KANDA DATA

The t-test is one of the most popular methods for comparing the means of two sample groups. In practice, the t-test can be used for two paired sample groups or two independent sample groups. We can perform this difference test using a paired-samples t-test or an independent-samples t-test, depending on the characteristics of the data being analyzed.

To use the t-test, there are assumptions that must be met, and one of them is that the data must be normally distributed. However, in reality, test results do not always turn out as we expect. It is possible that the results show the data are not normally distributed, meaning the required assumption is not fulfilled. So, can the t-test still be used? Or should we replace it with another alternative test?

Well, in this article, Kanda Data will discuss it for you. I will explain alternative tests when the results indicate that the data are not normally distributed. We will learn how to find the best alternative when this assumption is not met, especially by using a non-parametric statistical approach.

The Assumption of Normal Distribution in the t-test

In general, the t-test belongs to the group of parametric statistics, where several assumptions must be met so that the conclusions are robust and statistically defensible. One of the most frequently discussed assumptions—and the key one we must pay close attention to—is that our research data must be normally distributed.

What does it mean for data to be normally distributed? If we refer to statistical theory, normality means the data distribution forms a bell-shaped pattern. This normal distribution is important because the t-test calculates probabilities based on that distribution.

What happens if the data are not normally distributed? When the data are not normally distributed, the mean estimate can be biased. Especially if the research data contain outliers, the standard deviation can become unstable, and the p-value from the t-test can be misleading and not valid.

If we conduct a study with a small sample size and the data are highly skewed and not normally distributed, then it is safer to use a non-parametric statistical alternative. What is the difference between non-parametric and parametric statistics? Let’s discuss it further in more depth.

Alternative to the t-test Using Non-Parametric Statistics

If the test results show that the data are not normally distributed, then one of the best solutions is to use a non-parametric test. What is a non-parametric test?

A non-parametric test is a statistical method that does not require the assumption of normal distribution, is more robust to outliers, and usually uses ranks rather than the original data values.

That is why non-parametric tests are often chosen when the data are skewed to the right/left, contain many outliers, are ordinal (ranking scale), or the sample size is small. In other words, when the data—either through statistical testing or when visualized using a histogram—clearly show that they are not normally distributed.

Using the Wilcoxon Test and the Mann–Whitney Test

As I explained in the previous paragraph, what t-test alternatives can we use when the data are not normally distributed? The two most popular alternatives to replace the t-test are the Wilcoxon Test (Wilcoxon Signed-Rank Test) as an alternative to the paired-samples t-test, and the Mann–Whitney Test (Mann–Whitney U Test) as an alternative to the independent-samples t-test.

The Wilcoxon test is used as an alternative to the Paired t-test. We can use the Wilcoxon test when the two data groups come from the same subjects, for example: comparing pre-test and post-test scores, body weight before and after an intervention, or farmers’ productivity before and after a program.

We can use the Wilcoxon test when we want to compare two paired sample groups, the data are not normal, and the sample size is small. In the Wilcoxon test, what is tested is no longer the mean, but whether the median between pairs differs significantly.

Next, the Mann–Whitney test is used as an alternative to the Independent t-test. This test is used when we want to compare two different groups, for example a control group vs a treatment group, farmers in region A vs region B, or male vs female groups.

We can use the Mann–Whitney test to compare two independent groups when the data are not normal, there are extreme outliers, or the data are ordinal. The Mann–Whitney test compares the rank distributions between the two groups. In practice, this test is often considered as testing differences in the median between groups.

Conclusion

When data are not normally distributed, the t-test is not always the best choice, especially when the sample size is small or the data are highly skewed/not normally distributed. In this situation, a non-parametric statistical approach is a safer and more robust solution.

The two main alternatives most commonly used are the Wilcoxon test for paired data (a substitute for the paired t-test) and the Mann–Whitney test for two independent groups (a substitute for the independent t-test). By choosing the appropriate test, the analysis results will be more valid, more trustworthy, and will not force assumptions that are not met.

Alright, that is the article Kanda Data can write on this occasion. Hopefully it is useful and increases the added value of knowledge for all of you. If there is anything you would like to discuss, please leave your comment. Stay tuned for the next article update.