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Below is a histogram of some data, the bins are integers the other parameters are irrelevant.

Overlapping distributions

As you can see there seems to be two separate but overlapping normal distributions for odd and even numbers.

The probability of being an even number is 1/3, likewise 2/3 for an odd number.

I have no idea of actual statistical significance of this to be honest so I'm trying to find out what it even is to learn more, but I can't find anything, I've tried so many search terms to find this and even reverse image searches but all I get is information about multimodal distributions etc. and I can't find anything about when the multimodal distributions actually overlap in this manner

Is there a name for this?

For those interested the data is from 1,000,000 randomized games of goofspiel (N=13) using the matlab script

N = 1000000;
random = zeros(1,N);
for i = 1 : N
    pc = randperm(13);
    p1 = randperm(13);
    p2 = randperm(13);
    random(i) = sum(pc.*sign(p1-p2));
end
histogram(random,'BinMethod','integer')

A more general (though artificial) example would be the following

a = [1:50 50:-1:1];
b = normpdf(linspace(-2,2),0,0.5).*50;
c = a;
rng('default') %For reproducibility
d = logical(randi([0,1],1,length(a)));
for i = 1:length(c) %There's gotta be a way to do this without an explicit loop
    if(d(i)) 
        c(i) = b(i);
    end
end
bar(c)

General Example

Like the first example there's two distributions overlapped (triangular and normal), but in this case instead of alternating at each point, it's random.

I know this is an exaggerated example (and not even a histogram) but there has to be examples of this sort of thing actually happening with statistical data right? Then again maybe not, or it's completely irrelevant?

The actual question is two-fold:
The general question - What is this type of "thing" called, if anything? - so that I (or anyone else that might come across it) can learn more about it and if any adjustments need to be made.
The question as it specifically relates to my first dataset - should I separate the odd and even values or fit a normal distribution to the whole set?

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  • $\begingroup$ Looks like some funky mixture model, where the pdf is 1/3(pdf of evens)+2/3(pdf of odds). I don't know how to work the normal distribution into it though because it's clearly not continuous. $\endgroup$
    – Huy Pham
    Apr 7, 2019 at 18:51
  • $\begingroup$ What exactly is the question? You seem to simulate some data, that follows some strange distribution, but what exactly is the problem? $\endgroup$
    – Tim
    Apr 7, 2019 at 20:55
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    $\begingroup$ @Tim I've edited it to be a bit clearer. I suppose I'm under the assumption that this is less rare than it is, and has been studied before. If that's not the case then the question is simply how would I go about describing/modeling the distribution of my first set of data $\endgroup$ Apr 8, 2019 at 8:46
  • $\begingroup$ @BenjaminTilbury regarding your last question, fitting. You can more easily fit a normal density curve to the histogram when you increase the bin size to two. Another approach would be to fit the cumulative distribution. Which option you choose is a bit depending on what you are gonna do with it. Possibly your interest is more in the cumulative distribution function. $\endgroup$ Apr 9, 2019 at 8:51
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    $\begingroup$ It is called "aliasing." Moire patterns are one of (very many) examples. $\endgroup$
    – whuber
    Apr 11, 2019 at 20:52

2 Answers 2

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This answer is not a direct answer to your question, because it relates to a different cause of the pattern.

But it does relate to the same graphical appearance, and therefore I post it as an answer rather than a comment (before reading your Matlab script I actually thought the pattern in your histogram was due to this different cause).


Your question made me revisit a histogram that I plotted in an answer to a recent question.

old illustration

I used binsize 1, while the distance between the (discrete) results was 0.538. Making the bars of the histogram to be plotted occasionally with the counts for a single value instead of the counts for two values.

After adjusting the bin sizes the histogram appeared more typical

new illustration

In this case, we could call the pattern a Moiré pattern, which is the appearance of artificial light and dark bands due to a misalignment of two discrete scales.

In your case, however, the periodic pattern is not an artificial effect in the histogram but a truly periodic behavior in the probability mass function. Anyway, I thought it was useful to mention this related Moiré pattern.

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UPDATE: Re-reading this four years later, let me attempt to answer the first part too, what to call it. If it is due to two distinct distributions, as I suggest below, then the combination is called a mixture distribution; I more often hear the term mixture model, which is what we'd use to model it. If an artifact of how the data is displayed graphically, the other answer mentions moiré pattern. Or simply visual artifact? And in comments, whuber calls it aliasing, which I think I like best.


should I separate the odd and even values or fit a normal distribution to the whole set?

I think you should separate them. Your analysis has discovered that the most important factor/predictor is if the input is odd or even, so to merge them would be to blur both of the distributions, and make them less useful (*).

*: Of course, it really depends on your definition of useful. I'm approaching it from the viewpoint of you have some inputs and want to make a model to predict some output. Once we know it is significant, I'd want to give the model the hint that the parity of one/some of the inputs matters.

By the way, as in Martijn Weterings's answer, when I've had jagged histograms like this before, it has been related to the choice of bin size. It made me realize that experimentation with bin size is yet another tool in the Lying With Stats toolbox :-)

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    $\begingroup$ Dear downvoter: did you disagree with anything in particular? If so, please do educate me. $\endgroup$ May 16, 2019 at 16:27
  • $\begingroup$ It would be an extremely bad idea to treat this aliasing phenomenon as a mixture model. Redrawing the histogram using appropriate breaks would be simpler and more revealing. See the other answer in this thread. $\endgroup$
    – whuber
    Mar 19, 2023 at 13:43
  • $\begingroup$ @whuber The other answer actually says "In your case, however, the periodic pattern is not an artificial effect in the histogram but a truly periodic behavior in the probability mass function" And the OP says "The probability of being an even number is 1/3, likewise 2/3 for an odd number". So it seems to deliberately be a mixed distribution; and using larger histogram bars would hide this property. An example of how this could happen in real data is counts of bill sizes at a cafe, where everything on the menu is £2, £4 or £6, but there are a couple of types of cake for sale at £3. $\endgroup$ Mar 19, 2023 at 16:55
  • $\begingroup$ Did you see that the answer begins, "This answer is not a direct answer to your question, because it relates to a different cause of the pattern"? $\endgroup$
    – whuber
    Mar 19, 2023 at 17:11
  • $\begingroup$ @whuber. Yes. My point is the OP says odd and even numbers do actually have different probabilities. The jagged histogram is the truth, not an artifact of how the data is displayed. Just like it would be in my cafe example. $\endgroup$ Mar 19, 2023 at 17:34

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