Hello Jayde,

you could perhaps use Kruskal Wallis test, which does not requires that the data are normal.

Kind regards,

Dominique

Look at the histograms of the non-normally distributed variables. If these doe not look too much 'abnormal' (e.g. very skewed), run the t-test. This test is known to be quite robust.

see:

Article Robustness of the Two-Sample T-Test

https://thestatsgeek.com/2013/09/28/the-t-test-and-robustness-to-non-normality/

Use central limit theorem, which states that if you have a population with mean μ and standard deviation σ and take sufficiently large random sample ( n > 30) from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

Mohammad A. Amleh , the idea that you can ignore the distribution of the underlying population if the sample size is 30 or greater has always been untrue, and has been bad advice at least since we've had modern computers that allow us to easily fit generalized linear models, permutation tests, or conduct relatively complex nonparametric tests.

I present a quote from Rand Wilcox below.

Please don't continue to promote this antiquated advice.

Also note that while it's fair to summarize the central limit theorem as being about "sufficiently large random sample", it doesn't say anything about what sample size would be sufficiently large in a given situation. There's nothing magical about the number 30.

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Rand Wilcox, 2017, Modern Statistics for the Social and Behavioral Sciences. Section 7.3.4.

QUOTE

Three Modern Insights Regarding Methods for Comparing Means

There have been three modern insights regarding methods for comparing means, each of which has already been described. But these insights are of such fundamental importance that it is worth summarizing them here.

• Resorting to the central limit theorem in order to justify the normality assumption can be highly unsatisfactory when working with means. Under general conditions, hundreds of observations might be needed to get reasonably accurate confidence intervals and good control over the probability of a Type I error. Or in the context of Tukey's three-decision rule, hundreds of observations might be needed to be reasonably certain which group has the largest mean. When using Student's T, rather than Welch's test, concerns arise regardless of how large the sample sizes might be.

• Practical concerns about heteroscedasticity (unequal variances) have been found to much more serious than once thought. All indications are that it is generally better to use a method that allows unequal variances.

• When comparing means, power can be very low relative to other methods that might be used. Both differences in skewness and outliers can result in relatively low power. Even if no outliers are found, differences in skewness might create practical problems. Certainly there are exceptions. But all indications are that it is prudent not to assume that these concerns can be ignored.

Despite the negative features just listed, there is one positive feature of Student's T is worth stressing. If the groups being compared do not differ in any manner, meaning that they have identical distributions, so in particular the groups have equal means, equal variances, and the same amount of skewness, Student's T appears to control the probability of a Type I error reasonably well under nonnormality. That is, when Student's T rejects, it is reasonable to conclude that the groups differ in some manner, but the nature of the difference, or the main reason Student's T rejected, is unclear. Also note that from the point of view of Tukey's three-decision rule, testing and rejecting the hypothesis of identical distributions is not very interesting.

ENDQUOTE

Thanks everyone!