Hi Mark

Your sample has to be both around 30 odd values and a random sample from the population of interest. If so, feel free to use parametric statistics on what is going to be a normal distribution in the limit and close enough in practice.

You can have less than 30 odd values, randomly selected, from a population already known to be normally distributed and still use parametric statistics.

You can have arbitrarily many values, not randomly selected and you can't make inferences about the population at all.

Trust this helps

Best

Marco

Hi Mark

Your sample has to be both around 30 odd values and a random sample from the population of interest. If so, feel free to use parametric statistics on what is going to be a normal distribution in the limit and close enough in practice.

You can have less than 30 odd values, randomly selected, from a population already known to be normally distributed and still use parametric statistics.

You can have arbitrarily many values, not randomly selected and you can't make inferences about the population at all.

Trust this helps

Best

Marco

Thanks Marco - that is very helpful.

I agree with Marco Huesch that for N>30+ it is often safe to use parametric statistics. Asymptotic normality comes is surprisingly quick. In particular pre-testing for normality is ill-conceived.

But recall that many models estimate expected values of the underlying distributions on the given scale. These expected values may be not what you want if

* the distribution is very skewed (try transformation to a "natural" scale on which the distribution looks uni-modal and roughly symmetric; e.g. estimate mean on the log scale) or

* looks multi-modal.

CLT holds under suitable assumptions such as indepedence and short tails or finite moments.

Under serial dependence CLT deteriorates and fails for certain long memory processes.

In any case, optimality is lost under non-normal data and, more serious, parametric methods may lack robustness.

So if you have outliers using LS or similar methods may be dangerous and robust methods should be considered. Sensitivity analysis may help in understanding the influence of outliers, for example the celebrated Cook distance in regression analysis (https://www.researchgate.net/publication/236159079_Sensitivity_analysis_of_statistical_models?ev=prf_pub)

Chapter Sensitivity Analysis of Statistical Models

There are a lot of techniques you may use to correct for serial dependence like correction for auto correlations. Also one way to handle non parametric distributions is to do bootstrapping so even if your data is non normal you may use non parametric bootstrap. OLS is a basic regression technique and I guess there are lots of estimators(GMM,MLE,IV for instance) which have ways to handle problems which LS cannot solve.By the way having a large number of observations and not just ensuring that your sample size is greater than 30 is always good for your estimates.

I'd like to echo what Allesandro said: The CLT is not such guarantee of appropriateness, and many threats to its applicability are routinely ignored, such as serial dependence.

Some skew is probably not going to be a huge deal on the sample mean, but huge outliers such as are found in data such as reaction times/response latencies make the sample mean a poor estimator of location. If there is some skew, I'd suggest bootstrapping, robust estimation, or both. (That assumes some other model is inappropriate.)

Short of knowing the substance area it's hard to say more.

Dear Mark,

The decision for the method of analysis mostly arises from your rationale of research and the appropriateness of data that has been collected. Sometimes, you need to break data to smaller populations regarding a variable and analyze each population separately even though you have a large sample and normal distribution. It is the research question and acquired results by previous studies and scientific debates about your definite rough primary analysis that determines how to proceed to get the most accurate possible results.

Take Care

Younes

in fact estimates in the rates to CLT are indeed available (I mean probability bounds)

see eg. Petrov 75, revised 91

they prove that rates in fact depend on regularity of current observation distributions

this makes that many calculator use CLT to simulate Gaussian rvs based on only 12 uniform distributed randam variables

on the other hand the Binomial distributions B(n,p) do not converge really fast to the Gaussian (once recentered and renormalized)

A nice way to appreciate adequation to Gaussian in this cas is to consider the product np... if it is larger than 5 (said to be lard), statisticians usually think of Gaussian

now if np converges to a positive constant (p=p(n) should converge to 0) then you get the classical Poisson approximation

"CLT allows us to assume that if the sample used is large then it comes from a normally distributed population."

The CLT does not say much about the population. That's why we like it! :) It says that, so long as the population meets some widely but not universally true regularity conditions such as not having "too many" outliers, the sampling distribution of the sample mean behaves as if it were normally distributed when the sample size is "sufficiently large".

It has been extended to a number of other situations and frequently applies to the sampling distributions of other estimators, such as maximum likelihood estimators, that behave similar to weighted means. And lots of procedures are based on CLT arguments that don't make direct reference to the normal distribution, such as the Pearson chi square or the F test.

However, there are lots of circumstances in fairly common models where it does NOT apply. The most commonly cited one is the sample proportion when N is fairly small and p is near 0 or 1. Another is the sampling distribution of the correlation coefficient for relatively small samples. But violations of the CLT happen in larger samples, too, such as in an HLM where a variance parameter goes to 0.

Graphical tests are useful but by no means definitive. For instance, the T with 2 DF is not covered by the central limit theorem, but the T with 3 DF is (although convergence would be glacially slow). Graphically these are very hard to tell apart.

If you are dealing with a small sample problem, the best simple advice is to use bootstrapping or some other resampling method, which avoids the Central Limit Theorem entirely. But even so, in a high outlier situation the sample mean is dubious at best.

sure, in fact all my work addresses such questions

see eg. lecture notes in statistics 85 (1994) and 190 (2007)

among many others, let me also suggest my joint paper with Massart and Rio (Annales de l'IHP, 1994)

Non central limit situations arise from many possibilities either tail behaviors (see Jakubowski et al. (2012) for relevant conditions) or Long Range Dependent situations, see Doukhan, Oppenheim and Taqqu (2003) monograph Long Rang Dependence,; theory and applications for developments

I totally agree with George Halkos.

In theory, CLT allows us to assume that if the sample used is large then it comes from a normally distributed population and we usually say that a sample size > 30 ensures normality.

In practice, my opinion is that n=30 is not enough, we are never sure that samples were selected randomly (depends on the field you work).

So, I usually prefer to check descriptive statistics to see how the mean and median behave, and compute the coeficient of asymetry. Then perform a "batery" of normally tests.

If we cannot assume normality from all the testspreformed then I would prefer to use non-parametric method.

I agree with Jay here - the wonderful property of the CLT is that it allows the sampling distribution to be normal even if the parent population distribution isn't, provided the sample size is not tiny. It certainly does not follow that if your sample is large the distribution of the population from which it came will be normal. That is reverse causality!