A large sample will not converge to the Gaussian distribution. The distribution of a large sample converges to the distribution of the sampled population.

If you think of your experiment as one of hundreds of thousands of similar experiments that could have been done .... then, as the sample size gets large, the distribution of all of these possibilities will be Gaussian as the central limit theorem tells us.

I am not sure that a 5-point Likert scale could ever be Gaussian. I would choose nonparametric tests.

Thanks ........

If you have several 5-point items, all intended to measure the same underlying theoretical construct, summing the items will give you a scale that might be sufficiently continuous (and bell-shaped) to justify assuming normality.  Also, many parametric methods are fairly robust in the face of mild to moderate departures from normality. That said, on the specific question you asked, I agree with the answer by T. Ebert.

Thanks for valuable inputs...

One of the debates in statistics is deciding what it really means to say "fairly robust in the face of mild to moderate departures from normality." While I quoted Burke, some version of this phrase is seen over and over again in the literature.

Some argue that the whole premise is flawed. The only correct statistical methods are nonparametric. This is because it is nearly always possible to reject the assumption of normality (using a statistical test and the "magic" 0.05 value to determine significance) if you have a large sample size. Why use a methodology that assumes normality given that the assumption is always wrong? Part of the answer is parametric methods are what everyone is taught in "Introduction to Statistics" class, and comparatively few scientists get further through formal coursework. However, this is an excuse not an argument.  So in Burke's scenario with multiple Likert-type items combined, then the central limit theorem tells us that the distribution will converge to Gaussian (=Normal). Thus for each additional item I will need to increase my sample size in order to reject the null hypothesis that the data are Gaussian. So, if it takes 10^50 samples to reject the null hypothesis that the distribution is Gaussian, should we care about the difference?

Then there is a problem of quantifying what we mean by mild or moderate. At what point is the violation of the assumption so great that the test fails? What exactly constitutes failure? For fun assume that we are playing with a random number generator. I can now independently manipulate mean, standard deviation, skew, and kurtosis. So I now need to know if a skew of 1.2 is fine and 1.20001 is too much. However, the result will be a very small shift in the p-value of some test. So the answer will depend entirely on my decision that 0.05 is significant while 0.05000001 is not significant. So first we need to settle on some magic value for deciding to reject the null hypothesis. Does 0.05 include 0.050000001? Can we go to 0.054 (rounded off becomes 0.05), or 0.05499999999999 (which might be 0.05 with rounding depending on how one rounds). Some go higher and use 0.06,..., up to about 0.2 for rejection of the null hypothesis. I suggest that you don't do this, but that is your choice.

As before, I agree with all of Timothy's points.

In addition, there's a pragmatic answer to his question, "Why use a methodology that assumes normality given that the assumption is always wrong?"  Namely: parametric methods tend to have more statistical power, so it's more likely that a false null hypothesis will be rejected (at any given alpha). And they are often more flexible in application (e.g., making it more feasible to explore multivariate models.) Non-parametric methods generally depend on other kinds of assumptions that are also "always wrong" (e.g., error-free measurement).

Therefore, if your data are only "mildly" non-normal (by whatever standard), then using parametric methods can sometimes be the better choice.