Open AccessThe moments of the distribution for normal samples of measures of departure from normality
Author(s) -
Ronald Aylmer Fisher
Publication year - 1930
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1930.0185
Subject(s) - mathematics , normality , statistics , population , distribution (mathematics) , normality test , moment (physics) , normal distribution , sample (material) , zero (linguistics) , q function , cumulative distribution function , mathematical analysis , combinatorics , statistical hypothesis testing , physics , probability density function , demography , thermodynamics , linguistics , philosophy , classical mechanics , sociology
Ifx 1 ...x n are the values of a variate observed in a sample ofn , from any population, we may evaluate a series of statistics (K ) such that the mean value ofk p will be thep th cumulative moment function of the sampled population; the first three of these are defined by the equations;k 1 = 1/n S (x ),k 2 = 1/n -1 S (x -k 1 )2 ,k 3 =n /(n -1) (n -2) S (x -k 1 )3 ; then it has been shown (fisher, 1929) that the cumulative moment functions of the simultaneous distribution, in samples, ofk 1 ,k 2 ,k 3 ,..., may be obtained by the direct application of a very simple combination procedure. The simplest measure of departure from normality will the be γ =k 3 k 2 -3/2 , a quantity which is evidently independent of the units of measurements, and in samples from a symmetrical distribution will have a distribution symmetrical about the value zero. In testing the evidence provided by a sample, of departure from normality, the distribution of this quantity in normal samples is required.