Approximate distributions of the sample coefficient alpha under nonnormality as well as normality are derived by using the single- and two-term Edgeworth expansions up to the term of order 1/n. The case of the standardized coefficient alpha including the weights for the components of a test is also considered. From the numerical illustration with simulation using the normal and typical nonnormal distributions with different types/degrees of nonnormality, it is shown that the variances of the sample coefficient alpha under nonnormality can be grossly different from those under normality. The corresponding biases and skewnesses are shown to be negative under various conditions. The method of developing confidence intervals of the population coefficient alpha using the Cornish-Fisher expansion with sample cumulants is presented.