You’re probably familiar with data that follow the normal distribution. The normal distribution is that nice, familiar bell-shaped curve. Unfortunately, not all data are normally distributed or as intuitive to understand. You can picture the symmetric normal distribution, but what about the Weibull or Gamma distributions? This uncertainty might leave you feeling unsettled. In this post, I show you how to identify the probability distribution of your data.

You might think of nonnormal data as abnormal. However, in some areas, you should actually expect nonnormal distributions. For instance, income data are typically right skewed. If a process has a natural limit, data tend to skew away from the limit. For example, purity can’t be greater than 100%, which might cause the data to cluster near the upper limit and skew left towards lower values. On the other hand, drill holes can’t be smaller than the drill bit. The sizes of the drill holes might be right-skewed away from the minimum possible size.

Data that follow any probability distribution can be valuable. However, many people don’t feel as comfortable with nonnormal data. Let’s shed light on how to identify the distribution of your data!

We’ll learn how to identify the probability distribution using body fat percentage data from middle school girls that I collected during an experiment. You can download the CSV data file: body_fat.

**Related post**: Understanding Probability Distributions and The Normal Distribution

## Graph the Raw Data

Let’s plot the raw data to see what it looks like.

The histogram gives us a good overview of the data. At a glance, we can see that these data clearly are not normally distributed. They are right skewed. The peak is around 27%, and the distribution extends further into the higher values than to the lower values.

These data are not normal, but which probability distribution do they follow? Fortunately, statistical software can help us!

**Related posts**: Using Histograms to Understand Your Data and Assessing Normality: Histograms vs. Normal Probability Plots

## Using Distribution Tests to Identify the Probability Distribution that Your Data Follow

Distribution tests are hypothesis tests that determine whether your sample data were drawn from a population that follows a hypothesized probability distribution. Like any statistical hypothesis test, distribution tests have a null hypothesis and an alternative hypothesis.

- H
_{0}: The sample data follow the hypothesized distribution. - H
_{1}: The sample data do not follow the hypothesized distribution.

For distribution tests, small p-values indicate that you can reject the null hypothesis and conclude that your data were not drawn from a population with the specified distribution. However, we want to identify the probability distribution that our data follow rather than the distributions they don’t follow! Consequently, distribution tests are a rare case where you look for high p-values to identify candidate distributions.

Before we test our data to identify the distribution, here are some measures you need to know:

**Anderson-Darling statistic (AD):** There are different distribution tests. The test I’ll use for our data is the Anderson-Darling test. The Anderson-Darling statistic is the test statistic. It’s like the t-value for t-tests or the F-value for F-tests. Typically, you don’t interpret this statistic directly, but the software uses it to calculate the p-value for the test.

**P-value:** Distribution tests that have high p-values are suitable candidates for your data’s distribution. Unfortunately, it is not possible to calculate p-values for some distributions with three parameters.

**LRT P:** If you are considering a three-parameter distribution, assess the LRT P to determine whether the third parameter significantly improves the fit compared to the associated two-parameter distribution. An LRT P value that is less than your significance level indicates a significant improvement over the two-parameter distribution. If you see a higher value, consider staying with the two-parameter distribution.

## Goodness of Fit Test Results for the Distribution Tests

I’m using Minitab, which can test 14 probability distributions and two transformations all at once. Let’s take a look at the output below. We’re looking for higher p-values in the Goodness-of-Fit Test table below.

As we expected, the Normal distribution does not fit the data. The p-value is less than 0.005, which indicates that we can reject the null hypothesis that these data follow the normal distribution.

The Box-Cox transformation and the Johnson transformation both have high p-values. If we need to transform our data to follow the normal distribution, the high p-values indicate that we can use these transformations successfully. However, we’ll disregard the transformations because we want to identify our probability distribution rather than transform it.

The highest p-value is for the three-parameter Weibull distribution (>0.500). For the three-parameter Weibull, the LRT P is significant (0.000), which means that the third parameter significantly improves the fit.

The lognormal distribution has the next highest p-value of 0.345.

Let’s consider the three-parameter Weibull distribution and lognormal distribution to be our top two candidates.

## Using Probability Plots to Identify the Distribution of Your Data

Probability plots might be the best way to determine whether your data follow a particular distribution. If your data follow the straight line on the graph, the distribution fits your data. This process is very easy to do visually. Informally, this process is called the “fat pencil” test. If all the data points line up within the area of a fat pencil laid over the center straight line, you can conclude that your data follow the distribution.

These plots are especially useful in cases where the distribution tests are too powerful. Distribution tests are like other hypothesis tests. As the sample size increases, the statistical power of the test also increases. With very large sample sizes, the test can have so much power that trivial departures from the distribution produce statistically significant results. In these cases, your p-value will be less than the significance level even when your data follow the distribution.

The solution is to assess the probability plots to identify the distribution of your data. If the data points fall along the straight line, you can conclude the data follow that distribution even if the p-value is statistically significant.

The probability plots below include the normal distribution, our top two candidates, and the gamma distribution.

The data points for the normal distribution don’t follow the center line. However, the data points do follow the line very closely for both the lognormal and the three-parameter Weibull distributions. The gamma distribution doesn’t follow the center line quite as well as the other two, and its p-value is lower. Again, it appears like the choice comes down to our top two candidates from before. How do we choose?

## An Additional Consideration for Three-Parameter Distributions

Three-parameter distributions have a threshold parameter. The threshold parameter is also known as the location parameter. This parameter shifts the entire distribution left and right along the x-axis. The threshold/location parameter defines the smallest possible value in the distribution. You should use a three-parameter distribution only if the location truly is the lowest possible value. In other words, use subject-area knowledge to help you choose.

The threshold parameter for our data is 16.06038 (shown in the table below). This cutoff point is based on (but not equal to) the smallest value in our sample. However, in the full population of middle school girls, it is unlikely that there is a strict cutoff at this value. Instead, lower values are possible even though they are less likely. Consequently, I’ll pick the lognormal distribution.

## Parameter Values for Our Distribution

We’ve identified our distribution as the lognormal distribution. Now, we need to find the parameter values for it. Population parameters are the values that define the shape and location of the distribution. We just need to look in the distribution parameters table below!

Our body fat percentage data for middle school girls follows a lognormal distribution with a location of 3.32317 and a scale of 0.24188.

Below, I created a probability distribution plot of our two top candidates using the parameter estimates. You can see how the three-parameter Weibull distribution stops abruptly at the threshold/location value. However, the lognormal distribution continues to lower values.

Identifying the probability distribution that your data follow can be critical for analyses that are very sensitive to the distribution, such as capability analysis. In a future blog post, I’ll show you what else you can do by simply knowing the distribution of your data. This post is all continuous data and continuous probability distributions. If you have discrete data, read my post about Goodness-of-Fit Tests for Discrete Distributions.

Finally, I’ll close this post with a graph that compares the raw data to the fitted distribution that we identified.

**Note: I wrote a different version of this post that appeared elsewhere. I’ve completely rewritten and updated it for my blog site.**

Asmat says

Thank you so much for your detailef email. I really appriciate it.

Asmat says

Hi Jim,

Thanks for your detailed reply. I am using cross-sectional continuous data of inputs (11 variables) used in crop production system. Variabilities exist within the dataset due to different level of inputs consumptions in farming systems and in some cases some inputs are even zero. Should I go for any grouping of the data. If yes, what kind of grouping approach should I use. I am basically interested in the uncertainty analysis of inputs (fuel and chemicals consumptions) and sensitivity analysis of desired output and associated environmental impacts. It will be great if you can guide me.

Thanks.

Jim Frost says

Hi Asmat,

Given the very specific details of your data and goals for your study, I think you’ll need to discuss this with someone who can sit down with you and go over all of it and give it the time that your study deserves. There’s just not enough information for me to go on and I don’t have the time, unfortunately, to really look into it.

One thing I can say is that if you’re trying to link your inputs to changes in the output, consider using regression analysis. For regression analysis, you only need to worry about the distribution of your residuals rather than your inputs and outputs. Regression is all about linking changes in the inputs to changes in the output. Read my post about when to use regression analysis for more information. It sounds like that might be the goal of your analysis, but I’m not sure.

Best of luck with your study!

Andre Konski says

Thanks for the reply. No I am trying to determine the distribution of my survival curve from a published analysis. I was able to identify the survival probabilities from the published graph. The Minitab program only allows for the importation of one column. The distribution looks like a Weibull distribution but the Minitab results showed a normal distribution had the highest P value which didn’t make sense.

Jim Frost says

Ok, in your original comment you didn’t mention that you were using a published graph. I don’t fully understand what you’re trying to do, and it’s impossible for a concrete reply without the full details. However, below are some things to consider.

Analysts often need to use their process knowledge to help them determine which distribution is appropriate. Perhaps that’s a consideration here? I also don’t know how different the p-values are. Small differences are not meaningful. Additionally, in some cases, Weibull distributions can approximate a normal distribution. Consequently, there might be only a small difference between those distributions.

But, it’s really hard to say. There’s just not enough information.

Asmat says

Hi Jim,

Thanks for your reply.

Yes I on the basis of p-values only I am concluding that the data is not following any distribution.

My sample size is 1366 and 11 variables. None is following normal distribution. I tried Box Cox transformation and checked normality again following p-value.

After transformation, the data points of some variables largly follow the line but some data points deviate from the line either at the begging or at the end.

However some variabls, the data points largely follow the line even without transformation with some points deviations at the ends. Thanks.

Jim Frost says

Hi Asmat,

You have a particularly large sample size. Consequently, you might need to focus more on the probability plots rather than the p-values. I suggest you read the section in this post that is titled “Using Probability Plots to Identify the Distribution of Your Data.” It describes how the additional power that distribution tests have with large sample sizes can detect meaningless deviations from a distribution. In those cases, using probability plots might be a better approach.

After you look at the probability plots, if an untransformed distribution fits well, I’d use that, otherwise go with the transformed.

You didn’t mention what you’re using these data for but be aware that some hypothesis tests are robust to departures from the normal distribution.

Asmat says

Hi Jim,

after reading your blog, I have tried Minitab to check the distribution of my data but it surprisingly it does not follow any of the listed probability distribution. Could you please help me how should I move forward. Thanks.

Jim Frost says

Hi Asmat,

Before I can attempt to answer your question, I need to ask you several questions about your data.

What type of data are you talking about?

Did the Box-Cox transformation or Johnson transformation produce a good fit?

What is your sample size?

Are you primarily going by p-values? If so, do any of the probability plots look good? Good meaning that the data points largely following the line. There’s the informal fat pencil test where if you put a pencil over the line, do the data points stay within it.

Andre Konski says

Jim

I enjoyed this blog. I tried to determine the distribution and parameters of a survival curve by importing into minitab. Minitab only allows 1 parameter or column while the survival curve has time on x axis and y on the y. How does one find the type of curve and parameters of a survival curve.

Jim Frost says

Hi Andre,

Sorry about the delay in replying!

If I’m understanding your question correctly, the answer is that creating a survival plot with a survival curve is not a part of the process for identifying your distribution in Minitab that I show in this blog post. However, you can find the proper analyses in the Reliability/Survival menu in Minitab. In that menu path, there are distribution analyses for failure data specifically.

Additionally, there are other analyses in the Reliability/Survival path including the following:

Stat > Reliability/Survival > Probit Analysis.

And, if you’re using accelerated testing: Stat > Reliability/Survival > Accelerated Life Testing.

I hope this helps!

Jerry says

Hi Jim, in the next-to-last graph in your post (the distribution plots), you say the Weibull plot stops abruptly at the location value of 3.32. Yet it appears to stop at more like 13-ish. Did I misunderstand something, or is the graph incorrect? Also what is the ‘scale’ metric in Weibull plots? Thanks –

Jim Frost says

Hi Jerry, the Weibull distribution actually stops at the threshold value of ~16. The threshold value shifts the distribution along the X-axis relative to zero. Consequently, a threshold of 16 indicates the distribution starts with the lowest value of 16. Without the threshold parameter, the Weibull distribution starts at zero.

The scale parameter is similar to a measure of dispersion. For a given shape, it indicates how spread out the values are.

Here’s a nice site that shows the effect of the shape, scale, and threshold parameters for the Weibull distribution.

Alice zhang says

Hi, Jim. Thank you so much for your detailed reply. Does the Null Hypothesis in Minitab is the dsitribution follows the specific distribution? So larger p-value cannot reject the null hypothesis. Another question is about the correlation coefficient (PPCC value), does it can denote the goodness-of-fit of each dsitribution? Thanks.

Jim Frost says

Hi Alice, you’re very welcome!

Yes, as I detail in this post, the null hypothesis states that the data follow the specific distribution. Consequently, a low p-value suggests that you should reject the null and conclude that the data do not follow that distribution. Reread this post for more information about that aspect of these distribution tests.

Unless Minitab has changed something that I’m unaware of, you do not need to worry about PPCC when interpreting the probability plots. Again, reread this post to learn how to interpret the probability plots. With such a large sample size, it will be more important for you to interpret the probability plots rather than the p-values.

If you want to learn about PPCC for other reasons, here’s a good source of information about it: Probability Plot Correlation Coefficient.

Best of luck with your analysis!

Alice zhang says

Hi, Jim. Besides, I have tried the calculation with 1000 data, but the p-value is extremely small. Almost the p-value of all distributions are less than 0.005. Do you have any suggestions about this?

Jim Frost says

Yes, this is the issue that I described in my first response to you. With so many data points (even 1000 is a large dataset) these tests are very powerful. Trivial departures from the distribution will produce a low p-value. That’s why you’ll likely need to focus on the probability plots for each distribution.

Alice zhang says

Jim, Thanks for your detailed reply. Actually, I have tried the probability plots, and several distributions perform almost the same. And I used Minitab to calculate the p-value, but the software said that it is out of stock. There is no results for p-value. Do you know how to deal with it? Is the data (1,000,000) too much for the p-value calculation? Thanks.

Jim Frost says

I’m not sure. I think it might be out of memory. You should contact their technical support to find out for sure. They’ll know. Their support is quite good. You’ll reach a real person very quickly who can help you.

To answer your question, no, it’s not possible to have too many data points to calculate the p-value mathematically. But, it’s possible that the program can’t handle such a large dataset. I’m not sure about that.

Alice zhang says

Hi, Jim. Thanks for your detailed explanation. Actually, I have no experience in Minitab. I have a large matrix (1000000*13), but I found that when I went to Stat > Quality Tools > Individual Distribution Identification in Minitab, It only can do the single column data analysis. And it seems that it takes much time for 1,000,000 data. Do you have any sugggestions about how to find the appropriate distribution for 13 columns with 1,000,000 data?

Jim Frost says

Hi Alice,

Yes, that tool analyzes individual columns only. It assesses the distribution of a single variable. If you’re looking for some sort of multivariate distribution analysis, it won’t do that.

I think Minitab is good software, but it can struggle with extremely large datasets like yours.

One thing to be aware of is that with so many data points, the distribution tests become extremely powerful. They will be so powerful that they can detect trivial departures from a distribution. In other words, your data might follow a specific distribution, but the test is so powerful that it will reject the null hypothesis that it follows that distribution. For such a large dataset, pay particular attention to the probability plots! If those look good but the p-value is significant, go with the graph!

Rashmi Tiwari says

Hi Jim, very nicely explained. Thank you so much for your effort.

Are your blogs are available in printable version?

Jim Frost says

Hi Rashmi, thank you so much for your kind words. I really appreciate them!

I’m working on ebooks that contain the blog material plus a lot more! The first one should be available early next year.

Hani Hani says

Hello. I have a question. I have genome data has lots of zeros. I want to check the distribution of this data. It could be continuous or discrete. We can use ks.test, but this is for continuous distributions. Is there any way to check if the data follows a specific discrete distribution? Thank you

Jim Frost says

Hi, there are distribution tests for discrete data. I’d start by reading my post about goodness-of-fit tests for discrete distributions and see if that helps.

Rainy says

Hi Jim,

Great sharing. I have perform an identification of distribution of my nonnormal data, however none of the distribution have good fit to my data. All the p-value < 0.05. What are the different approach I can use to study the distribution model before I can perform capability analysis ðŸ™‚ Thanks for your help

Alice Pettersson says

Hi Jim! I’m trying to test the distribution of my data in SPSS and have used the One-Sample Kolmogorov-Smirnov Test which test for normal, uniform, poisson or exponential distribution. Non of them fit my data… How do I preoceed, I don’t know how to work in R or MiniTab, so do you know if there’s another test in SPSS I can do or do I have to learn a new program? I need to know the distribution to be able to choose the right model for the GLM test I’m gonna do.

Jim Frost says

Hi Alice! Unfortunately, I haven’t used SPSS in quite some time and I’m not familiar with its distribution testing capabilities. The one additional distribution that I’d check is the Weibull distribution. That is a particularly flexible distribution that can fit many different shapes–but I don’t see that in your list.

Maria says

Very good explanation!!

Jim Frost says

Thank you, Maria!

Olayan Albalawi says

Great explanation, thanks Jim.

Jim Frost says

Thank you, Olayan!

Wilbrod Ntawiha says

Thanks Jim. I am going to try the same implementation using stata and/or R.

Jim Frost says

You’re very welcome Wilbrod! Best of luck with your analysis!

Charles says

Hi Jim,

Great article! ðŸ™‚

What software are you using to evaluate the distribution of the data?

Jim Frost says

Hi Charles, thanks and I’m glad you found it helpful! I’m using Minitab statistical software.

Ricardo says

Hello Jim, what kind of statistics software do you use?

Jim Frost says

Hi Ricardo, I’m using Minitab statistical software.

Muhammad Arif says

what a fantastic example Jim!

Jim Frost says

Thank you so much, Muhammad!