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Dear Professor Mean, How does the central limit theorem affect the statistical tests that I might use for my data?
The Central Limit Theorem tells you about how an average from a random sample behaves. For most situations, the average from a random sample will tend towards a normal distribution (bell-shaped curve) as the sample size increases, even if the individual data values follow a different distribution.
How quickly the average converges to a normal distribution depends on what the individual data look like. Highly skewed data and/or data with outliers will tend to converge more slowly. If your data has no outliers and is reasonably symmetric, then convergence will be very fast.
If your sample size is large, then you should be more comfortable with using parametric statistics, like a t-test or analysis of variance, because you can be reasonably confident that the averages used in the t-test or analysis of variance, are reasonably close to a normal distribution.
The first four references give nice computer simulations of the central limit theorem. The last two references gives the detailed mathematical conditions for the central limit theorem.