ESTIMATING THE MEAN AND STANDARD DEVIATION FROM A BRUCETON STATISTICAL ANALYSIS

The maximum likelihood equations used to determine estimates of the mean and standard deviation for the Bruceton method of statistical analysis were solved numerically. A digital computer algorithm was developed to reduce the time associated with the numerical calculations. The method of successive approximations Newton-Raphson method was suitable. Test data was evaluated to determine the relative difference between values of the means and standard deviation as obtained from the numerical solution of the maximum likelihood equations and as obtained from formulae approximating these equations. Results of the analysis showed that the differences between means were slight while those between standard deviations were more pronounced. Calculated high and low probability levels, because of their dependence on the mean and standard deviation, differed widely from one method of calculation to the other. The analysis also showed that the most adverse effects occurred for the smallest samples. Because the maximum likelihood equations yield better estimates of the mean and standard deviation than can be obtained from equations approximating them -- especially for small samples -- solution of the maximum likelihood equations by the Newton-Raphson method is advocated for determining the best estimates of the population mean and standard deviation.