biased in one direction – the direction away from the mean! The wikipedia article has examples in the first section of this type of systematic error.

19

**iamnotbatman **:
As I said, my original curiosity arose from the ubiquitous assumption of Gaussian errors. The financial crisis added some additional motivation though. Most analysts are aware of these issues, but I at least don’t have a solid grasp of them.

I like the systematic error concept. I’m inclined to abandon my “No observed Gauss anywhere” notion: there are solid reasons to believe in Gaussian processes in certain physics contexts. Let me propose another conceptual handle: "In practice, most dataset errors reflect some sort of distribution mixture. Following the central limit theorem, the sum of lots of equally weighted distributions will be Gauss. But in practice, Gauss will be an approximation, since the weights won’t be equal.[1] " That should encompass the systematic error concept to some extent. One of the problems in the social sciences is that there typically aren’t solid theoretic reasons for believing in any particular exact error distribution (even if there are plausible arguments for Gaussian approximations or whatever). Furthermore your dependent variable typically reflects a lot of unmeasurables and even unponderables.

I agree, but fear that financial returns are somewhat more complicated than that. Still, “Gauss plus a swan every 5-10 years” is probably a lot better than simple Gauss.

FWIW, financial returns typically have a negative skew as well.

I downloaded a dataset of 15,000 observations from yahoo. It consists of daily percentage price changes of the S&P Composite, a weighted sum of large capitalization stocks.[2] I’ll compare it to Gauss in an upcoming post.
[1] (Whether the sample size is sufficient to distinguish your empirical distribution from pure Gauss is a separate matter.)

[2] It’s the S&P 500, except there were fewer companies in the index during the 1950s.

20

So here’s what the distribution of the S&P Composite looks like. As a control, I specified a normally distributed random variable with the same mean and standard deviation:



    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
dailypret500 |     15474    .0003306      .00967   -.204669      .1158
     normrnd |     15475    .0002605    .0097017   -.036254   .0388416

Resolved mean daily returns differed, even with a sample size exceeding 15,000. Standard deviation differed as well. Daily price returns ranged from -20.5% to 11.6%: the Gauss random varible had a range of only -3.6% to +3.9%.

Now consider kurtosis and skew:



                        DailyPret500
-------------------------------------------------------------
      Percentiles      Smallest
 1%     -.025709       -.204669
 5%     -.014333        -.09035
10%     -.009873       -.089295       Obs               15474
25%     -.004123       -.088068       Sum of Wgt.       15474

50%     .0004635                      Mean           .0003306
                        Largest       Std. Dev.        .00967
75%      .004963        .070758
90%      .010133        .090994       Variance       .0000935
95%      .014507         .10789       Skewness      -.6616936
99%      .025685          .1158       Kurtosis       25.22477

                           normrnd
-------------------------------------------------------------
      Percentiles      Smallest
 1%    -.0224537       -.036254
 5%    -.0157981      -.0340847
10%    -.0121544      -.0337052       Obs               15475
25%    -.0062004        -.03304       Sum of Wgt.       15475

50%     .0002771                      Mean           .0002605
                        Largest       Std. Dev.      .0097017
75%     .0068832       .0339581
90%     .0126438       .0363308       Variance       .0000941
95%      .016328       .0382705       Skewness      -.0217871
99%     .0227815       .0388416       Kurtosis        3.00358

The SP500 had a kurtosis of 25.2. In contrast Gauss random variables have kurtosis of 3 while LaPlace R.Vs have kurtosis of 6. The SP500 also has a negative skew. Yes the kurtosis and skew is significantly different than Gauss:



                   Skewness/Kurtosis tests for Normality
                                                 ------- joint ------
    Variable |  Pr(Skewness)   Pr(Kurtosis)  adj chi2(2)    Prob>chi2
-------------+-------------------------------------------------------
dailypret500 |      0.000         0.000               .            .
     normrnd |      0.268         0.904            1.24       0.5379

Here’s a histogram of the SP500:
http://wm55.inbox.com/thumbs/44_130b7a_fda38365_oP.png.thumb

It looks a little like the Burj Dubai. Here it is with a overlaid perfect normal distribution:
http://wm55.inbox.com/thumbs/45_130b79_1532517c_oP.png.thumb

The Burj 500 is pointier, has longer tails, and is somewhat skewed.

Are outliers driving this effect? Let’s see what happens if we remove the 30 most negative and positive returns. That would be one day per year on average, so we are removing swans of all colors.



                        DailyPret500
-------------------------------------------------------------
      Percentiles      Smallest
 1%     -.024287        -.04356
 5%     -.014088       -.043181
10%     -.009789        -.04279       Obs               15414
25%     -.004107       -.042532       Sum of Wgt.       15414

50%     .0004635                      Mean           .0003492
                        Largest       Std. Dev.      .0087627
75%      .004948        .040826
90%      .010061        .041729       Variance       .0000768
95%      .014279        .041867       Skewness      -.0402395
99%      .024358         .04241       Kurtosis       5.482383

                           normrnd
-------------------------------------------------------------
      Percentiles      Smallest
 1%    -.0224537       -.036254
 5%     -.015802      -.0340847
10%    -.0121599      -.0337052       Obs               15414
25%    -.0062022        -.03304       Sum of Wgt.       15414

50%     .0002828                      Mean           .0002593
                        Largest       Std. Dev.      .0097034
75%     .0068737       .0339581
90%     .0126438       .0363308       Variance       .0000942
95%     .0163285       .0382705       Skewness      -.0216746
99%     .0227815       .0388416       Kurtosis       3.004975

                   Skewness/Kurtosis tests for Normality
                                                 ------- joint ------
    Variable |  Pr(Skewness)   Pr(Kurtosis)  adj chi2(2)    Prob>chi2
-------------+-------------------------------------------------------
dailypret500 |      0.041         0.000               .       0.0000
     normrnd |      0.272         0.876            1.23       0.5400

Well it’s improved. Kurtosis is down to 5.5, which isn’t Gauss but is at least close to LaPlace. Both skew and kurtosis are significantly different from Gauss at the 5% level.

In short, Gauss is a pretty rough approximation for financial returns and while black swans are a big part of the story, they are not the only part. Note that I picked a period of relative economic stability. 1890-1950 was far more tumultuous. For that matter 1830-1890 wasn’t exactly smooth sailing either.