Tables of the Probability Integral of the Studentized Range

Abstract

"Denote by x 1, . . . , xn a random sample of n observations arranged in ascending order of magnitude and drawn from a normal population with standard deviation o. The range, or spread, in the sample is Xn[long dash]X1 and may be expressed in units of the standard deviation by the ratio [image] Let a 2d independent sample X1 . . . , xv+1 be drawn from the same population. [image] Tables are given which enable one to obtain the probability integral of q and the lower and upper percentage points of the studentized range q. The probability integral v P n (Q) is expressed approx. by the quadratic equation [image] Table 1 contains values of P n (Q), an(Q) and bn (Q) for value of Q from 0.00-6.00 by intervals of 1/4 and for values of n from 3 to 20. Table 2 provides gauge values for the actual range when some information about s is known. Applications pertaining to accuracy of chemical analyses, quality control limits and manufactoring certain articles are given.