It is most important to understand that the requirement for the majority of parametrical tests is the normality of the distribution of the mean, not of the original data! Of course, if the original data is normally distributed, so the mean distribution is certainly normal. But means of samples from an uniform distribution also presents normal distributions, since sample size is at least five or six. This property comes from the central limit theory.

As a general rule, the more skewed the data distribution, the greater the sample size needed to garantee the normality of mean distribution. If you have big samples (at least 30, but better greater than 50), you don't need to worry about normality, unless you have very skewed distribution.

To think about central limit theory and use visual inspection of histograms, as sugested above, is better than to use normality tests as Kolmogorov-Smirnov or Chi-squared. The first has no power to reject null hypothesis with small sample sizes and is focused in only one point of the distribution. The second is usefull only with great sample sizes, where you don't need to worry about it.

So, forget about normality tests and focus on the visual examination. If you have data aproximately simetrically distributed, go futher with parametric tests!

There is no data which are 100 per cent normal. If one wants to examine whether data are normally distributed, one has to plot a frequency histogram with the equal class intervals. If the histogram shows a peak in the middle and other bars are almost symmetrically distributed with small differences in height (frequencies), data for all analytical purposes should be considered normal. In the case, peak of the histogram reaches earlier and a larger number of bars are on the right side, the data are positively skewed, while peak reaches latter with larger number of bars on left and a small number on the right, the data are negatively skewed. In both the situations data need to be normalised as by taking square or cube of the data or conversely square root or cube root of the data. Log or anti-log may also be taken of data to normalise them. If data are otherwise distributed, that is, increase or decrease with class intervals or show some other pattern, data have a different distribution other than the normal.

It is most important to understand that the requirement for the majority of parametrical tests is the normality of the distribution of the mean, not of the original data! Of course, if the original data is normally distributed, so the mean distribution is certainly normal. But means of samples from an uniform distribution also presents normal distributions, since sample size is at least five or six. This property comes from the central limit theory.

As a general rule, the more skewed the data distribution, the greater the sample size needed to garantee the normality of mean distribution. If you have big samples (at least 30, but better greater than 50), you don't need to worry about normality, unless you have very skewed distribution.

To think about central limit theory and use visual inspection of histograms, as sugested above, is better than to use normality tests as Kolmogorov-Smirnov or Chi-squared. The first has no power to reject null hypothesis with small sample sizes and is focused in only one point of the distribution. The second is usefull only with great sample sizes, where you don't need to worry about it.

So, forget about normality tests and focus on the visual examination. If you have data aproximately simetrically distributed, go futher with parametric tests!

If the post is about normality test of the data, the person who posted the query might be knowing what for normality is required.

Calculate two first moment with the statistical data using the moment's estimators for the finite sample and use them for scaling to the standard distribution. Then compare.

Often, inferential statistics involves regression and this means being concerned about the normality of the error term in the model, i.e. the residuals, and not the normality of the response variable(s) per se. As Sandro says, visual examination is called for and years ago used to be done with normal probability graph paper. Nowadays, your software package should support this type of plot. (If not, you can apply the inverse of the normal cdf function to the data and you should then get data that plots like a straight line.)

You should perform a test for normality.

these tests esan source conditional on data extraction.

If the data come from a "population", the procedures include:

the Kolmogorov-Smirnov test for one sample

P-P plot

Q-Q plot

Goodness of Fit test.

If the data are for multiple populations

Kolmogorov-Smirnov

Shapiro-Wilk

P-P plot for each population

Q-Q plot for each population.

First way:

You have n data. Order your data. Put a weight of 1/n to each datum. So the first (smallest) datum gets 1/n. The second gets 2/n and so on. Now you have the empirical cumulative distribution (EDF) of the data. Is it Normal? If you form the derivative of your EDF you get something like the probability density function (PDF) - which should look like the bell shaped 'Normal' curve. But that's rarely works. Therefore, you need to retrive the relevant parameters of your n data: mean value and standard deviation. Now compare your n data with the respective Normal distribution using the Kolmogorov-Smirnov test. The book 'Numerical Recipes in ...' (... can be BASIC, Pascal, Fortran, C etc.) by Press et al. at Cambridge University Press (still) is an essential companion for all people having to deal with complex numerical data at computers. Good luck.

Second way:

Create Normal data (using Box-Muller algorithm) in the range of your available data. Compare the original data and the computer-simulated data whether they are obtained from a sufficiently similar distribution. For this procedure, various tests exist. Check with a good text book in statistics. Here also a computer is a helpful tool.

There is no dispute that there are several statistical methods to test the normality of a distribution. However, in our good old days 9Now software for everything are available) we used to draw histogram or plot data point according to their respective ranks. That is on the X-axis ranks are marked at equal interval and on the Y-axis values of the variable according to scale are plotted. The theory behind this practice is that if the variable is 100% (an impossibility) normally distributed, all the points will form an exact straight line, otherwise very small deviations from the straight line passing through the points will be observed. This method has the added advantage to find out outliers in the dependent variable which may be excluded from the regression analysis for special investigation. Obviously, if a straight line is drawn from these points, at the beginning or at the the end side some data will not conform to the straight line. If deviation is considered too large, these data may be considered as outliers. If the variable is not normally distributed points will not conform to a straight line and distribution of values of the variable may be deduced from the shape of curve described by data points.

Simply you don't know. Normality is a convention that requires number of data N>30 in order to work. It depends on the problem you have to solve. What exactly do you want to solve?

There are several hypothesis tests for normality, like Shappiro-Wilk or Kruskal-Wallis. Observation of histograms and Q-Q plots are also recommended. As for the N>30 rule, it does not make your population normal, but by the central limit theorem, the MEANS of the samples from the population do follow a normal distribution when the sample size is large enough (N>30 use to be an accepted threshold), no matter what distribution follows the original population.

do you think about?

now you must face the challenge of avoiding the normality assumption analysis or analyze towards compliance and therefore accept the answer.

For many it is easy, but new research incorporate SKEWNORMAL Distribution.

leading to a deeper analysis of the behavior of the data.

e.g.

http://www.sciencedirect.com/science/article/pii/S0167947310000940

http://www.scielo.cl/scielo.php?pid=S0716-09172011000300008&script=sci_arttext&tlng=e

what works for approximation, deserves to be more careful now.

Thank you everyone for your valuable suggestions:)

Thank you for all for good information