SALE! Use code blackfriday for 40% off everything!
Hurry, sale ends soon! Click to see the full catalog.

# How to Code the Student’s t-Test from Scratch in Python

Last Updated on August 8, 2019

Perhaps one of the most widely used statistical hypothesis tests is the Student’s t test.

Because you may use this test yourself someday, it is important to have a deep understanding of how the test works. As a developer, this understanding is best achieved by implementing the hypothesis test yourself from scratch.

In this tutorial, you will discover how to implement the Student’s t-test statistical hypothesis test from scratch in Python.

After completing this tutorial, you will know:

• The Student’s t-test will comment on whether it is likely to observe two samples given that the samples were drawn from the same population.
• How to implement the Student’s t-test from scratch for two independent samples.
• How to implement the paired Student’s t-test from scratch for two dependent samples.

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. How to Code the Student’s t-Test from Scratch in Python
Photo by n1d, some rights reserved.

## Tutorial Overview

This tutorial is divided into three parts; they are:

1. Student’s t-Test
2. Student’s t-Test for Independent Samples
3. Student’s t-Test for Dependent Samples

### Need help with Statistics for Machine Learning?

Take my free 7-day email crash course now (with sample code).

Click to sign-up and also get a free PDF Ebook version of the course.

## Student’s t-Test

The Student’s t-Test is a statistical hypothesis test for testing whether two samples are expected to have been drawn from the same population.

It is named for the pseudonym “Student” used by William Gosset, who developed the test.

The test works by checking the means from two samples to see if they are significantly different from each other. It does this by calculating the standard error in the difference between means, which can be interpreted to see how likely the difference is, if the two samples have the same mean (the null hypothesis).

The t statistic calculated by the test can be interpreted by comparing it to critical values from the t-distribution. The critical value can be calculated using the degrees of freedom and a significance level with the percent point function (PPF).

We can interpret the statistic value in a two-tailed test, meaning that if we reject the null hypothesis, it could be because the first mean is smaller or greater than the second mean. To do this, we can calculate the absolute value of the test statistic and compare it to the positive (right tailed) critical value, as follows:

• If abs(t-statistic) <= critical value: Accept null hypothesis that the means are equal.
• If abs(t-statistic) > critical value: Reject the null hypothesis that the means are equal.

We can also retrieve the cumulative probability of observing the absolute value of the t-statistic using the cumulative distribution function (CDF) of the t-distribution in order to calculate a p-value. The p-value can then be compared to a chosen significance level (alpha) such as 0.05 to determine if the null hypothesis can be rejected:

• If p > alpha: Accept null hypothesis that the means are equal.
• If p <= alpha: Reject null hypothesis that the means are equal.

In working with the means of the samples, the test assumes that both samples were drawn from a Gaussian distribution. The test also assumes that the samples have the same variance, and the same size, although there are corrections to the test if these assumptions do not hold. For example, see Welch’s t-test.

There are two main versions of Student’s t-test:

• Independent Samples. The case where the two samples are unrelated.
• Dependent Samples. The case where the samples are related, such as repeated measures on the same population. Also called a paired test.

Both the independent and the dependent Student’s t-tests are available in Python via the ttest_ind() and ttest_rel() SciPy functions respectively.

Note: I recommend using these SciPy functions to calculate the Student’s t-test for your applications, if they are suitable. The library implementations will be faster and less prone to bugs. I would only recommend implementing the test yourself for learning purposes or in the case where you require a modified version of the test.

We will use the SciPy functions to confirm the results from our own version of the tests.

Note, for reference, all calculations presented in this tutorial are taken directly from Chapter 9 “t Tests” in “Statistics in Plain English“, Third Edition, 2010. I mention this because you may see the equations with different forms, depending on the reference text that you use.

## Student’s t-Test for Independent Samples

We’ll start with the most common form of the Student’s t-test: the case where we are comparing the means of two independent samples.

### Calculation

The calculation of the t-statistic for two independent samples is as follows:

or

Where X1 and X2 are the first and second data samples and sed is the standard error of the difference between the means.

The standard error of the difference between the means can be calculated as follows:

Where se1 and se2 are the standard errors for the first and second datasets.

The standard error of a sample can be calculated as:

Where se is the standard error of the sample, std is the sample standard deviation, and n is the number of observations in the sample.

These calculations make the following assumptions:

• The samples are drawn from a Gaussian distribution.
• The size of each sample is approximately equal.
• The samples have the same variance.

### Implementation

We can implement these equations easily using functions from the Python standard library, NumPy and SciPy.

Let’s assume that our two data samples are stored in the variables data1 and data2.

We can start off by calculating the mean for these samples as follows:

We’re halfway there.

Now we need to calculate the standard error.

We can do this manually, first by calculating the sample standard deviations:

And then the standard errors:

Alternately, we can use the sem() SciPy function to calculate the standard error directly.

We can use the standard errors of the samples to calculate the “standard error on the difference between the samples“:

We can now calculate the t statistic:

We can also calculate some other values to help interpret and present the statistic.

The number of degrees of freedom for the test is calculated as the sum of the observations in both samples, minus two.

The critical value can be calculated using the percent point function (PPF) for a given significance level, such as 0.05 (95% confidence).

This function is available for the t distribution in SciPy, as follows:

The p-value can be calculated using the cumulative distribution function on the t-distribution, again in SciPy.

Here, we assume a two-tailed distribution, where the rejection of the null hypothesis could be interpreted as the first mean is either smaller or larger than the second mean.

We can tie all of these pieces together into a simple function for calculating the t-test for two independent samples:

### Worked Example

In this section we will calculate the t-test on some synthetic data samples.

First, let’s generate two samples of 100 Gaussian random numbers with the same variance of 5 and differing means of 50 and 51 respectively. We will expect the test to reject the null hypothesis and find a significant difference between the samples:

We can calculate the t-test on these samples using the built in SciPy function ttest_ind(). This will give us a t-statistic value and a p-value to compare to, to ensure that we have implemented the test correctly.

The complete example is listed below.

Running the example, we can see a t-statistic value and p value.

We will use these as our expected values for the test on these data.

We can now apply our own implementation on the same data, using the function defined in the previous section.

The function will return a t-statistic value and a critical value. We can use the critical value to interpret the t statistic to see if the finding of the test is significant and that indeed the means are different as we expected.

The function also returns a p-value. We can interpret the p-value using an alpha, such as 0.05 to determine if the finding of the test is significant and that indeed the means are different as we expected.

We expect that both interpretations will always match.

The complete example is listed below.

Running the example first calculates the test.

The results of the test are printed, including the t-statistic, the degrees of freedom, the critical value, and the p-value.

We can see that both the t-statistic and p-value match the outputs of the SciPy function. The test appears to be implemented correctly.

The t-statistic and the p-value are then used to interpret the results of the test. We find that as we expect, there is sufficient evidence to reject the null hypothesis, finding that the sample means are likely different.

## Student’s t-Test for Dependent Samples

We can now look at the case of calculating the Student’s t-test for dependent samples.

This is the case where we collect some observations on a sample from the population, then apply some treatment, and then collect observations from the same sample.

The result is two samples of the same size where the observations in each sample are related or paired.

The t-test for dependent samples is referred to as the paired Student’s t-test.

### Calculation

The calculation of the paired Student’s t-test is similar to the case with independent samples.

The main difference is in the calculation of the denominator.

Where X1 and X2 are the first and second data samples and sed is the standard error of the difference between the means.

Here, sed is calculated as:

Where sd is the standard deviation of the difference between the dependent sample means and n is the total number of paired observations (e.g. the size of each sample).

The calculation of sd first requires the calculation of the sum of the squared differences between the samples:

It also requires the sum of the (non squared) differences between the samples:

We can then calculate sd as:

That’s it.

### Implementation

We can implement the calculation of the paired Student’s t-test directly in Python.

The first step is to calculate the means of each sample.

Next, we will require the number of pairs (n). We will use this in a few different calculations.

Next, we must calculate the sum of the squared differences between the samples, as well as the sum differences.

We can now calculate the standard deviation of the difference between means.

This is then used to calculate the standard error of the difference between the means.

Finally, we have everything we need to calculate the t statistic.

The only other key difference between this implementation and the implementation for independent samples is the calculation of the number of degrees of freedom.

As before, we can tie all of this together into a reusable function. The function will take two paired samples and a significance level (alpha) and calculate the t-statistic, number of degrees of freedom, critical value, and p-value.

The complete function is listed below.

### Worked Example

In this section, we will use the same dataset in the worked example as we did for the independent Student’s t-test.

The data samples are not paired, but we will pretend they are. We expect the test to reject the null hypothesis and find a significant difference between the samples.

As before, we can evaluate the test problem with the SciPy function for calculating a paired t-test. In this case, the ttest_rel() function.

The complete example is listed below.

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

We will use these values to validate the calculation of our own paired t-test function.

We can now test our own implementation of the paired Student’s t-test.

The complete example, including the developed function and interpretation of the results of the function, is listed below.

Running the example calculates the paired t-test on the sample problem.

The calculated t-statistic and p-value match what we expect from the SciPy library implementation. This suggests that the implementation is correct.

The interpretation of the t-test statistic with the critical value, and the p-value with the significance level both find a significant result, rejecting the null hypothesis that the means are equal.

### Extensions

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

• Apply each test to your own contrived sample problem.
• Update the independent test and add the correction for samples with different variances and sample sizes.
• Perform a code review of one of the tests implemented in the SciPy library and summarize the differences in the implementation details.

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

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

## Summary

In this tutorial, you discovered how to implement the Student’s t-test statistical hypothesis test from scratch in Python.

Specifically, you learned:

• The Student’s t-test will comment on whether it is likely to observe two samples given that the samples were drawn from the same population.
• How to implement the Student’s t-test from scratch for two independent samples.
• How to implement the paired Student’s t-test from scratch for two dependent samples.

Do you have any questions?

## Get a Handle on Statistics for Machine Learning! #### Develop a working understanding of statistics

...by writing lines of code in python

Discover how in my new Ebook:
Statistical Methods for Machine Learning

It provides self-study tutorials on topics like:
Hypothesis Tests, Correlation, Nonparametric Stats, Resampling, and much more...

### 39 Responses to How to Code the Student’s t-Test from Scratch in Python

1. Tim Levine August 3, 2018 at 6:48 am #

Jason,

Great post, I enjoy them.

A source of confusion for me for a long time was that the t-test doesn’t require the two populations to be normally distributed, but rather the *sampling distribution* be normal which happens with a larger number of samples thanks to the central limit theorem

So taking 200 samples each of size 30, the distribution of those means will approach normality.

A large sample will just reproduce the parent population. This is why the t-test is so useful, because you could have a freaky triangular parent distribution and it still works.

• Jason Brownlee August 3, 2018 at 2:20 pm #

Great note Tim, thanks for sharing.

2. Sanjan TP February 20, 2019 at 12:45 pm #

Hi Jason,

Thanks for this well-written and informative blog-post ! I had two minor queries:

1) In the above post, you have recommended to make use of abs(t-statistic). Does the sign of t-statistic matter in a one-sided t-test or will it be always positive?

2) Does the interpretation made based on “observed value of statistic > t-critical” always match the one drawn by p-value? [What surprises me is that the formula for the former cv = t.ppf(1.0 – alpha, df) needs alpha or significance level to be mentioned apriori whereas the latter p = (1 – t.cdf(abs(t_stat), df)) * 2 doesn’t]

• Jason Brownlee February 20, 2019 at 2:10 pm #

It really depends on the specific finding you’re looking to explore with the test, e.g. one or two tailed. We’re keeping things really simple in this case – e.g. assuming a right tailed test.

The interpretations must always match.

3. sanju March 9, 2019 at 5:01 pm #

I am new to hypothesis testing . So for the one tailed test how could we modify this code

• sanju March 9, 2019 at 6:59 pm #

For the t-test one tailed shouldn’t the p value = p/2/

• Jason Brownlee March 10, 2019 at 8:15 am #

You can pick one side or the other around the critical value, e.g. above or below.

Or you can use the library:
https://machinelearningmastery.com/parametric-statistical-significance-tests-in-python/

4. Imerdar2323 October 22, 2019 at 3:19 am #

Have you got a tutorial on A/B-testing. It must be similar I do assume or not? A dedicated tutorial could bring clarification.

• Jason Brownlee October 22, 2019 at 5:57 am #

Thanks for the suggestion, but A/B testing is not really machine learning.

It is more stats or experimental design.

• Imerdar2323 October 22, 2019 at 10:27 pm #

Thanks for the answer and your excellent work general. You are right with AB of course, but this is true for the stuff above, isn’t it?

• Jason Brownlee October 23, 2019 at 6:46 am #

A/B testing is a methodology, the comparison of results involves the comparison of data samples.

The broader question of comparing data samples is called statistical hypothesis testing, and I do cover it with many tutorials:
https://machinelearningmastery.com/start-here/#statistical_methods

5. Lloyd Courtenay November 9, 2019 at 8:13 pm #

Fantastic post Jason as always.
Quick question:
I’m carrying out a study on a massive data set with 1500 observations and 270 variables. Ive seen that the standard deviation of each variable is huge yet the means of each sample overlap greatly. Im doing some statistical experiments and have seen differences in other aspects of the sample that aren’t the mean.
Is it possible to substitue the mean for the median or any other measurement of variance among the samples for this test? Instead of comparing differences in mean can i substitute it for something else? Or does the test not work like that? It might be a stupid question, so I apologise! If not do you know any other test that compares the distribution of samples without using the mean?
Ive read some work using the biweight midvariance but am searching for alternatives just to experiment with.
Thanks

• Jason Brownlee November 10, 2019 at 8:20 am #

If the variables are not Gaussian, perhaps use a rank test instead (e.g. non-parametric)?
https://machinelearningmastery.com/nonparametric-statistical-significance-tests-in-python/

• Lloyd Courtenay November 11, 2019 at 7:28 pm #

Fantastic, will give it a try! Thanks for the reply!

• Jason Brownlee November 12, 2019 at 6:36 am #

You’re welcome.

• Lloyd Courtenay November 18, 2019 at 8:38 pm #

Update:
It worked! The kruskal test worked really well on my particular data set and joint with the wilcoxon test I was able to detect some interesting patterns that has really made the entire study very interesting.

On another note I also applied the Levene’s test (which I discovered surfing random blogs) and that also helped me overcome issues that were conditioning my study with regards to the mean.

Thanks a lot Jason, great advice!

• Jason Brownlee November 19, 2019 at 7:40 am #

Well done!

6. Pawel November 26, 2019 at 10:25 pm #

Thank you Jason!
Finally found someone who explained the topic so intuitive.

Regards,

• Jason Brownlee November 27, 2019 at 6:08 am #

Thanks, I’m happy that it helped!

7. Michelle February 13, 2020 at 5:23 am #

Hi, Jason

when I compare two regressors , would it be better to apply the t-test to scoring skill like r2 score, or better to MSE? will they show different results whether to reject the null hypothesis?

Thanks and Regards,
Michelle

• Jason Brownlee February 13, 2020 at 5:44 am #

MSE I believe.

8. Michelle February 14, 2020 at 12:37 am #

thank you Jason, I tried it, when I apply the t test to r2 score, collected from each folds of 10fold cross validation, the result turns different from using RMSE or MSE, do you know why?

thanks.

• Jason Brownlee February 14, 2020 at 6:37 am #

I don’t know about the expected distribution of r^2 scores and whether the test is appropriate.

Perhaps post to crossvalidated?

• Michelle February 14, 2020 at 8:55 pm #

Hi Jason,

what do you mean, post to crossvalidated?
btw, i have noticed when I apply t test to MSE to compare SVR and Decision tree, if I do 10fold cv with each MSE sample of 10 varibales, the result if to reject the null hypothesis is different compare with that, when I do 20fold cv and get each MSE smaple of 20 varibales. Do you know how to handle this kind of situation? Thank you.

• Jason Brownlee February 15, 2020 at 6:28 am #

Sorry, the Cross Validated Stack Exchange:
https://stats.stackexchange.com/

Yes, focus on the mean value and the variance. Differences across runs is to be expected:
https://machinelearningmastery.com/faq/single-faq/why-do-i-get-different-results-each-time-i-run-the-code

• Michelle February 17, 2020 at 5:11 am #

thanks Jason, what I meant is, if use 10fold CV and get 10 MSE values for SVR and decision tree(DT) respecitively, I apply these 10 values for t test get SVR and DT are significantly different. Again i do 20fold cv and get 20 MSE values for SVR and DT, again apply to t test but get the result SVR and DT are NOT significantly different. Which result shall I take then?

thank you, Jason

• Jason Brownlee February 17, 2020 at 7:53 am #

Perhaps the result based on the largest sample?
Perhaps use a version of the t-test that is modified the dof to account for the reuse of samples? See this:
https://machinelearningmastery.com/statistical-significance-tests-for-comparing-machine-learning-algorithms/

• Michelle February 17, 2020 at 8:32 pm #

Hi Jason, actually I used shapiro algorithm to check the distributions of the MSEs, which is not normal distributed, therefore I have applied wilcoxon test rather than students t test. Thank you!

• Jason Brownlee February 18, 2020 at 6:18 am #

Nice work.

9. uma February 24, 2020 at 10:53 am #

Do you share how to preform power analysis on ttest one sample test.

• Jason Brownlee February 24, 2020 at 1:26 pm #
10. Nick May 8, 2020 at 2:32 am #

Hi Jason,
could you demonstrate how to do a one tailed t test? The two tailed tests if they are = or not equal, but what if you want to see which is greater or smaller than the other?

• Jason Brownlee May 8, 2020 at 6:38 am #

Thanks for the suggestion, hopefully I can give an example in the future.

11. Jason Brownlee May 17, 2020 at 6:34 am #

Thanks.

It is a print statement that outputs floats and integers. Perhaps see this:
https://docs.python.org/3/tutorial/inputoutput.html

12. asdf June 29, 2020 at 5:08 pm #

When i encode categorical data to numerical data, would i use pair t test? How about ordinal data(good, fair, bed) ?
When i find that the p value is larger than alpha, which means i do not reject the null hypothesis? And those two variables are same?
Thanks.

13. hank cooper October 17, 2020 at 8:09 am #

i would like to see the actual code used within scipy to calculate the function call scipy.stats.t.ppf(q, df).

I.e, the code that calculates the particular t value to provide column/tow entry in the table with row (1-alpha), column degrees of freedom.

i want to see this because i want to calculate it from first principles. I have the gamma function but it foes not provide the correct valve.

• Jason Brownlee October 17, 2020 at 1:42 pm #

All of the code for scipy is available:
https://github.com/scipy/scipy

• hank cooper October 20, 2020 at 2:04 pm #

you may be wondering about my initial question, but I guess for me it’s really a case of trying to code and lend (import library) code to do this from 1st principles.

thanks for your reply and yes you are right, it’s there but for the life of me I can’t find it. According to the documentation, scipy.special.stdr returns the integral from minus INF to t according to the calculation

gamma((df+1)/2)/(sqrt(df*pi)*gamma(df/2) * intergral((1+x**2)/df)**(-df/2-1/2), x=-inf..t)
this being the CDF…

the PPF uses stdtrit (df, q) to return the inverse of the above CDF via stdtr(df.f).

Still can’t find the code so I can’t substitute the actual code (scipy repository) to replace a simple t.ppf(1 – alpha, df) call.

i know your busy, but if you find time to locate the file and line number I would love to know.

maybe its really hard to extract because of the class abstractions !