Modification of Bartlett's statistic

Bartlett's statistic is widely used in test for homogeneity of variances. It is well known that Bartlett's statistic is an approximate χ² distribution. However, Bartlett's statistic departs far from its χ² distribution and results in low test precision in small sample situation. Based on mean equality of both Bartlett's statistic and its χ² distribution, a new modified formulation of Bartlett's statistic is proposed. It can not only be well distributed as χ² distribution in both large and small sample situation, but also effectively increase test precision. Comparison between modified and traditional Bartlett's statistic is also presented. Furthermore, test methods for simulation results are established, including standard deviation ratio test for two normal populations, mean ratio test for two exponent populations, and two-step simulation method of standard deviation ratio test for multiple populations.