A: Non-normality is often less of a problem than researchers imagine. In an ordinary regression analysis, for example, the independent variables are not assumed to be normal. In fact, even the dependent variable is not assumed to be normal. Only the residual error term is assumed to be normal. This assumption can be checked by looking at the estimated residuals on a normal quantile plot.
Even if your residuals are far from normal, a large-enough sample can still assure you that your parameter estimates have a normal sampling distribution. For estimating a mean, a useful rule of thumb is that you can use a normal or t approximation if your sample size is 25 times the square of the skew (Boos & Hughes-Oliver 2000). (Larger samples may be required if there is very serious kurtosis.)
Nonparametric methods that do not assume normality are described in Hollander & Wolfe (1999), among other sources. If you need to use normal-theory methods, you can sometimes transform your variable to a better appproximation of normality. A brief summary of normalizing transformations is given by von Hippel (2003).
When a variable is skewed or has a restricted range, the most serious problem may not be the distribution per se. The most serious problem may be that the variable is likely to have nonlinear relationships with other variables. For tips on finding and fitting nonlinear relationships, click here.
Hollander, M., & Wolfe, D.A. (1999). Nonparametric statistical methods. 2nd ed. New York: Wiley.
Boos, D. D., and Hughes-Oliver, J. M. (2000), "How Large Does n Have to Be for Z and t Intervals?," The American Statistician, Vol. 54, 121-128.
von Hippel, P.T. (2003). Normalization. Encyclopedia of Social Science Research Methods (M. Lewis-Beck, A. Bryman, T.F. Liao, eds.). Thousand Oaks, CA: Sage.