Can you share a histogram of the values ? You might also mention the reason why you want to know if the data are normally distributed.

Hello Salvatore, this is the histogram without any transformation. the reason why i want to check for normality is to do parametric test (t-test specifically, and maybe an ANOVA later on). but in order to use these tests, you need to check that the parametric assumption of data normality is met right?

for this histogram, values of skewness and kurtosis remain within -1 and 1. but normality tests come up as significant.

Your data is strictly positive - so it cannot be normal distributed.

Further, the histogram shows that the normal distribution is an extremely bad model to descibe the distribution of the data.

Are the data from a homogeneous group? If the values are assumed to depend on other variables, it's more appropriate to look at the distribution of the residuals from a fitted model.

For a sample size of 2000, however, the distribution of the values (or residuals) does not matter much. What is relevant is the sampling distribution of the mean. You can use resampling to get an impression of the sampling distribution. But depending on the complexity of your model this might be slightly difficult.

Hello Jochen,

These are all independent observations. Would you say then that for me to conduct a t-test or ANOVA I shouldn’t really worry about the distribution of the data as this is a large sample? Why is this?

sorry if its too many questions,

thank you very much

Read about the central limit theorem.

Joel Coll Ferrer This should help you understand what Jochen is hinting at: https://www.youtube.com/watch?v=YAlJCEDH2uY

The data are certainly not normally distributed. But it's not an assumption of the t-test or anova that the pooled data are normally distributed.

(EDIT: Given that these are results of the PHQ-9 questionnaire, you can disregard the following: ... If I had to think about your data, it looks like it may be log-normally distributed, but with excess 0 values. ... Think about those 0 values. What do they mean ? Can they be pulled out and assessed separately ? Would a zero-inflated model make sense ? ... Or maybe those zeros make sense in a standard anova. )

Sal Mangiafico , maybe it's a lack of ones, not an excess of zeroes...

So I have read about the central limit theorem and actually it seems I shouldnt worry much about how the data looks like as i have a large sample, so the assumption of normality is automatically met. That’s how I understand it.

Yes, that's essentially correct. But please do not confuse the distribution of the data and the sampling distribution of the mean. Just because you have a large sample, the distribution of the data does not miracuously become normal. The distribution remains what it is, and the assumtions remains violated. But the relevant thing for the test is the sampling distribution of the mean. This is exactly normal when the distribution of the data is exactly normal (i.e. when the assumtion is meat). Otherwise it is only approximately normal. The approximation is good when the distroiution of the data is symmetric. If the distribution of the data is skewed, the sampling distributions is also skewed, but less so. It requires larger sample sizes to get a reasonably symmetric sampling distribution in such cases. When, for one of these reasons, tha sampling distribution is close to normal, then the violation of the assumption is not very relevant.

Please look up the K-S test you have the interpretation backwards. Not everything works like a t test. Here large p-values favor the normal. I have attached a short excerpt from Wikipedia. Best wishes, David Booth

Maybe as illustration for the CLT, see the small (quick and dirty) simulation in R, to visualize how the sampling distribution behaves for different sample sizes. I used an exponential distribution and a log normal with additional zeros as population (just comment/uncomment the specific lines) and tried to get a similar population as your sample (just roughly for illustration).

Have fun.

verifying the data distribution is not always an easy task. However, to verify abnormality, graphic images of the data distribution (Gaussian distribution - bell shape) must be taken into account, as well as the sample size. In this way, you can be guided to a suitable choice. Kurtosis and skewness are more flexible, while tests of normality such as KS, SW, and D'Agostino are more rigid. Regarding the sample, the larger it is, the more it interferes with normality tests. This happens because outliers deviate from sample normality in calculations, leading to a non-parametric distribution. This can be seen in the article that I attach in the link below. In summary, you have a robust sample of 2000 people and your kurtosis and skewness data indicate normality. In this case, I assume a normal distribution. For small samples, it's worth looking at distribution plots and normality tests. I hope I have contributed. Available for any queries.

Victor Guimarães have you seen the histograms of the data? You would consider this as normally distributed?

I agree and quote the Jochen Wilhelm statement: "Your data is strictly positive - so it cannot be normal distributed".

It looks like the distribution of pesticide residues in food: obviously they don't have negative values but “zero” can be defined in different ways (LOD, LOQ, etc).

Therefore, depending on your hypothesis, it might be worth thinking about frequency tests.

Roberto Molteni , it's results from the PHQ-9 questionnaire for depression ( https://www.mdcalc.com/calc/1725/phq9-patient-health-questionnaire9 ). The results are integers that range from 0 to 27. ... It looks like this kind of distribution is typical for these scores, at least for a population that isn't specifically chosen to have depressive symptoms ( e.g. Article Normative data and psychometric properties of the Patient He...

I don't mean to throw a wet blanket over all the wonderful explanations. But by Occam's Razor it seems like he got the goodness of fit tests interpretation backwards just as my PhD students used to do. And almost every post here supports my experience with my own students. Can't we ask him for the p-values of his goodness of fit tests and if they are less than0.05 then everyone else's conclusion is supported. David Booth

Thanks Sal Mangiafico

My humble answer was to point out that the data is not normal and there's no point in wasting your time asking how to test for that normality.

Joel Coll Ferrer, just focus on using the data and think about your hypothesis.

Roberto Molteni I agree. but sorry for asking this again Im just a bit confused. so even though the sample size is large enough to accept the central limit theorem where i should accept normality, i still cannot accept the parametric assumption of normality?

Joel Coll Ferrer , your data is clearly not normal distributed, and assuming that it would be a sample from a normal distribution is very clearly wrong. This is independent of the sample size.

However, you want to make inference abot the mean of that (unknown, clearly not normal) distribution. For this inference the distribution of the statistic (the sample mean) is relevant.

If -hypothetcally!- the data were taken from a normal distributuon, then we would know that the sample mean could be interpreted as a value taken from a normal distribution, too. And if this would be the case, then we could make inference based on the normal distribution model. This is what linear models (t-test, ANOVAs, liner regressions) do. If the assumption that the data are approximately normal distributed is quite reasonabble, then the inference made are also quite reasonable or approximately correct. This is again (quite) independent of the sample size.

You data is very clearly not from a normal distribution. Assuming that it is from a distribution that would be at least approximately normal is clearly not reasonable. Hence, inference based on such assumptions may be considerably wrong. To be able to make some reasonably correct inference would require to know the distribution of the sample mean. It might be possible to derive this, but this might be difficult and require the help of a mathematical statistician.

The distribution of the sample mean depends on the distribution of the data. The CLT tells us that for large sample sizes it will approximate the normal distribution reasonably well. Hence, large samples allow you to do reasonable inference about the mean using the standard linear models although we know that the distribution of the data is not normal.

The symmetries can also be found, in the uniform function, or triangular function (with suitable parameters), which clearly do not have a NORMAL probability behavior.

Rainer Duesing Thanks, I hadn't seen the graph, it was in another part of the conversation. By the distribution, it would not be normal. As I said earlier, I follow the visual criteria of graphs, kurtosis and skewness, sample size, and normality test. As described in the article. It is not an independent review. All must be analyzed together. A normality test alone cannot determine the context of the sample.

Thanks, bro!

One way is to make a histogram of the data. If the shape of the distribution resembles a bell curve, the data is likely normal.

Another way is to create a Q-Q plot. If the points in the plot roughly fall along a straight diagonal line, then the data is assumed to be normally distributed.