# Determining how well given real-life data fits to a given probability distribution

Let's say I have some data, i.e. a collection of real numbers.

Are there any good general methods to determine whether this data conforms to a given probability distribution, e.g. the normal distribution, the log-normal distribution, or any other distribution? If so, what are some of the best methods out there?

• Maybe you should ask this on stat.stackexchange.com instead of math.stackexchange.com – Dilip Sarwate Oct 29 '11 at 19:43
• Short answer to your question: you want a "likelihood ratio test" or a "chi-squared test". – Michael Lugo Oct 29 '11 at 19:51
• @Two Cents: I'm going to migrate this question to the Statistics.SE site. There will be a link that appears below the question here that you can follow to the new location of your question. If you need help associating an account on Statistics.SE, you can flag your question for moderator attention, and someone over there will help out. – Zev Chonoles Oct 29 '11 at 20:22
• @rolando2: This is a recently migrated post. If you were, by chance, the one that downvoted it, you might reconsider since the OP has not yet visited here and may not even (yet) be aware of the existence of this particular SE site. Also, if you have a potential duplicate in mind, it would be useful to provide a link and/or flag it appropriately. Cheers :) – cardinal Oct 29 '11 at 20:53
• @cardinal - thanks for the message; I've reversed that vote. – rolando2 Oct 30 '11 at 0:19

There are statistical tests that allow you to check if your data are an inappropriately poor match to a given distribution as @ChillPenguin has noted. However, I think graphical techniques are best for this task.

Typically, the best approach is to use a qq-plot. A somewhat less-used, but similar approach is to use a pp-plot. Note that a qq-plot gives you better resolution in the tails of the distribution, while a pp-plot gives you better resolution in the middle of the distribution. As I said, people usually go with a qq-plot, because typically deviations in the tails are more important.

These plots make it easy to see that your data differ from a theoretical distribution, but sometimes it is hard to interpret how they are deviating. If you have checked a qq-plot, and are concerned that your data don't fit, but want a clearer picture of how that manifests, one approach is to make a kernel demsity plot of your data, possibly overlaid with a theoretical distribution that has the same mean and SD.

Note that none of these approaches necessarily tells you which distribution your data come from, they would only tell you that the fit is reasonable or poor. If they are poor, then you need to use your knowledge of your data and the range of distributions that exist to pick another contender to explore. For example, if you had a distribution of counts for, say, the number of auto accidents at different locations, and checked it against a normal, you would most likely find a poor fit. However, nothing there would tell you that you should be checking your data against a Poisson distribution instead; you would need to know about that yourself.

Given a model (that is, a parametric family of distributions, such the family of normal distributions parametrized by mean and variance), the most straightforward thing to do is use maximum likelihood estimation to estimate the parameters, then use the probability density function to assess how typical the data are. If the model is conventional and parsimonious (rather than being tailored to the data, or something), and it makes the data look reasonably typical, you can argue that the model is good enough.

Goodness-of-fit tests are often recommended for this sort of thing, but all they're good for is justifying a statement that the data doesn't come from a given distribution. Failure to reject the null hypothesis of a goodness-of-fit test isn't evidence that the data does in fact come from that distribution.

• The crux of your reply is to check whether the "model is conventional and parsimonious ... and it makes the data look reasonably typical." How precisely does one check these things? – whuber Jan 19 '12 at 14:52