A Gentle Introduction to Normality Tests in Python

An important decision point when working with a sample of data is whether to use parametric or nonparametric statistical methods.

Parametric statistical methods assume that the data has a known and specific distribution, often a Gaussian distribution. If a data sample is not Gaussian, then the assumptions of parametric statistical tests are violated and nonparametric statistical methods must be used.

There are a range of techniques that you can use to check if your data sample deviates from a Gaussian distribution, called normality tests.

In this tutorial, you will discover the importance of checking whether a data sample deviates from the normal distribution and a suite of techniques that you can use to evaluate your data sample.

After completing this tutorial, you will know:

  • How whether a sample is normal dictates the types of statistical methods to use with a data sample.
  • Graphical methods for qualifying deviations from normal, such as histograms and the Q-Q plot.
  • Statistical normality tests for quantifying deviations from normal.

Kick-start your project with my new book Statistics for Machine Learning, including step-by-step tutorials and the Python source code files for all examples.

Let’s get started.

  • Update May/2018: Updated interpretation of results for Anderson-Darling test, thanks Elie.
  • Update May/2018: Updated language about “reject” vs “failure to reject” H0.
A Gentle Introduction to Normality Tests in Python

A Gentle Introduction to Normality Tests in Python
Photo by Ramoun Cabuhay, some rights reserved.

Tutorial Overview

This tutorial is divided into 5 parts; they are:

  1. Normality Assumption
  2. Test Dataset
  3. Visual Normality Checks
  4. Statistical Normality Tests
  5. What Test Should You Use?

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Normality Assumption

A large fraction of the field of statistics is concerned with data that assumes that it was drawn from a Gaussian distribution.

If methods are used that assume a Gaussian distribution, and your data was drawn from a different distribution, the findings may be misleading or plain wrong.

There are a number of techniques that you can check if your data sample is Gaussian or sufficiently Gaussian-like to use the standard techniques, or sufficiently non-Gaussian to instead use non-parametric statistical methods.

This is a key decision point when it comes to choosing statistical methods for your data sample. We can summarize this decision as follows:

There is also some middle ground where we can assume that the data is Gaussian-enough to use parametric methods or that we can use data preparation techniques to transform the data to be sufficiently Gaussian to use the parametric methods.

There are three main areas where you may need to make this evaluation of a data sample in a machine learning project; they are:

  • Input data to the model in the case of fitting models.
  • Model evaluation results in the case of model selection.
  • Residual errors from model predictions in the case of regression.

In this tutorial, we will look at two classes of techniques for checking whether a sample of data is Gaussian:

  • Graphical Methods. These are methods for plotting the data and qualitatively evaluating whether the data looks Gaussian.
  • Statistical Tests. These are methods that calculate statistics on the data and quantify how likely it is that the data was drawn from a Gaussian distribution.

Methods of this type are often called normality tests.

Test Dataset

Before we start looking at normality tests, let’s first develop a test dataset that we can use throughout this tutorial.

We will generate a small sample of random numbers drawn from a Gaussian distribution.

The choice of Gaussian random numbers for the test dataset means that we do expect each test to correctly identify the distribution, nevertheless, the small-ish sample size may introduce some noise into the results.

We will use the randn() NumPy function to generate random Gaussian numbers with a mean of 0 and a standard deviation of 1, so-called standard, normal variables. We will then shift them to have a mean of 50 and a standard deviation of 5.

The complete example is listed below.

Running the example generates the sample and prints the mean and standard deviation of the sample.

We can see that the mean and standard deviation are reasonable but rough estimations of the true underlying population mean and standard deviation, given the small-ish sample size.

Visual Normality Checks

We can create plots of the data to check whether it is Gaussian.

These checks are qualitative, so less accurate than the statistical methods we will calculate in the next section. Nevertheless, they are fast and like the statistical tests, must still be interpreted before you can make a call about your data sample.

In this section, we will look at two common methods for visually inspecting a dataset to check if it was drawn from a Gaussian distribution.

Histogram Plot

A simple and commonly used plot to quickly check the distribution of a sample of data is the histogram.

In the histogram, the data is divided into a pre-specified number of groups called bins. The data is then sorted into each bin and the count of the number of observations in each bin is retained.

The plot shows the bins across the x-axis maintaining their ordinal relationship, and the count in each bin on the y-axis.

A sample of data has a Gaussian distribution of the histogram plot, showing the familiar bell shape.

A histogram can be created using the hist() matplotlib function. By default, the number of bins is automatically estimated from the data sample.

A complete example demonstrating the histogram plot on the test problem is listed below.

Running the example creates a histogram plot showing the number of observations in each bin.

We can see a Gaussian-like shape to the data, that although is not strongly the familiar bell-shape, is a rough approximation.

Histogram Plot Normality Check

Histogram Plot Normality Check

Quantile-Quantile Plot

Another popular plot for checking the distribution of a data sample is the quantile-quantile plot, Q-Q plot, or QQ plot for short.

This plot generates its own sample of the idealized distribution that we are comparing with, in this case the Gaussian distribution. The idealized samples are divided into groups (e.g. 5), called quantiles. Each data point in the sample is paired with a similar member from the idealized distribution at the same cumulative distribution.

The resulting points are plotted as a scatter plot with the idealized value on the x-axis and the data sample on the y-axis.

A perfect match for the distribution will be shown by a line of dots on a 45-degree angle from the bottom left of the plot to the top right. Often a line is drawn on the plot to help make this expectation clear. Deviations by the dots from the line shows a deviation from the expected distribution.

We can develop a QQ plot in Python using the qqplot() statsmodels function. The function takes the data sample and by default assumes we are comparing it to a Gaussian distribution. We can draw the standardized line by setting the ‘line‘ argument to ‘s‘.

A complete example of plotting the test dataset as a QQ plot is provided below.

Running the example creates the QQ plot showing the scatter plot of points in a diagonal line, closely fitting the expected diagonal pattern for a sample from a Gaussian distribution.

There are a few small deviations, especially at the bottom of the plot, which is to be expected given the small data sample.

Q-Q Plot Normality Check

QQ Plot Normality Check

Statistical Normality Tests

There are many statistical tests that we can use to quantify whether a sample of data looks as though it was drawn from a Gaussian distribution.

Each test makes different assumptions and considers different aspects of the data.

We will look at 3 commonly used tests in this section that you can apply to your own data samples.

Interpretation of a Test

Before you can apply the statistical tests, you must know how to interpret the results.

Each test will return at least two things:

  • Statistic: A quantity calculated by the test that can be interpreted in the context of the test via comparing it to critical values from the distribution of the test statistic.
  • p-value: Used to interpret the test, in this case whether the sample was drawn from a Gaussian distribution.

Each test calculates a test-specific statistic. This statistic can aid in the interpretation of the result, although it may require a deeper proficiency with statistics and a deeper knowledge of the specific statistical test. Instead, the p-value can be used to quickly and accurately interpret the statistic in practical applications.

The tests assume that that the sample was drawn from a Gaussian distribution. Technically this is called the null hypothesis, or H0. A threshold level is chosen called alpha, typically 5% (or 0.05), that is used to interpret the p-value.

In the SciPy implementation of these tests, you can interpret the p value as follows.

  • p <= alpha: reject H0, not normal.
  • p > alpha: fail to reject H0, normal.

This means that, in general, we are seeking results with a larger p-value to confirm that our sample was likely drawn from a Gaussian distribution.

A result above 5% does not mean that the null hypothesis is true. It means that it is very likely true given available evidence. The p-value is not the probability of the data fitting a Gaussian distribution; it can be thought of as a value that helps us interpret the statistical test.

Shapiro-Wilk Test

The Shapiro-Wilk test evaluates a data sample and quantifies how likely it is that the data was drawn from a Gaussian distribution, named for Samuel Shapiro and Martin Wilk.

In practice, the Shapiro-Wilk test is believed to be a reliable test of normality, although there is some suggestion that the test may be suitable for smaller samples of data, e.g. thousands of observations or fewer.

The shapiro() SciPy function will calculate the Shapiro-Wilk on a given dataset. The function returns both the W-statistic calculated by the test and the p-value.

The complete example of performing the Shapiro-Wilk test on the dataset is listed below.

Running the example first calculates the test on the data sample, then prints the statistic and calculated p-value.

The p-value is interested and finds that the data is likely drawn from a Gaussian distribution.

D’Agostino’s K^2 Test

The D’Agostino’s K^2 test calculates summary statistics from the data, namely kurtosis and skewness, to determine if the data distribution departs from the normal distribution, named for Ralph D’Agostino.

  • Skew is a quantification of how much a distribution is pushed left or right, a measure of asymmetry in the distribution.
  • Kurtosis quantifies how much of the distribution is in the tail. It is a simple and commonly used statistical test for normality.

The D’Agostino’s K^2 test is available via the normaltest() SciPy function and returns the test statistic and the p-value.

The complete example of the D’Agostino’s K^2 test on the dataset is listed below.

Running the example calculates the statistic and prints the statistic and p-value.

The p-value is interpreted against an alpha of 5% and finds that the test dataset does not significantly deviate from normal.

Anderson-Darling Test

Anderson-Darling Test is a statistical test that can be used to evaluate whether a data sample comes from one of among many known data samples, named for Theodore Anderson and Donald Darling.

It can be used to check whether a data sample is normal. The test is a modified version of a more sophisticated nonparametric goodness-of-fit statistical test called the Kolmogorov-Smirnov test.

A feature of the Anderson-Darling test is that it returns a list of critical values rather than a single p-value. This can provide the basis for a more thorough interpretation of the result.

The anderson() SciPy function implements the Anderson-Darling test. It takes as parameters the data sample and the name of the distribution to test it against. By default, the test will check against the Gaussian distribution (dist=’norm’).

The complete example of calculating the Anderson-Darling test on the sample problem is listed below.

Running the example calculates the statistic on the test data set and prints the critical values.

Critical values in a statistical test are a range of pre-defined significance boundaries at which the H0 can be failed to be rejected if the calculated statistic is less than the critical value. Rather than just a single p-value, the test returns a critical value for a range of different commonly used significance levels.

We can interpret the results by failing to reject the null hypothesis that the data is normal if the calculated test statistic is less than the critical value at a chosen significance level.

We can see that at each significance level, the test has found that the data follows a normal distribution

What Test Should You Use?

We have covered a few normality tests, but this is not all of the tests that exist.

So which test do you use?

I recommend using them all on your data, where appropriate.

The question then becomes, how do you interpret the results? What if the tests disagree, which they often will?

I have two suggestions for you to help think about this question.

Hard Fail

Your data may not be normal for lots of different reasons. Each test looks at the question of whether a sample was drawn from a Gaussian distribution from a slightly different perspective.

A failure of one normality test means that your data is not normal. As simple as that.

You can either investigate why your data is not normal and perhaps use data preparation techniques to make the data more normal.

Or you can start looking into the use of nonparametric statistical methods instead of the parametric methods.

Soft Fail

If some of the methods suggest that the sample is Gaussian and some not, then perhaps take this as an indication that your data is Gaussian-like.

In many situations, you can treat your data as though it is Gaussian and proceed with your chosen parametric statistical methods.


This section lists some ideas for extending the tutorial that you may wish to explore.

  • List two additional examples of when you think a normality test might be useful in a machine learning project.
  • Develop your own contrived dataset and apply each normality test.
  • Load a standard machine learning dataset and apply normality tests to each real-valued variable.

If you explore any of these extensions, I’d love to know.

Further Reading

This section provides more resources on the topic if you are looking to go deeper.




In this tutorial, you discovered the importance of checking whether a data sample deviates from the normal distribution and a suite of techniques that you can use to evaluate your data sample.

Specifically, you learned:

  • How whether a sample is normal dictates the types of statistical methods to use with a data sample.
  • Graphical methods for qualifying deviations from normal such as histograms and the Q-Q plot.
  • Statistical normality tests for quantifying deviations from normal.

Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.

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128 Responses to A Gentle Introduction to Normality Tests in Python

  1. Avatar
    Elie Kawerk May 11, 2018 at 5:43 am #

    Hi Jason,

    Thanks for this nice post. In the Anderson-Darling Test paragraph, should the last sentence be:

    ‘We can see that at each significance level, the test has found that the data follows a normal distribution’

    instead of:

    ‘We can see that at each significance level, the test has found significant departure from normal.’


  2. Avatar
    Nouroz May 11, 2018 at 7:40 am #

    Failed to reject a hypothesis doesn’t mean you accept it. 😀

  3. Avatar
    Vinay May 11, 2018 at 11:52 am #


    Brilliantly put together. Thank you for the post. Now that we know if the data is normally distributed or not, how do we transform data that is not normally distributed into normally distributed? Is boxcox the only way or do you prefer any other method?

    • Avatar
      Jason Brownlee May 11, 2018 at 2:31 pm #


      Awesome question! I have a post on this exact question scheduled. Stand by for a few days.

      • Avatar
        Vinay May 15, 2018 at 9:42 pm #

        Sweet!, thank you.

    • Avatar
      Sumithra July 29, 2021 at 3:13 am #

      you can log transformation also and some other techniques also there

  4. Avatar
    Raj May 11, 2018 at 2:47 pm #

    Thanks once again for nice post.

  5. Avatar
    Akhilesh May 13, 2018 at 6:08 pm #

    Excellent article. I liked the section where you highlighted the use of parametric of non-parametric model for fitting a ML model depending upon whether the data at hand is close to normal or not. I think most of the books do not really state this explicitly.

  6. Avatar
    Tarlan Mammadov May 20, 2018 at 6:59 am #

    Jason, this is a great article thanks! so in the case that the data set has multiple real valued features we should be doing this test for each and then if one or more of these features turn out to be non-Gaussian does it mean that whole data set is non-Gaussian? Or should we try to address the non-Gaussian features via the data preparation step?

    Also, do you have a similar post about the inductive biases of various machine learning algorithms?

    • Avatar
      Jason Brownlee May 21, 2018 at 6:20 am #

      We would address it one variable at a time.

      Inductive bias is is more of an abstraction to understand how algorithm design decisions impact the search/choice of mapping hypotheses, it’s not really a measurable quantity.

  7. Avatar
    Bhaskar May 24, 2018 at 4:45 pm #

    Shouldn’t the Null Hypothesis (H0) be assuming our data to be normal?

  8. Avatar
    Mamta May 25, 2018 at 7:08 pm #

    Great Article ..
    I have tried further extension “Load a standard machine learning dataset and apply normality tests to real-valued variable.”

    Link : https://github.com/Mamtasadani/Normality-Test-v

    Please give your valuable suggestions..

  9. Avatar
    Krzysztof June 22, 2018 at 3:30 pm #

    Hi Jason,

    I’m currently working on project for time series predictions. Basically I use RNN (LSTM) machine learning models for making predictions. Does the normality (or not) of time series I work with affect the way I should construct my models or LSTM can work well even with data that is not normal? Simply, should I always make my time series normal where work with LSTM?


    • Avatar
      Jason Brownlee June 23, 2018 at 6:12 am #

      It may, it really depends on the problem. I’d recommend evaluating multiple different preparations of your input data.

  10. Avatar
    Dom July 10, 2018 at 3:09 pm #

    How would you analyze highly right skewed data like salary data where few people earn a lot and rest is somewhere in the middle / lower range?

    • Avatar
      Jason Brownlee July 11, 2018 at 5:51 am #

      Sounds like a power/exponential distribution, not Gaussian.

  11. Avatar
    Vips August 4, 2018 at 12:22 pm #

    Possibly the only logical and best explained tutorial that I could find after hours of searching. Bookmarked this and will be referring to this over and over again. Can’t thank you enough for your efforts, Dr. Brownlee!

  12. Avatar
    Jorge August 15, 2018 at 7:23 pm #

    I did the Shapiro-Wilk test for alpha = 0.05 and I got a p value = 0.04, the null hypothesis is rejected. The D’Agostino K-square test gave a value of p = 0.191, the null hypothesis is not rejected, and the Anderson-Darling test for four critical values the null hypothesis is rejected and for one critical value the null hypothesis is not rejected . What do you recommend me to do next?


    only confirming, if my data reject the null hypothesis (not normal), should not I use any model (MAchine Learning) that is based on Gaussian distributions, such as Gaussian processes or exponential smoothing?

    and if I want to use them, should I process the data ??

    Thank you

    • Avatar
      Jason Brownlee August 16, 2018 at 6:02 am #

      Each test gives a different perspective on the normality of your data, you must interest the result in the context of your project/goals.

      See the section: “What Test Should You Use?”

      • Avatar
        Marcos February 4, 2020 at 1:21 pm #

        I’m incredibly greatful for this introduction, I’ve implemented them all to test my data. I’m in doubt about the interpretation of significance levels in the Anderson-Darling Test. Coloquialy speaking: what would it mean that H0 is rejected for 2.5% but not for 5%? What are the percentages? Say, at which level would you be confident that the input data is Gaussian? 5%? 10%?

        Maybe it might be useful to have a “not so Gaussian” example, so that it passes say 2 out of 5 significance levels

        • Avatar
          Jason Brownlee February 4, 2020 at 2:17 pm #

          Thanks, I’m happy it helps!

          The probabilities represent how likely the event is real vs a statistical fluke in the data. We can never know for sure, but we can use a statistical test to talk about the likelihood.

          Typically you pick the strongest level available, e.g. 5%. Beyond 5% (1 in 20) we often give up and say it’s probably not real.

  13. Avatar
    xkcd August 19, 2018 at 12:18 am #

    Excellent article!

    Do you have some pointers for how to test if the components of a Multimodal distributed data are Normal or not?

    • Avatar
      Jason Brownlee August 19, 2018 at 6:25 am #

      Ouch, why?

      Well, if forced, you could re-define the distribution and cut out an element you think is bell shaped and test that. Sounds dangerously invalid to me.

      I expect there are far better approaches if you spoke to a real statistician! Perhaps post on the math stackoverflow re best practices in this situation.

  14. Avatar
    Makis September 14, 2018 at 7:16 pm #

    Hello Jason,

    Amazing tutorial.

    I was wondering if there is any function in python that assesses multivariate normality.

    In R, there is the MVN package that does that (https://cran.r-project.org/web/packages/MVN/vignettes/MVN.pdf).


    • Avatar
      Jason Brownlee September 15, 2018 at 6:04 am #


      Good question, there might be. I’m not sure off the cuff.

      Let me know if you find something.

  15. Avatar
    Jk November 11, 2018 at 8:17 pm #

    I understand the importance of testing residuals in a regression model. But the other 2 usages you mention are unclear to me (Input data to the model in the case of fitting models.
    Model evaluation results in the case of model selection). Can you point to a resource for further explanation?

  16. Avatar
    Emily Stander December 19, 2018 at 11:59 pm #

    Hi Jason,

    Is this post available yet? Could you perhaps link to it?


  17. Avatar
    Anirban January 4, 2019 at 2:24 am #

    In which cases it will be helpful to know about distribution of our data? Please share some examples!

    • Avatar
      Jason Brownlee January 4, 2019 at 6:32 am #

      Almost every time – to aid in data preparation and model selection.

    • Avatar
      RAJESH November 24, 2020 at 3:37 pm #

      Hi Jason,

      All of your articles are very informative and helpful.

      I just wanted to know does KS test is sensitive to data normalisation? Because I’m getting a totally different results when my data is normalised and non-normalized.

      Does it expects the mean and std dev of my dataset should be 0 , 1.

      However, other tests like AD and D’agastino doesn’t get affected.

      • Avatar
        Jason Brownlee November 25, 2020 at 6:39 am #


        I guess it is – from your results. I recommend checking the documentation.

  18. Avatar
    Antonio February 7, 2019 at 2:59 am #


    Thank you for the good explanation of how to perform normality test on own data.

    How can I transform a non-normally data in normally data?

    Thanks again!

    • Avatar
      Jason Brownlee February 7, 2019 at 6:44 am #

      It depends on how non-normal it is.

      E.g. perhaps remove outliers, or fix simple issues with a power transform like a boxcox.

  19. Avatar
    Nasim February 8, 2019 at 6:36 am #

    Hi Jason,
    Thanks for the article. I was wondering if I want to apply these tests on streaming data, what would be the proper window size (window size = number of samples)? can I apply these tests on streaming data with a sliding window?

    • Avatar
      Jason Brownlee February 8, 2019 at 8:07 am #

      Good question, I don’t know off hand. I expect there might be specific methodologies for handing streaming data.

      If not, then a representative sample collected over a large interval might be a good start.

  20. Avatar
    dongliang February 16, 2019 at 11:37 am #

    helpful, thank you!

  21. Avatar
    Raz March 8, 2019 at 2:09 am #

    Hi Jason,

    Great tutorial. I am using Shapiro-Wilk Test to check if my data is normally distributed. I have two sample of distributions, when i test them individually the test says they have a Gaussian distribution. But after combining both into a single larger sample the test says otherwise.

    Should i be looking for the normality of the combined data?
    I am trying to compare the two sample distributions to check if they are significantly different.

    • Avatar
      Jason Brownlee March 8, 2019 at 7:54 am #

      If they are separate samples then it perhaps does not make sense to combine them.

  22. Avatar
    Samik March 11, 2019 at 8:41 pm #

    Hi Jason,

    Any pointers regarding correctional measures if data is not normally distributed. Asking particularly for to residual plots.

  23. Avatar
    AK March 14, 2019 at 1:00 am #

    Hi jason,

    How do we test normality of data for multi-dimensional features? Suppose I have 20 features, how should I know whether I run (plain) linear regression or whether I should take interaction terms or transform some features in squares etc??? How do we get to know about it.

  24. Avatar
    Mattia Ducci May 11, 2019 at 10:19 pm #

    Just thank you for your tutorials like this one! Very clear as always

  25. Avatar
    B Chandrashekar May 30, 2019 at 9:37 pm #

    It’s a wonderful article. Helped a lot to understand those concepts.
    Hope to get some more good articles from u.
    Thank you

  26. Avatar
    AR May 31, 2019 at 4:17 am #

    Thanks for the excellent tutorial. I have a follow-up question that I hope you can help me with.

    I have several datasets where I need to calculate the sample statistics like mean, variance, etc. After applying the tests you suggested, I found that many of the datasets aren’t normally distributed, and I wanted to know if you have any recommendations for the steps I should follow. The majority of the remaining appear to have an exponential distribution. Thanks.

    • Avatar
      Jason Brownlee May 31, 2019 at 7:55 am #

      It depends on what you want to do with the data as to the action you might want to take.

      E.g. modeling with a linear algorithm requires the inputs are gaussian. Modeling with something else does not have this requirement.

  27. Avatar
    Sahil June 7, 2019 at 1:11 am #


    Thanks for such an informative post. Is there any way to check the normality of entire dataset. Suppose I have 50 columns in my dataset, and I want to check if my data deviates from a Gaussian distribution, Can I do it ?
    Thanks in advance

    • Avatar
      Jason Brownlee June 7, 2019 at 8:05 am #

      Column by column.

      Stay away from multivariate gaussians if you can, it gets very complex (for me).

  28. Avatar
    Alok June 18, 2019 at 10:23 pm #

    What to do if the number of observation is very large nearly 40,000 observations?
    Is normality test still hold ?

    • Avatar
      Jason Brownlee June 19, 2019 at 7:56 am #

      Wonderful question.

      Probably, but almost all statistical tests were designed for small data. I think the validity on big data is a great open problem – e.g. the increased likelihood of rare events/outliers.

      • Avatar
        Tushar July 20, 2019 at 10:38 pm #

        so in that case, should we break the data in small chunks and test the “gaussianity” of those chunks, and those chunks which do not conform to the tests, should be reviewed and if required, deleted from the dataset on which we are working on ?

        • Avatar
          Jason Brownlee July 21, 2019 at 6:30 am #

          I would not advocate that approach, the results would have too much variance.

  29. Avatar
    Tushar July 20, 2019 at 10:35 pm #

    Hi Json
    Thanks for this post. Its really informative. Could you please tell as to why you have not included Dickey Fuller test? I mean from other posts which I have read, Dickey Fuller test is most widely talked about. So, is there specific reason that you chose these three tests over Dickey Fuller to mention about?

  30. Avatar
    Masfiq July 23, 2019 at 4:16 pm #

    This was very helpful. Now I have a dataset. This is not normal. The histogram of my data set looks this..


    Now how can I design a model (that will regenerate this data) .

    • Avatar
      Jason Brownlee July 24, 2019 at 7:48 am #

      Perhaps select a probability distribution that fits the data or model with with density estimation, then use the model to probabilistically generate new instances.

  31. Avatar
    William G. August 26, 2019 at 2:49 am #

    Thanks, great article. How can I use those tools with and input of a Pandas Dataframe?

    • Avatar
      Jason Brownlee August 26, 2019 at 6:22 am #

      Extract the numpy array from the dataframe (.values) and away you go!

      • Avatar
        William G. August 26, 2019 at 6:02 pm #

        Thank you for your answer Jason I tried, for example, with this one

        results = pd.DataFrame.from_records(all_stats, columns=[‘Sales’,
        stat, p = shapiro(results.values)
        print(‘Statistics=%.3f, p=%.3f’ % (stat, p))
        # interpret
        alpha = 0.05
        if p > alpha:
        print(‘Sample looks Gaussian (fail to reject H0)’)
        print(‘Sample does not look Gaussian (reject H0)’)

        But the p is always 0.000 . I hope you can help me, really appreciate it. Thank you!

        • Avatar
          Jason Brownlee August 27, 2019 at 6:36 am #

          Perhaps double check that you have loaded the data correctly?

          • Avatar
            William G. August 27, 2019 at 7:41 pm #

            I really appreciate your help, thank you. The output of my data seems correct:

            [[82024750. 2809815. 82350000.]
            [86901250. 2970088. 87000000.]
            [81341500. 2776092. 81375000.]

            [85586000. 2933965. 85875000.]
            [82237250. 2742930. 83025000.]
            [82130000. 2748805. 82875000.]]

            And I always get a result like that:
            ” Statistics=0.642, p=0.000 ”

            Should I change something?
            Thank you!

          • Avatar
            Jason Brownlee August 28, 2019 at 6:32 am #

            The result suggests the data is not gaussian.

  32. Avatar
    Yoon October 3, 2019 at 7:29 am #

    Regarding use of the QQ plot to check for normal distribution, can we simply plot the datapoints or should we take means of random samples to plot against the theoretical? Per the CLT, isn’t it saying that sample means would eventually take the form of a guassian distribution? And for the shapiro test, I get a warning for when sample is over 5000. Is there a better test for high sample?

  33. Avatar
    Daniel W. October 9, 2019 at 9:43 pm #

    Nice article! I am studying the shapiro-wilk test, but i can’t understant.
    What statistics, p and alpha stands for exactly?

    thank you for the answer man!

  34. Avatar
    Ravi Kumar October 24, 2019 at 3:44 am #


    I’m testing if my two samples have different means or not. For that, I’m first testing the normality of two samples. Let’s say they are not normal and after transformation they have become normal. Now if I perform a two-sample t-test on the transformed data, does the conclusion about the same/different means applicable to the original data?

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    AJILISA O A November 22, 2019 at 4:59 pm #

    Excellent Tutorial Sir….when I am working imbalanced classification problem where the majority of instance belongs to one class, several balanced datasets are formed using random undersampling method and other techniques. Can we use any method to evaluate the significance of the sample generated from each method? whether we need to check the sample follows Gaussian distribution or not?

    • Avatar
      Jason Brownlee November 23, 2019 at 6:44 am #


      No, I don’t think such analysis is required or meaningful.

      • Avatar
        AJILISA O A November 25, 2019 at 4:04 pm #

        whether we need to check the new sample generated using our proposed method follows the distribution of the original dataset

        • Avatar
          Jason Brownlee November 26, 2019 at 5:58 am #

          I assume they are multivariate examples. Perhaps look into density based methods, e.g. one class classification to see if new examples fit or don’t fit.

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    Venkatesh Gandi November 25, 2019 at 12:52 am #

    Hi Jason, Nice article. Thanks for sharing the knowledge. Is the alpha-value to all the above tests is 0.05? How can we change the alpha value to desired value. I have seen the documentation, I couldn’t find any answer. Can you please help me on this?

    • Avatar
      Jason Brownlee November 25, 2019 at 6:30 am #

      You can specify the alpha value directly, use any value you wish such as 0.01 0.05 or even 0.1.

  37. Avatar
    Venkatesh Gandi November 27, 2019 at 9:09 pm #

    Hi Jason, Can you please let us know how can we perform the Cramér–von Mises test for normality in python?

  38. Avatar
    Goodman February 12, 2020 at 9:07 am #

    This was brilliant, so easy to understand and systematic! This one post has taught me more about normality testing than three years of university! Just excellent thank you!

  39. Avatar
    San March 16, 2020 at 6:03 pm #

    What are the graphical & statistical normality tests used for categorical variables?

    I have a dataset that has only categorical variables. There’s an age column which represents 3 categories. The other attributes have only true, false values. The target attribute has 22 categories.


  40. Avatar
    Ahmet March 23, 2020 at 9:36 pm #

    Hi Jason, you are great. I have a question. What is the mean of W-Statistic in the Shapiro-Wilk test? How can we interpret this?

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    Rishi May 12, 2020 at 5:08 pm #

    Hi Jason , Great article. Quick question on the critical values in the Anderson-Darling test. Are the critical values equivalent to the confidence levels for examples say 15 = 85% confidence , 5 = 95% confidence and so on.

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    Borja June 27, 2020 at 9:32 pm #

    Thank you very much for this article. I was ready to spend ten times as long as I did researching about normality tests. I will still have to see more things by myself to understand better, but this was incredibly useful.
    What a time to be alive where you can Google the answer to so many questions.

  43. Avatar
    Levani October 16, 2020 at 6:29 am #

    A minor comment. Q-Q plots are not diagonal, but simply linear. Even the plot shown for QQ plot as an example here is not diagonal.

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    Ankush Jamthikar October 28, 2020 at 4:08 pm #

    I am a beginner in statistics and machine learning. You have given a very good explanation about normality tests! I just have one question. In machine learning algorithms, where we use features to perform a classification task, how do these normality tests matter? I mean even if the features are not normally distributed, they can still contribute to the classification process and perhaps may be important as well?

    • Avatar
      Jason Brownlee October 29, 2020 at 7:55 am #


      They may only matter if you are using algorithms that assume or require input or output variables to have a Gaussian distribution.

  45. Avatar
    Deepak Mongia November 20, 2020 at 6:41 am #

    Super helpful Jason ! Straightforward and ready to use.

  46. Avatar
    Borjakas January 30, 2021 at 3:47 am #

    Thanks for this helpful post!

    Just a small suggestion:
    Do you think if, in the Anderson’s example, using the “cv” variable into the comparison “if result.statistic < cv" instead of "if result.statistic < result.critical_values[i]:"

  47. Avatar
    Henry Jacob April 16, 2021 at 3:54 pm #

    Hai Jason,
    Excellent work, I appreciate it.
    Why did you mention a variable “p = 0” in Anderson Darling test?

  48. Avatar
    Steven P. June 9, 2021 at 3:39 am #

    In case anyone else runs into this issue (getting a p value of exactly 0.000 seems odd), adjust your print statement to be able to include more decimal points. (ie. a quick and easy way to do this is something like changing to ‘ print(‘Statistics=%.3f, p=%.10f’ % (stat, p)) ‘. )

  49. Avatar
    Aashish Rana November 26, 2021 at 10:59 pm #

    learnt alot from this
    thanks jason
    your blog is gem

    • Adrian Tam
      Adrian Tam November 29, 2021 at 8:37 am #

      Thanks. Glad you like it.

  50. Avatar
    Hamzah March 13, 2022 at 2:40 am #

    I am a PhD student from Yemen, and I am a follower of your posts, I really do not know how I should thank you. Thank you so much for your time and efforts and sharing your knowledge.

    • Avatar
      James Carmichael March 13, 2022 at 1:16 pm #

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