The Conversion of Range to Standard Deviation

It frequently happens that we have a series of samples of n individuals drawn from a population and we wish to know their standard deviation. We can of course calculate the variance for each sample and take the average (since the samples are all of n individuals the simple average variance will be the same as that obtained by pooling the sums of squares and the degrees of freedom). The 

It will be seen that except when the sample size is 2 we are sacrificing some accuracy in our estimate of o, the compensating gain being the ease of computation, of course. This loss of information is not very serious, however, being less than 20% for samples of size 10 and less for smaller samples. Under the conditions in which we use this method this does not really matter, as since we have such a large sample our estimate will be accurate enough for almost all purposes. 

The reader will note that it is this factor dn which is the basis of the factor A o-oss and A o.ooi used for the formulation of the control lines for means. Thus it is clear, that for a sample of size n, control lines for 1 in 40 limits (1 in 40 about the upper limit and 1 in 40 below the lower limit, i.e, 1 in 20 outside the limits) should be at 1.96e/Vn, where o is the standard deviation, 1.96 is the value of t for infinite degrees of freedom (it is assumed that a is known exactly as we have supposedly taken a large number, preferably greater than 20, of samples) and the 5% probability level, and n is the size of sample. If for example n is 4, and w is the mean range, then the limits should be at 1.96(w/2.0S9)/v 4 = 0.476 w, where 2.059 is the value of dn for n = 4. It vrill be seen that this is the value for A o.ojs given in Table V of the Appendix. 

A problem that arises frequently is to determine whether an apparent relation between two variables is significant, and having shown it to be significant, to determine the best form of representation. 

The statistical methods available for dealing with this problem are not very satisfactory. The treatment in general is to test whether the data can be represented by the equation for the simple straight line, y = a -h bx, whether the deviations from this straight line are significant and if so can these deviations be represented by the equation y = a -h bx -1- cx or y = a -f- bx -f- cx -1- dx , etc. In practice, the computational labour is excessively heavy for any of the steps but the first, namely, seeing whether the data can be approximately represented by a straight line of the form y = a 4- bx.( ) 

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