Be careful not to mix up normality of the sample and normality of the average of the values in your sample. The Central Limit Theorem (CLT) says that the average of a large number of independent observations (themselves non necessarily normal) is normal. 

So, to summarise, you can assume your individual observations are normal if their histogram says so, but this has nothing to do with the CLT (unless by considering that theses observations themselves are made up as as sums of unobserved independent contributions).

In my opinion, central limit theorem is just a theorotical approach and it doesn't fit with the reality. I studied on triglyceride data having sample size 59.274 and it obviously was not normal. You also mentioned about normality tests. Current normality tests are sensitive to sample size. We developed a new vision for parametrization, independent of the sample size

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Be careful not to mix up normality of the sample and normality of the average of the values in your sample. The Central Limit Theorem (CLT) says that the average of a large number of independent observations (themselves non necessarily normal) is normal. 

So, to summarise, you can assume your individual observations are normal if their histogram says so, but this has nothing to do with the CLT (unless by considering that theses observations themselves are made up as as sums of unobserved independent contributions).

What the Central Limit Theme is saying is if you construct a histogram that is made of samples say 30 and greater in size, the resulting histogram will take on the shape of "Normal" distribution. When you confirm this then you could use the tests that revolve around a Normal distribution, being careful with your conclusions as the results relate to large sample sizes you used, not to the individual population.

I believe, in most cases is better to determine the fundamental distribution (make a histogram) you are working with and and select your tests accordingly.

The normal distribution is a theoretical construct. Nothing real will show a normal distribution. However, the normal distribution model can be a good approximation.

Many statistical procedures are based on the assumption that the data or a statistic calculated from the data is normal distributed. This is only an assumption that is required to make the theory work. It is not the real question is this assumption is "truely fulfilled" with your data (it won't, as I explained above!), the question is if the assumed model is a sufficiently good approximation.

If the assumption is about the statistic (that is calculated from the data), the central limit theorem comes into play. It sais that the approximation will be better with increasing sample size. In many practical cases, n>30 guarantuees a sufficiently good approxiamtion.

And as Earl already mentioned, it is usually a good idea to think about the data. Often, the data is binomial, counts, a concentration, a proportion, a rate, or a life-time. There are tailor-made models for such data, and they should be used.

1.  Just  looking at the istogram and/or plots is not that precise as the skewness-, Wilk-Shapiro- , Kolmogorov- , Pearson-  or any other  tests, whenever appropriately applied. Thus, if the result of the tests is that the sample is not of normal distribution, then we further procedure should take this into account. One of possibilities (however no always appropriate) is suggested by Guven.

2. The distribution of the variables do not change with the sample size. Thus the statement that the histogram becomes more close to normal with increas of the sample size is not correct. Indeed, imagine that all your patients get illness at two ages only - half of them say  at 65  and  the othe half at 70. Then the histogram will have only two almost equal frequences  at  65  and at 70, and this is NOT similar to any bell-shaped curve. Contrarily, the arithmetic mean of 305 data will obey approximately the normal distribution  with mean  67.5  and standard deviation 2.5/ \sqrt{305}. The latter statement is to be understood as follows: In one takes 1000 times the sample of 305 patients each time independently from other times, then the histogram of the 1000 arithmetic means will follow the indicated normal pd.

3.  Some tests like t-student for the expectation or other moments can be used independently of the distribution in the population in the same way as it were normal, provided the sample is large enough, under some mild restrictionsd (like finiteness of the second moment in the general population) - this is what has been mentioned by Pierre

4. Choosing a theoretical pd well suited for the concrete problem is often an art (in the domain of the Applied Probability). Mainly, since it requires both - the mathematical preparation AND  deep knowledge about the investigated phenomenon. E.g. the current problem of Siew requires a knowledge in gerontology (in the wide sense), if the theoretical model is to be formulated before the observations. Otherwise a search has to be performed  via trying best fitting among known distributions. Then the more distributions are known, the better chance to get the best one. In any case one can use the histogram as the theoretical distribution.

Regards

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as indicated by Pierre above, you seem to be mixing normality of a distribution and the rule of thumb (more than 30 samples) for assuming the empirical mean distribution of a sample to be normally distributed (with the asymptotic justification given by the CLT)

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assume you have a variable X distributed as an exponential law ; draw (iid) a sample of size N in this distribution

obviously, no matter the size of N, the empirical distribution of your sample will look like an exponential law (and the larger N, the better the approximation) ;  larger N will no change an exponential distribution into a normal !

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now consider the variable Y = (X1 + X2 + ... +XN)/N, that is, the empirical mean of your sample ; this is indeed a random variable, just because your sample itself is randomly drawn from the distribution ; what the CLT states is that the distribution of this random variable Y is asymptotically (that is, for N going to infinity) normal (under the condition that the distribution of X has a finite variance)

and because "asymptotically" is not very easy to use in practice, it often boils down to the rule of thumb that for "large enough N" the distribution of Y can be considered normal

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what "large enough" means indeed depends of the variance of the distribution of X : for very large variances, beware of the N > 30 rule of thumb

(EDIT : the above should read

what "large enough" means indeed depends of the higher moments of the distribution of X : in particular, for distributions with strong skewness, beware of the N > 30 rule of thumb

see the comment below ; thanks Joachim !)

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for more details, see

It’s Time To Retire the “n≥30” Rule

Tim Hesterberg

https://static.googleusercontent.com/media/research.google.com/fr//pubs/archive/34906.pdf

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"what "large enough" means indeed depends of the variance of the distribution of X : for very large variances, beware of the N > 30 rule of thumb"

The above  remark requires comments.  If the population pd. is normal, then the arithmetic mean is also normal, independently how large is the variance and the size of the sample. The convergence in the CLT  depends on moments higher than 2.

The Berry Essen Theorem presents suitable convergence rate for pd-s with finite moment of order 3, cf:

Regards, Joachim

https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem

The central limit theorem does not indicate anything about the distribution of your sample. As sample size becomes large the distribution of your sample will converge to the distribution of your population (whatever that might be). At the point where the sample equals the population, the distribution of the "sample" will equal the distribution of the population. No other outcome is possible.

The central limit theorem does apply to the distribution of all possible samples. So I run an experiment with 20 replicates per treatment, and a thousand other people run the same experiment. The distribution of all 1001 outcomes will converge to normally distributed as the number of replicates increases as per the central limit theorem.

The biggest issue is that each sample must be independent of all samples before and after. It must also have the same distribution. So if the first sample is from a normal (min=2, max=14) distribution then all values must be from this distribution. As long as all 1001 scietists are experimenting on the same population, the distribution of that population makes no difference (subject to the above restrictions).