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1 Biometrika Trust The Mean and Coefficient of Variation of Range in Small Samples from Non- Normal Populations Author(s): D. R. Cox Source: Biometrika, Vol. 41, No. 3/4 (Dec., 1954), pp Published by: Oxford University Press on behalf of Biometrika Trust Stable URL: Accessed: :40 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust, Oxford University Press are collaborating with JSTOR to digitize, preserve and extend access to Biometrika

2 [ 469 ] THE MEAN AND) COEFFICIENT OF VARIATION OF RANGE IN SMALL SAMPLES FROM NON-NORMAL POPULATIONS BY D. R. COX Statistical Laboratory, UJniversity of Cambridge By examining special populations, a table is obtained for predicting approximately the mean and coefficient of variation of the range of randon samples of n (n < 5), drawn from a population of specified kurtosis, fl2. The effect of non-normalit.y on various statistical methods that use the range is then considered. 1. INTRODUCTION Since Tippett (1925) tabulated the mean range of random samples from a normal population, the range has been extensively used for the rapid estimation of dispersion. Moreover, much work has been done recently on quick significance tests in which the root-mean-square estimate of dispersion in the t-test, analysis of variance, etc., is replaced by an estimate derived from the range. All these uses of the range rest on the assumption of normality,, and so it is of interest to examine the distribution of range from non-normal populations. This was first done by E. S. Pearson & Adyanthaya (1929), and their work, and later work by Shone (1949), has been summarized, discussed and extended by Pearson (1950). The conclusion was that in small samples the ratio of mean range to populiation standard deviation is not much affected by the form of the population, but that the coefficient of variation of range depends fairly critically on the population. In large samnples the distribution of range is, of course, determined bv the tails of the population and so is very sensitive to nonnorlnality. The main object of the present paper is to predict the mean and coefficient of variation of range in small random samples of n (n < 5) fromn a population of given skewness and kurtosis and then to show how these results can be uised to assess the effect of non-normality on the comiimon applications of the range. Such a prediction can only be approximate because there is no functional relation between the distribution of range and population skewness and kurtosis. There are four ways of proceeding: (i) by evaltuating numerically the single and double integrals for the mean and mean square of range for a representative selection of non-normal populations; (ii) by sampling experiments; (iii) by a derivation of upper and lower limnits for the miiean andl coefficient of variation of range of populations with given properties. This was first done by Plackett (1947), who obtained an upper bound to the ratio of mean range to population standard deviation in samples of given size from an arbitrary population. An imiportant extension of Plackett's work (Hartley & David, 1954) appeared as the present paper was being completed; (iv) by finding a set of populations with widely varying,/,, f2 for which the propertie of range can be expressed exactly or approximately in terms of known quantities. While (iii) is the most satisfying approach from a mathematical point of view, mnethod (iv) seems to lead more directly to a working answer to the problem and it will be followedl here.

3 470 Mean and coefficient of variation of range 2. MIXED NORMAL POPULATIONS Consider a population formed by combining in proportions l-p, p normal distributions of zero mean and unit standard deviation and of positive mean,t and standard deviation r. We call such a population an unsymmetrical mixed population.* The drawing of an observation from such a population can be thought of as a two-stage process, first the selection with probabilities 1 -p, p of a normal distribution and then the drawing of an observation from the appropriate normal distribution. We shall assume that It, r are such th samples of the size considered no observation from the second distribution falls below an observation from the first distribution, i.e. that the constituent normal distributions are effectively non-overlapping. It is then easy to write down the mean and mean square of the range of random samples of n from the mixed population. For let dr, er be the mean and mean square of range in samples of r from the unit normal population (Pearson & Hartley, 1951) and mr = Id,, nr the mean an(d miean square of the largest value of s r from the unit normal population (Karl Pearson, 1931, Table XXI). To illustrate the method, let n = 3; then Table 1 shows the various possibilities. Thus in the second line the probability that only one observation is taken from the upper distribution is given by the binomial expression 3(1 -p)2 p, and given the allocation of observations, the range is the difference of a single observation from the upper distribution and the smaller of two observations from the unit normal distribution. The expressions for the conditional mean and mean-square range follow immediately. The overall mean range is the sum of products of the third and fourth columns of Table 1, and the overall mean-square range is the sum of products of the third and fifth columns. Table 1. Samples of three from unsymmetrical mixed populations No. of No. of observations observations Probability Conditional Conditional mean in lower in upper mean range square range distribution distribution 3 0 (I-p)3 d3 e (1- p)2 p +M2 F2+2/M2+r2+n2 1 l 2 3(1 -p) p2 /I+m2T 2 + 2/Im2T+T2m + l 0 3 p3 d3rt e,3t2 Similar formulae can be obtained for other values of n; there is no point in writing down a general formula. More refined formulae making allowance for the overlap of the two distributions can be found. They will not be given here, although they have been used to check that overlap produces no appreciable error in the conclusions set out below. We can deal in the same way with symmetrical mixed populations, i.e. mixtures in proportion 1-2p, p, p of the unit normal distribution with normal distributions of means,u, -,u and standard deviation r. Again we take Iu, r so that the constituent distributions a effectively non-overlapping. The mean and mean square of range can be obtained by a method similar to that used in Table 1, the details being a little more complicated. For both types of mixed population, the mean, standard deviation, /,8 and fl2 can be worked out in a straightforward way. The mean and coefficient of variation of range have * The use of mixed normal populations is by no means new.

4 D. R. Cox 471 been found for samples of size n = 2,3,4,5 from the following mixed populations: = 3, T = 0; It = 4, T = 0, 1; it = 5, T = 0, i, 1, taken in combination with p = 0-01, 0-05, 05. Each combination of p, It, T gives two mixed populations, one symmetric unsymmetrical. The value of fl2 goes from 1 to 8-2 and the value of/h, from 0 to The main justification for the use of mixed normal populations is that they form a sufficiently flexible set of manageable populations. However, some types of non-normality occurring in practice can be represented by a single normal population contaminated by occasional 'outliers'. If the outliers have a normal distribution, a mixed normal population results. 3. SOME MATHEMATICAL RESULTS The numerical work outlined in? 2 will be the main basis for the proposed relation between the properties of range and the form of the population, but it is very desirable to supplement this work by results for populations differing as much as possible from the mixed normal form. In random samples of n from a population with frequency function p(x) and distribution function P(x), the probability density of range, fn(w), is given by fn(w) = n(n- 1) f p(x) p(x + w) [P(x + w) -P(x)]-2 dx. (1) Consider the exponential population with p(x) = e-x (x > 0, p(x) = 0, x < 0). Then (1) gives (Maguire, Pearson & Wynn, 1952) fn(w) = (n-1) (1-e-w)n-2 e,-w (2) and the moment-generating function of range, MJ(t) Mn(t) = etwfn(w) dw = P(n) P(t+ 1)/P(n+t). (3) The cumulants of range are obtained by taking logs and expanding in powers of t, so that n-1 n-1 the mean and variance of range are E l/r and E I/r, on using well-known properties r=1 r=1 (Jeifreys & Jeffreys, 1946) of the digamma and trigamma functions. The above exponential population has standard deviation o- = 1 and,81 = 4, f2 = 9. Similar calculations can be done for populations with frequency functions x7 e-x/r!, where r is a small integer, although the results get rapidly more cumbersome as r, n increase. A few results for the population r = 1, o = V2,,81 = 2, f2 = 6, are given in Table 2, together with corresponding results for the exponential population, and, for comparison, the standard values for a normal population.* Plackett (1947) showed that the ratio of mean range to the population standard deviation o can, for given n, be made arbitrarily small by choice of the population. It is of interest to examine in more detail the conditions under which this ratio is zero. Now if the distribution function of the population is P(x), the distribution functions of the largest and smallest values in a random sample of n are respectively Pn(x) and 1- (1 -P(x))n. Since the range is the difference between the largest and smallest values, it follows that the mean * [These two population distributions are also considered, among others, by H. A. David earlier in this issue of Biometrika (see pp )-ED.].

5 472 Mean and coefficient of variation of range range, En(w), is the difference between the corresponding mean values and s the Stieltjes integral E.(w) = f xd{p?(x) + (1- P(x))- 1}. (4) This converges at -xo if and only if f xdp(x) converges, and at + oo if and only if 00~~~~~~~~~~~~~~~ 00 J'xd(I - P(x)) converges, i.e. (4) converges if and onlyiff xdp(x) converges.* Therefore the mean range is finite if and only if the population mean is finite. Table 2. Properties of range of samples of n from populations (a) e-x, (b) x e-x, (c) normal Population (a) Popuilation (b) Poputlation (c) Mean c.v. of Mean c.v. of Mean c.v. of range/aj range range/o range range/o range : * } * In this case lim x(l -P(x)) = lim xp(x) = 0, X-a>0.'00 and we miay ilitegttte (4) by parts to give E,1() = rn f {- PI'(x) (5)(1- P(x))7} dx. (5) This is the fo rn used by siderably shorter thani t We have proved, therefore, that there are many populations, in particular those with tiinite me(n anid infinite variance, for which E,v(w)/ is zero. Further, such a population should be cosidereed as havring infinite fl2. For although fl2 is strictly not defined, the original population is for practical purposes equivalent to one truncated. at + X, where X is very large, and such a truncate(d population has a very large fl2. Hence for some sets of populations E,,(wv)f tends to zero as fl2 tends to infinity, although this is not true in full generality. 4. NUM1ERICAL RESULTS We shall discuss in detail the results for n = 5, since this enables earlier work suimmarized by Pearson (1950) to be used. Fig. 1 gives mean range/(d,), where d5 is Tippett's divisor for samples of five, plotted against f32. As indicated in the legend, results for five types of population are included in the diagram: (i) theoretical; rectangular (82 = 1 8), normal (f2 = 3), x e-x (2 = 6) and exponential (1i2 = 9); (ii) symmnetrical mixed normal populations; (iii) insyi;flcnvetrical iiixed norenai populations; * Rill convergence is required and not just the3 convergencee of a principal value.

6 D. R. Cox 473 1* , bo Fig. 1. Samples of five. Mean range/(d5o-). + theoretical. 0 symmetrical mixed normal populations. x unsymmetrical mixed normal populations. * sampling experiments. O Shone's discrete populations theoretical upper bound. 2 0* C~~~~~~~~~~~~~,05 9t?? o~~~~~~~~~~~o x 0 x >04 - x0 0 >: 0-4- x?? X x 0o 3 x/ <~~~~~~~~~ Fig. 2. Samples of five. Coefficient of variation of

7 474 Mean and coefficient of variation of range (iv) results for various Pearson type populations obtained by sampling experiments (Pearson, 1950). A 95 % confidence interval for the true value is marked by a vertical line; (v) Shone's (1949) numerical results for five populations each with seven equally spaced discrete values. Fig. 2 is the corresponding diagram for the coefficient of variation of range. The dotted line in Fig. 1 is Plackett's upper bound over all possible populations of E5(w)/(d5 o-). As noted by Plackett some of the points obtained by experimental sampling lie very slightly above the theoretical maximum. The following conclusions may be drawn from Figs. 1 and 2: (i) There is no point in introducing /,% into the formulae for predicting the properti of range. This is obvious in a general way from the diagrams and can be confirmed by drawing smooth curves through the points and plotting the residuals against I1, I. (ii) Although there are certain systematic differences between the five types, the general agreement is good. A smooth curve drawn in Fig. 1 gives a maximum error of 5 %. Similar graphs have been drawn for the smaller values of n and smooth curves drawn. Table 3 is constructed from these smooth curves and predicts the mean and coefficient of variation of range for n < a, fl2 < 9- Table 3. Approximate properties of the range in random samples of n from a population with kurtosis /2 Mean range/(dr oj)=bn(2) Coefficient of variation of range=cn(p2) fi2 n= n= *375 0* *923 * *482 *302 * *969 *947 *924 *814 * * *985 *981 * *796 *464 *319 * *992 *987 * *786 * * * *964 * * * * *412 * *988 *986 *985 * * * *980 *980 *980 *982 *805 *563 *482 * *971 * *980 *822 * * *962 *963 * *840 *609 * *0 * *960 *969 *858 *629 * *932 *938 *946 *956 *893 *665 *581 * *918 *923 *931 *941 *929 *696 *602 * *910 *915 *925 *964 *723 *622 *560 9o0 I *900 * *748 *636 *573 The tabulated values of mean range/(dn o) for A?2 = 3 are unity for a normal population, and the tabulated values for coefficient of variation of range slightly above the exact values in normal theory. This is clearly justified from Figs. 1 and 2 and happens because the mean range for a normal population is, in small samples, very near the theoretical maximum, and any dispersion in the ratio (mean range)/(dno) must tend to reduce its value. A case can be made for using in practice a range divisor lower than the usual dn; this and other practical implications of Table 3 are discussed in? 6.

8 D. R. Cox NOTES ON THE DISTRIBUTIONS OF RANGE AND OF THE RATIO OF RANGES In the above work attention has been concentrated on the mean and scatter of range. The whole distribution function could be investigated in a similar way, and the results would be of some interest in indicating the effect of non-normality on the conventional 'controlchart' technique of setting limits at, say, the 5 % point of the distribution of range. This will not be discussed in detail here, but since simple results can be obtained for the exponential population, these will be given briefly. The distribution function of range in samples from the unit exponential population is, on integrating (2), (1- e-w)n-1. This is given in Table 4 for n = 4 and is compared with the corresponding function for the unit normal population (Pearson & Hartley, 1942). Table 4. Distribution function of range of samples of four Range _ Population Unit exponential Unit normal * * _ 7* Range/oC: 5 % point % point Range/mean range: 5 % point F765 1 % point The main comparison of practical importance is the one at the foot of the table, since it shows the effect of an extreme form of non-normality on the range control chart. It is discussed more fully in? 6. A further distribution connected with the exponential population and that can be investigated easily is the distribution of the ratio of ranges from independent samples. If wl, w2 are ranges of independent samples of n from the unit exponential population, their joint distribution is (n-1)2 (1 -e-w e)n-2 (1 - ew2)n-2 eww2 dw1 dw2. If we define r = w1/w2, make the change of variables to r, w2 and integrate out w2, we find for q(r), the frequency function of r, q(r) = (n-1)2 f (1- e-rv2)n-2 (1 - e-w2)n-2 w2 e(r+l) dw2. (6)

9 476 Mean and coefficient of variation of range Now it is easy to show that e-rw,,(l -e-rw2)n-2dr = ( ) {1 -(1e-rw2)n-l}. Hence we may determine the distribution function, Q(r), of r by integrating (6) under the integral sign, thus giving 1 -Q(r) = (n-1)f (1 ( 1e-t2)n-2 {1-(1 1-e-ru2)}fl e-"2 dw2 rx = (n-1)f {1(I ur),,l.-(1 u)n-2du, on putting u e-'c2. This is a rational function of r and for small n takes a simple form; thus for n=2, 1 -Q(r)=r/(r 1); n = 3, 1-Q(r) = (7r+2)/(r+L1)(r+2)(r+3). From the corresponding formula for n = 4, which is rather more complicated, the upper 10, 5 and 1 % points for the ratio of two ranges of four are found to be 3-3, 4 7, 9-6. The corresponding values for a normal population are (Link, 1950) , 5*6. There is a discussion in? APPLICATION We now consider the practical implications of the above results by discussing in turn four types of application of the range. The aim is to indicate the methods for investigating the effect of non-normality rather than to present comprehensive conclusions. (i) Point estimation of dispersion. Suppose that there are available a large number N of samples of n observations from populations of constant shape, and that it is required to estimate the standard deviation, o', from the variation within samples. Consider the estimates based on (a) the mean square within samples and on (b) the mean range within samples, using the usual divisor d,. The first estimate has bias O(N-1), which we ignore, and approximate standard error ff (7) 2JNt n n-if The second has approximate bias (B,(,f2) -1) oc and approximate st n (AB)C Bn(AG15N, where Bn(f62), Cn(fi2) are the quantities given in Table 3. The first thing to look at is the relative magnitude of the bias and the standard error of the estimate based on the range; the bias is the larger if N> C(fl 2) B(n/32) (8) The critical value of N is given for a few special ca N has to be large before the bias is comparable with the standard error. It is fair to say that in most applications the standard error will be the larger.

10 D. R. Cox 477 Table 5. Approximate value of N at which bias equals standard error i o 40 5* Suppose now that N is such th error of the estimate based on the range to that of the first estimate. This ratio is 2C,(ft2)B,(ft2) n ( n-l (9) It is given for a few values of n and f2 in Table 6. Table 6. Ratio of standard errors of estimate from range and root-mean-square estimate i o * l The general conclusion is that for the s square estimate is the better one, but that for the larger values of f82 the estimate based on the range, even though biased, is to be preferred. The reason is essentially that for populations with large f2 very extreme observations are common, and these have relatively less effect on the range than on the other estimate. The conclusion only holds when N is sufficiently small for the bias to be neglected (Table 5), although if the population f2 were known approximately, much of the bias could be removed by using a divisor Bn(f82) dn instead of dn. This leads to the point noted in? 4 that the curves fitted to the mean range pass at f2 = 3 below the theorotical point for the normal population. In fact if little were known about the population except that f2 lay within rather wide limits, it would be sensible to use a divisor 2-5 % lower than dn depending on n and the range of possible values of f2. For this would not lead to appreciable bias if the population were normal and would reduce the bias for extreme values of ft2, both large and small. If f2 were known one could, instead of producing an unbiased estimate, choose a diviso as a function of n, N and f2 in order to minimize, say E(s,-()2/.2, where s. is the estimat of o-. This would be relevant if some irrevocable decision rested on the value of the estima and if the 'loss function' were known, but in more usual cases I do not think anything wou be gained by this. The criterion of unbiasedness, while certainly arbitrary, leads to an estimate with known distributional properties and centred near the true parameteric value; the point estimate is therefore near the true value, and the confidence distribution of the unknown parameter can be found and this is all that is required. (ii) Control chart for ranges. In the usual control-chart technique for ranges, the mean range is found for a preliminary run, and say 5 and 1 % limits set at multiples of the mean

11 478 Mean and coefficient of variation of range range determined from the distribution function of the range in samples from the unit normal population. If the population is not normal two problems arise: (a) If the population 82 is approximately known, where should the limits be placed? To answer this, we approximate to the distribution of range either by a X distribution (Patnaik, 1950), or by a %2 distribution (Cox, 1949). These approximations have, of course, only been tested for the range in normal samples, but are here, nevertheless, employed as approximations to range in non-normal samples. Although we have checked this approximation in the case of the exponential below this method must, as yet, be regarded as tentative. An advantage of the x2 approximation over that using X is that it is rat and deals more easily with mean ranges. Limits based on the x2 approximation will here; the corresponding limits for the X approximation can be obtained similarly. range/(mean range) in the form x;iv; this assures agreement of the first moment, while to get agreement of the second moment we must take v = v.(32) = 2/Cn(/2). Thus limits for range/(mean range) should be set at 1/v (V.2) times the appropriate percentage point of X with vn(ft2) degrees of freedom. Example. For the exponential population C4(/J2) = (Tables 2 and 3), so that v = 4.94 and the approximate 5 and 1 % limits are at 1/4.94 x 10-98, 14-98, i.e. at 2-22 and 3 03, interpolating v-wise in the x2 table. These compare with exact values of 2-22 and 3 11 (Table 4). If a control chart is set up for the mean of m ranges of n the degrees of freedom become mfvnf(2). (b) If there is no information on which to base a correction for non-normality, what errors is non-normality likely to introduce? If ft2 is small, the upper limits are set too high and it will be difficult to detect lack of control; conversely, if it is found that few or no points fall above the limits, this suggests that the population ft2 may be low. If f2 is large, too many ranges will fall above the limits; thus for the exponential population it was shown in? 5 that more than 5 % of ranges of four lie above the normal 1 % point. This is, however an extreme case of non-normality. Conversely, if an excessive number of ranges fall above the control lines, it is possible that the ranges arise from sampling a stable but non-normal population and that the process is 'in control'. Indeed, it might require a lengthy analysis to decide by purely statistical means whether or not such a set of ranges are consistent with a state of control. (iii) Comparison of several variances. In the first two applications, only one unknown population standard deviation is involved. Now consider the problem of testing the hypothesis that two populations have the same variance, given the mean ranges of random samples from the two populations. If we continue to use the x2 approximation to the distribution of range, we refer the range ratio R = w1dn2 li2 dn). to the F tables with (ml i1, im2 i2) degrees of freedom. Here ii is the mean of mi ranges of ni from the ith population and vi are the effective degrees of freedom for a range of ni from a normal population (Cox, 1949). Suppose now that the populations are not normal. Provided that the populations have approximately the same 82, the bias factors Bn(ft2) affect, for ni < 5, numerator and d inator almost equally, even if n, $ n2. If the population ft2's were very different this

12 D. R. Cox 479 not be true, but a test of the equality of the variances of populations of very different shapes is probably not a common problem, and this case will not be considered further. Therefore the only effect of non-normality is that the effective degrees of freedom should be (mlv n(/?2), m2vn2(fi2)), where Vn(/12) = 2/Cn(f2) Table 7 gives some conclusions, based on? 5, for the case n1 = n2 = 4, ml = m2 = 1, i.e. for the ratio of the ranges of single samples of four, where the population is (a) normal, Table 7. Percentage points of range ratio test for single samples of four 10% 5% 1 % Normal population Exact 2* X2 approximation X approximation Exponential population* Exact X2 approximation * The X approximation is difficult to compute (b) exponential. The conclusions are first that while th for the normal population, the x2 approximation is adequate in all cases, particularly when it is remembered that in nearly all applications the means of several ranges will be used, and not single ranges, and that the approximate percentage points will then be closer to the exact ones. Secondly, as would be expected, the use of the normal percentage points would be considerably in error with a population as non-normal as the exponential; the error would be much greater if the means of several ranges were used. Box (1953) has shown the serious effect of non-normality on the conventional tests for comparing variances. His results are not, however, directly comparable with the present ones, because his are concerned with the comparison of single large samples. The best way of comparing the relative stability of tests based on the range and on the standard deviation, for numbers of small samples, is to compare: (a) Cn(,f2)/Cn(3), which shows how the coefficient of variation of an estimate based a mean of ranges of n, changes relative to its value for B2 = 3; (b) the quantity { ( n1 f2 I3)+ ), (10) which by (7) is the corresponding quantity for the estimate of standard deviation based on a large number of mean squares of samples of n. This is done in Table 8. The conclusion is that comparative tests based on ranges of small samples are appreciably less affected by non-normality than the corresponding tests based on mean squares. (iv) Use of the range in the t-test. In the preceding applications, the population standard deviation oc is of direct interest; we now consider a case in which o is a nuisance parameter. Suppose that x is the mean of r observations from a population of mean It and standar deviation o and that o is estimated from iv-m, n the mean of m ranges of samples of n. test the hypothesis that,u = 0, the natural analogue of the t criterion is U = r (11) Wm, n Biometrika 41 3I

13 to the t table with bias of the estimate 480 Mean and coefficient of variation of range Table 8. Relative variation with f82 of coefficient of variation of estimates of crfrom (a) range, (b) root-mean-8quare l *0 4* (a) 1* V27 (b) * (a) O V (b) *73 4 (a) (b) * (a) 0* V (b) P [ 161 1l84 Table 9. Percentage points of Size No. 10% point 5 % point 1 % point O f o f I - _ - sample samples l l % 2 % x2 n m Exact XExact X X Exact X approx. approx. approx. approx. approx. approx * * *15 2* * *32 2*33 2* *50 3* *71 2*07 2*07 2*08 2*83 2* The exact percentage points of u for a normal population have been found by numerical integration by Lord (1947), and approximations based on the X approximation to the r distribution have been given by Patnaik (1950). The percentage points based on the x2 approximation to the range distribution are obtained by referring - 1/ ) (12) mvn, of the mean of variate, to the X variate occurring in the denominator of t. Table 9 compares the exact percentage points of xdn Vr/Iwm,n with those obtained by the two methods of approximation. The e2 approximation is less accurate than the X approximation, although both are ver good; the advantage of the x2 approximation is that its use requires only a table of v", whereas the X approximation requires a double table of scale factors and degrees of freedom as functions of m, n.

14 D. R. Cox 481 If the population is not normal it is natural to replace (12) by testing ;db n(,#2)vr (13) with imv~?h) degrees of Vm, n (1 - V as t withimvn(182) degrees of freedom. of correlation between the mean and the mean range; the general effect of this will be to increase the dispersion of (13) when /2 is small and to decrease the dispersion when /62 is high. Except in very small samples a change in the degrees of freedom is not important, and the only effect of non-normality is the insertion of the factor Bn(/32); this will not be important except in extreme cases and so as a general rule this test is not appreciably affected by non-normality. The broad conclusions of? 6 are that iff2 is known, it is possible to make a rough correction for non-normality in methods that use ranges of small samples, and that on the whole these methods are less affected by non-normality than corresponding methods using variances of small samples. I am grateful to Mr D. A. East and Miss Patricia Johnson for their care with the numerical work. Box, G. E. P. (1953). Biometrika, 40, 318. Cox, D. R. (1949). J.R. Statist. Soc. B, 11, 101. REFERENCES HARTLEY, H. 0. & DAVID, H. A. (1954). Ann. Math. Statist. 25, 85. JEFFREYS, H. & JEFFREYS, B. S. (1946). Methods of Mathematical Physic8. Cambridge University Press. KENDALL, M. G. (1943). The Advanced Theory of Stati8tics, 1. London: Griffin and Co. LINK, R. F. (1950). Ann. Math. Statist. 21, 112. LORD, E. (1947). Biometrika, 34, 41. MAGUIRE, B. A., PEARSON, E. S. & WYNN, A. H. A. (1952). Bionmtrika, 39, 168. PATNAIK, P. B. (1950). Biometrika, 37, 78. PEARSON, E. S. (1950). Biometrika, 37, 88. PEARSON, E. S. & ADYANTHAYA, N. K. (1929). Biometrika, 21, 259. PEARSON, E. S. & HARTLEY, H. 0. (1942). Biometrika, 32, 301. PEARSON, E. S. & HARTLEY, H. 0. (1951). Biometrika, 38, 463. PEARSON, K. (1931). Tables for Statisticians and Biometricians, 2. Cambridge University Press. PLACKETT, R. L. (1947). Biometrika, 34, 120. SHONE, K. J. (1949). J.R. Statist. Soc. B, 11, 85. TIPPETT, L. H. C. (1925). Biometrika, 17,