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I have some data points and, when I plot them on R, it looks like a normal distribution. I want to know how well my data fits the normal distribution. What kind of test should I do ?

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(I see that Eric Tressler already mentioned the Wikipedia article before I posted my answer.) Just what sort of plot you did in R is not specified. – Michael Hardy May 5 '11 at 19:40

There's actually a much broader question that you should be asking yourself here- does it matter whether your data really is normally distributed, or will the procedures that you're going to perform on the data be reasonably robust in the presence of a distribution that is only approximately normal?

There are various classical statistical tests for normality that you can use, such as the Anderon-Darling test. These tests return a p-value, which is the probability that a sample of your size would exhibit behavior as extreme or more extreme than the behavior of your sample. Small p-values (e.g. less than 0.05) tell you that it is unlikely that this data came from a normal distribution. High p-values (e.g. greater than 0.05) are consistent with normality but do not necessarily imply normality- these tests can be fooled by a small sample of data from a non-normal distribution.

The Anderson-Darling test is extremely strict- in practice, virtually any real world data set that is sufficiently large (hundreds to thousands of data points) will fail the test. In many statistical applications a failure of this sort can safely be ignored, because the procedures used are not terribly sensitive to data distributions that are not quite normal.

Graphical methods such as normal probability plots or Q-Q plots are another very good way to assess whether your data are normally distributed or close enough to normally distributed.

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Great response in the first paragraph. Robustness is more fundamental than goodness of fit. – John D. Cook May 6 '11 at 10:29

The Wikipedia article titled Normality test lists these frequentist tests: D'Agostino's K-squared test, the Jarque–Bera test, the Anderson–Darling test, the Cramér–von-Mises criterion, the Lilliefors test for normality (itself an adaptation of the Kolmogorov–Smirnov test), the Shapiro–Wilk test, the Pearson's chi-square test, and the Shapiro–Francia test.

All but one of those link to Wikipedia articles.

Some Bayesian tests are also mentioned.

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