# [R] identify the distribution of the data

Spencer Graves @pencer@gr@ve@ @end|ng |rom e||ect|vede|en@e@org
Wed Feb 8 19:24:44 CET 2023

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On 2/8/23 12:06 PM, Ebert,Timothy Aaron wrote:
> IMO) The best approach is to develop a good understanding of the individual processes that resulted in the observed values. The blend of those processes then results in the distribution of the observed values. This is seldom done, and often not possible to do. The alternatives depend on why you are doing this.
>
> 0) Sometime the nature of the data suggest a distribution. You list integer values. If all observations are integer (counts for example) then Poisson may be appropriate. With two values then maybe the Binomial distribution. Continuous data might be normally distributed (Gaussian distribution). If I roll one six-sided die many times I will have a uniform distribution (assuming a fair die). I could then try the same task but roll 2 dice and add the result. I still have discrete values, but the shape is closer to Gaussian. The distribution looks more and more Gaussian as I add more dice together in each roll.

I concur:  The application will often suggest a distribution, e.g.,
Poisson, binomial or negative binomial for nonnegative integers, Weibull
for lifetime data, etc.

I love normal probability plots -- the qqnorm function.  This can
identify outliers or multimodality or the need for a transformation.
Continuous data that are always positive are often log-normal -- or a
mixture of log-normals.

x <- rnorm(100)
X <- exp(x)
qqnorm(X, datax=TRUE, log='x')

The central limit theorem says that the distribution of almost any
sum of random variables will be more nearly normal than the
distributions of individual summands.  It also says that almost any
product of positive random variables will be more nearly log-normal than
the distributions of individual components of the product.  This
application to products is less well known and occasionally controversial.

https://en.wikipedia.org/wiki/Gibrat%27s_law

Spencer Graves

>
> 1) Try a simulation. Draw 5 values from a normal distribution, make a histogram. Then do it again. Is it easy to see that both samples are from the same distribution? Personally, the answer is no. So increase the sample size until you are happy with a decision that any two draws are from the same distribution. For my part, at 1 million most people would not be able to detect any difference between the two histograms. This helps calibrate the people. How does your sample size compare to your choice in this exercise?
>
> 2) Given that you have sufficient data (see above), can you see the distribution in your data? Is that good enough?
>
> 3) Are you doing this as part of following the assumptions of statistical models? In such tests for normality, we tend to assume that a failure to reject the null hypothesis is sufficient proof that the null hypothesis is true. However, in most other cases we are told that a failure to reject the null hypothesis is not sufficient to prove the null hypothesis. You need to work this out, but the importance, consequences, and alternatives of testing model assumptions is a large body of literature with (sometimes) widely divergent viewpoints.
>
> 4) There are hundreds of distributions. https://cran.r-project.org/web/views/Distributions.html but the common distributions are seen in sites like this one:  https://www.stat.umn.edu/geyer/old/5101/rlook.html. Given so many choices, you can probably find one that will fit your data reasonably well. Depending on how many data points you have will determine the reliability of that answer. Is that really informative to the problem you are trying to solve? Answering "what distribution do these data follow?" is not usually the goal.
>
> Regards,
> Tim
>
>
> -----Original Message-----
> From: R-help <r-help-bounces using r-project.org> On Behalf Of Bert Gunter
> Sent: Wednesday, February 8, 2023 12:00 PM
> To: Bogdan Tanasa <tanasa using gmail.com>
> Cc: r-help <r-help using r-project.org>
> Subject: Re: [R] identify the distribution of the data
>
> [External Email]
>
> 1. This is a statistical question, which usually is inappropriate here:
> this list is about R language (including packages) programming.
>
> 2. IMO (so others may disagree), your question indicates a profound misunderstanding of basic statistical issues. While maybe you phrased it poorly or I misunderstand, but "identify the type of distribution" is basically a meaningless query. Explaining why this is so and what may be more meaningful would require a deep dive into statistics. You might try referencing a basic statistical text and/or online tutorials. Try searching on "Goodness of fit", "statistical modeling" or the like.
>
> Cheers,
> Bert
>
> On Wed, Feb 8, 2023 at 8:35 AM Bogdan Tanasa <tanasa using gmail.com> wrote:
>
>> Dear all,
>>
>> I do have dataframes with numerical values such as 1,9, 20, 51, 100
>> etc
>>
>> Which way do you recommend to use in order to identify the type of the
>> distribution of the data (normal, poisson, bernoulli, exponential,
>> log-normal etc ..)
>>
>> Thanks so much,
>>
>> Bogdan
>>
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