Correction of Data for Errors of Averages Obtained from Small Samples

Recent contributions to the theory of statistics make possible the calculation of the error of the average of a smsll sample--something that cannot be done accurately with customary error theory. Obviously, these contributions are of very general importance, because experimental and engineering sciences alike rest upon averages which in a majority of cases are determined from small samples, and because an average cannot be used to advantage without its probable error being known. The present paper attempts to show in a simple way why we cannot use customary error theory to calculate the error of the average of a small sample and to show what we should use instead. The points of interest are illustrated with actual data taken for this purpose. The paper closes with applications of the theory to four types of problems involving samples of small size for each of which numerous examples arise in practice. These types are: 1. Determination of error of average. 2. Determination of error of average difference. 3. Determination of most probable value of the root mean square deviation of the universe when only one sample of n pieces has been examined. 4. Determination of most probable value of the root mean square deviation of the universe when several samples of n pieces each have been examined. USEFUL THEORY OVERLOOKED: WHY?