The shape of the sampling distribution of the mean depends on both the sample size and the shape of the population distribution (of raw scores).

• If the population distribution is normal (something that is impossible in the real world), the sampling distribution of the mean is guaranteed to be normal. Even with sample size n =1, it is normal, because it is an exact copy of the population distribution.
• Some authors have suggested that for ugly population distributions that are far from normal, n >= 300 will (in many cases) yield a sampling distribution of the mean that is approximately normal.

The variables many of us deal with have population distributions that are somewhere between those two extremes. If the population distribution is reasonably symmetrical with no outliers, the sampling distribution of the mean will probably look pretty good even for n < 30. (E.g., I do an example in one of my lectures where the population distribution is very right-skewed, but the means of 10,000 samples of n=16 approximate a normal distribution quite well.)

I think there is also much confusion about the so-called rule of 30. Older textbooks often included two separate sections in the t-test chapter, inference for small samples, and inference for large samples. And the dividing line between small and large was often 30 (or sometimes 50). But if you study those books carefully, you'll find that the authors did not say, "Thou shalt not do a t-test if thy sample size is less than 30." In both the small and large inference sections, the computation of the observed t-value was identical. The only difference was that for small sample inference, readers were advised to look up the critical value in a table of the t-distribution, and for large sample inference, they were directed to use the critical value of z (from the standard normal distribution). So again, the message was not to refrain from doing t-tests when n < 30; rather, it was to use a critical value from the t-distribution rather than the standard normal distribution. (Nowadays, with software that spits out p-values, that advice would be to compute p using the appropriate t-distribution, not the standard normal distribution.)

Remember that those books were written before everyone had a desktop or laptop computer with statistical software. Test values were computed "by hand", perhaps with the aid of a calculator of some sort; and observed test values had to be compared to critical values. In that context, publishers did not want to needlessly publish pages and pages of critical values, and pretty much everyone agreed that for n (or df) > 30, the critical value of z was close enough to the critical value of t. IMO, that is the real origin of the so-called rule of 30. YMMV. ;-)

Hello Shuvajit,

Yes, you may run a t-test. You'd be better off, however, using a bootstrap/resampling estimation for the difference and error term rather than hoping for robustness to violations of normality based on the sample size.

Realistically, chances are good that an N > 750 would give results that weren't wildly biased, but the fact is, that this depends entirely on just how badly departed from the normal distribution your model residuals would actually be. So, for me, the safer approach is bootstrap/resampling methods. These are available in a growing number of statistical software packages.

Good luck with your work.

Thanks David Morse for your suggestion.

Indeed the Welch t-test is known to be quite robust. When still in doubt, you could use the Mann-Whtney U test, the robust rankorder test, the median test, the Kolmogorov-Smirnov two-sample test, or the Siegel-Tukey test for scale differences.

If your sample size is that large it has to be normally distributed. You can do t-test and then compare with Mann-Whitnay U-test and you see that you get the same answer. Why not first do a principal component analysis and you get rid of your outliers and so on.

Béatrice Marianne Ewalds-Kvist – what has to be normally distributed?

And comparing a t-test with a Wilcoxon Mann-Whitney test is problematic because they do not test the same, or even equivalent hypotheses.

The important questions are what is the data generation process and what is the hypothesis. Whatever test you use will have to take both of these questions into account.

Hi Ronán,

We discussed this previously and there is a discrepancy between the tests in the two lines of thinking. In psychology in Finland, groups > 30 were considered normally distributed. Of course Kolmogorov-Smirnov test can always be used. Our textbook at that time was Sidney Siegel's Nonparametric statistics for the behavioral sciences printed 1956. I do not know why we had such an old book. Yes, the U-test tests if two at least ordinal scale independent samples are drawn from the same population, which is not exactly the same as a t-test does, but the p-values end up to be the same for both methods.

you can do t-test but it is not recommended when you have normality issues. if you are to keep the outliers, then nonparametric tests can be useful as others have described here.

best of luck

Béatrice Marianne Ewalds-Kvist why should a large sample be normally distributed? A sample will approximately have the distribution of the population and there is no reason to believe that variables from behavioral sciences will be generally normally distributed in the population. So, as already mentioned, you should consider the data generating process and switch to appropriate methods if necessary.

If U-test and t-test show the same result, why bothering thinking about which test to use at all? How do you know they are drawn from the same population? Isn't that the point of the tests in the first place, to get evidence that they are not drawn from the same population?

P.S.: can you elaborate on the point with the principal component analyses? I do not see how this would help here.

Rainer Dusing,

I am from Finland where n >30 is considered normally distributed. In Sweden they prefer nonparametric tests even for very big samples. The traditions between countries are different. I have been teaching in both countries and this fact has surprised me many times. Here is a paper on the issue.

Principal component analysis for data containing outliers and missing elements - ScienceDirect

I am pretty sure that they do not consider n>30 as normally distributed, as the form of the distribution of the sample has nothing to do with the sample size (except that small samples may not approximate the population distribution very well..).

And I am not sure if your impression how do you think scientists in countries handle data answers the questions and gives any support for one or the other method.

I think Rainer Duesing is prompting me to post (once again) what George Box famously said about normality in statistics. See the first attached image. (A PDF of the entire article can be found here: http://mkweb.bcgsc.ca/pointsofsignificance/img/Boxonmaths.pdf.)

EDIT: Ronán Michael Conroy alerted me that RG includes the period and closing bracket on that link, which causes it to fail. Try this instead:

• http://mkweb.bcgsc.ca/pointsofsignificance/img/Boxonmaths.pdf

The other thing that is important to remember, I think, is that all z- and t-tests have a common format, as shown on the 2nd attached image. As I said on that slide, it is the sampling distribution of the statistic in the numerator that must be approximately normal.

People often state that it is the errors for OLS models (including t-tests) that must be normally distributed. Normality of the errors is a sufficient condition, but not a necessary condition. The necessary condition is approximate normality of the sampling distribution of the statistic in the numerator. The shape of the sampling distribution of the statistic is affected by both sample size and the shape of the error distribution. Given a large enough sample, the sampling distribution of the statistic can be approximately normal even when the error distribution is not all that close to normal.

In case anyone thinks I've made all this up on my own, I assure you that I have not. See these slides that summarize textbook author Jeff Wooldridge's discussion of the assumptions for (cross-sectional) OLS regression, for example:

• https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/files/OLS_regression_assumptions_Wooldridge.pdf

HTH.

😁😍

KS is known to be inappropriate when the parameters are estimated from the data

Béatrice Marianne Ewalds-Kvist – I have to correct an assumption that "p-values end up the same for both [t-test and Wilcoxon Mann-Whitney] tests. Because they test different hypotheses, this is not necessarily true. And indeed, I use a teaching dataset in which one is significant and the other not because the means are indeed different but the distributions are not.

I am fascinated by the statistical habits of finns and swedes! Surely practice should be evidence based, and not based on a textbook as old as the kalevala (which I used in college too!)

Ronán and Stephen,

The Central Limit Theorem

For samples of size 30 or more, the sample mean is approximately normally distributed, with mean μX−−=μμX-=μ and standard deviation σX−−=σ/n−−√σX-=σ/n, where n is the sample size. The larger the sample size, the better the approximation. The Sampling Distribution of the Sample Mean (saylordotorg.github.io)

I am tired of this question and of the differences between the two Nordic countries, I am happy to learn from you.

The shape of the sampling distribution of the mean depends on both the sample size and the shape of the population distribution (of raw scores).

• If the population distribution is normal (something that is impossible in the real world), the sampling distribution of the mean is guaranteed to be normal. Even with sample size n =1, it is normal, because it is an exact copy of the population distribution.
• Some authors have suggested that for ugly population distributions that are far from normal, n >= 300 will (in many cases) yield a sampling distribution of the mean that is approximately normal.

The variables many of us deal with have population distributions that are somewhere between those two extremes. If the population distribution is reasonably symmetrical with no outliers, the sampling distribution of the mean will probably look pretty good even for n < 30. (E.g., I do an example in one of my lectures where the population distribution is very right-skewed, but the means of 10,000 samples of n=16 approximate a normal distribution quite well.)

I think there is also much confusion about the so-called rule of 30. Older textbooks often included two separate sections in the t-test chapter, inference for small samples, and inference for large samples. And the dividing line between small and large was often 30 (or sometimes 50). But if you study those books carefully, you'll find that the authors did not say, "Thou shalt not do a t-test if thy sample size is less than 30." In both the small and large inference sections, the computation of the observed t-value was identical. The only difference was that for small sample inference, readers were advised to look up the critical value in a table of the t-distribution, and for large sample inference, they were directed to use the critical value of z (from the standard normal distribution). So again, the message was not to refrain from doing t-tests when n < 30; rather, it was to use a critical value from the t-distribution rather than the standard normal distribution. (Nowadays, with software that spits out p-values, that advice would be to compute p using the appropriate t-distribution, not the standard normal distribution.)

Remember that those books were written before everyone had a desktop or laptop computer with statistical software. Test values were computed "by hand", perhaps with the aid of a calculator of some sort; and observed test values had to be compared to critical values. In that context, publishers did not want to needlessly publish pages and pages of critical values, and pretty much everyone agreed that for n (or df) > 30, the critical value of z was close enough to the critical value of t. IMO, that is the real origin of the so-called rule of 30. YMMV. ;-)

Bruce Weaver

Very very good:) Thank you. I checked Siegel's book. More than 50 pages comprised tables for hand calculations of results. On the other hand when you calculate by hand these methods you really understand them as opposed to when you press enter on the PC.