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Is there such a formula? Given a set of data for which the mean, variance, skewness and kurtosis is known, or can be measured, is there a single formula which can be used to calculate the probability density of a value assumed to come from the aforementioned data?

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For any normal (Gaussian) distribution, the skewness is $0$ since it is symmetric, and the excess kurtosis is also $0$ from the properties of a normal distribution. For other distributions, the mean, variance, skewness and kurtosis are not enough to define the distribution, though examples can be usually be found. –  Henry Apr 17 '11 at 11:46
@Henry Actually, in most $k$-parameter families of distributions with $k \le 4$, the first four moments--which can be recovered from the mean, variance, skewness, and kurtosis--are usually enough to identify the distribution. –  whuber Apr 17 '11 at 20:05
@whuber: That reads to me as being slightly circular: restricting distributions to a family where there are four or fewer parameters, knowing four statistics of the distribution often identifies the parameters. I agree. But one of my points was essentially that unrestricted there are different possibilities of distributions with substantially varying probability densities at particular points even with the same first four moments overall. –  Henry Apr 18 '11 at 2:12
I see what you mean, Henry: by "other distributions" you meant in a widely general sense, whereas my response took it to mean in the sense of distributions commonly used in statistics (which rarely have more than four parameters). I think your codicil--"though examples can usually be found"--may have suggested my narrower interpretation. –  whuber Mar 2 '12 at 14:44

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There are many such formulas. The first successful attempt at solving precisely this problem was made by Karl Pearson in 1895, eventually leading to the system of Pearson distributions. This family can be parameterized by the mean, variance, skewness, and kurtosis. It includes, as familiar special cases, Normal, Student-t, Chi-square, Inverse Gamma, and F distributions. Kendall & Stuart Vol 1 give details and examples.

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This sounds like a 'moment-matching' approach to fitting a distribution to data. It is generally regarded as not a great idea (the title of John Cook's blog post is 'a statistical dead end').

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D’Agostino’s K2 test will tell you whether a sample distribution came from a normal distribution based on the sample's skewness and kurtosis.

If you want to do a test assuming a non-normal distribution (perhaps with high skewness or kurtosis), you'll need to figure out what the distribution is. You can look at the skew normal distribution and the generalized normal distribution. If you do this, you consider other distributions too.

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