Algebraic boundedness of sample statistics

Some puzzling results from a computer simulation study are explained by showing that the sample skew coefficient has population-independent bounds depending only on the sample size. Similar results are obtained for the coefficient of variation of positive data, the maximum standardized deviate, and the standardized range. The upper bounds of these statistics are (n − 2)/(n − 1)1/2, (n − 1)1/2, (n − 1)1/2, and (2n)1/2.