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An Analysis of Variance Test for Normality (Complete Samples) Author(s): S. S. Shapiro and M. B. Wilk Source: Biometrika , Dec., 1965, Vol. 52, No. 3/4 (Dec., 1965), pp. 591-611 Published by: Oxford University Press on behalf of Biometrika Trust Stable URL: https://www.jstor.org/stable/2333709
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Biometi-ika (1965), 52, 3 and 4, p. 591 591 With 5 text-figures Printed in Great Britain
An analysis of variance test for normality (complete samples)t
BY S. S. SHAPIRO AND M. B. WILK General Electric Co. and Bell Telephone Laboratories, Inc.
1. INTRODUCTION The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6.
2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work
This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis?
In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order models can be fitted and an F-type ratio used to evaluate the adequacy of the linear fit.
t Part of this research was supported by the Office of Naval Research while both authors were at Rutgers University.
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592 S. S. SHAPIRO AND M. B. WILK
This approach was investigated in preliminary work. While some promising results were obtained, the procedure is subject to the serious shortcoming that the selection of the higher-order model is, practically speaking, arbitrary. However, research is continuing along these lines.
Another analysis of variance viewpoint which has been investigated by the present authors is to compare the squared slope of the probability plot regression line, which under the normality hypothesis is an estimate of the population variance multiplied by a constant, with the residual mean square about the regression line, which is another estimate of the variance. This procedure can be used with incomplete samples and has been described elsewhere (Shapiro & Wilk, 1965b).
As an alternative to the above, for complete samples, the squared slope may be compared with the usual symmetric sample sum of squares about the mean which is independent of the ordering and easily computable. It is this last statistic that is discussed in the remainder of this paper.
2-2. Derivation of the W statistic
Let M' = (in1, i2, ...,Mn) denote the vector of expected values of standard normal order statistics, and let V = (vi>) be the corresponding n x n covariance matrix. That is, if xi< x2 < ... xn denotes an ordered random sample of size n from a normal distribution with
mean 0 and variance 1, then
E(x)i = mni (i = 1, 2, * *, n), and cov(xi,xj)=vij (i,j= 1,2,.. ., n).
Let y' = (Yi' , Yn) denote a vector of ordered ra
to derive a test for the hypothesis that this is a sample from a normal distribution with unknown mean 4a and unknown variance o2.
Clearly, if the {yi} are a normal sample then yi may be expressed as
Yi = It+ rxi (i-=1, 2, ..., n).
It follows from the generalized least-squares theorem (Aitken, 1935; Lloyd, 1952) that the
best linear unbiased estimates of It and o are those quantities that minimize the quadrat form (y-jtl--om)' V-1 (y- II-o-m), where 1' = (1, 1, ..., 1). These estimates are, res
tively, _ Vn' -F (Ml' - lm') V-1y
1 'V-11 lm' V-lm (1 'V-1M)2
a'V-1(lm'-Ml') V-ly
and 1 'V lm'V-1m (1'V-'m)2

For symmetric distributions, 1' V-lm

/t2=-n Yi=
n
Let S2 -E (y_9)2 1
denote the usual symmetric unbiased estimate of (n - 1) -2. The W test statistic for normality is defined by
R4--2 2 (a'y)2 (n 2 2n 02SC=2S~2~S=2 =S2a i(=iyJh i~2=1

y,

and

C

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An analysis of variance test for normality 593
where R2 = M' V-1m,
2 = M V-1 V-'M,
a = (a1,... an)= (M'V--V-lM)2
and b= - /
Thus, b is, up to the normalizing constant C, the best linear unbiased estimate of the slope of a linear regression of the ordered observations, yi, on the expected values, mi, of the standard normal order statistics. The constant C is so defined that the linear coefficients are normalized.
It may be noted that if one is indeed sampling from a normal population then the numerator, b2, and denominator, S2, of W are both, up to a constant, estimating the same quantity, namely o.2. For non-normal populations, these quantities would not in general be estimating the same thing. Heuristic considerations augmented by some fairly extensive empirical sampling results (Shapiro & Wilk, 1964a) using populations with a wide range of 4fl1 and
A2 values, suggest that the mean values of W for non-null distributions tends to shift
to the left of that for the null case. Further it appears that the variance of the null distribution of W tends to be smaller than that of the non-null distribution. It is likely that this is due to the positive correlation between the numerator and denominator for a normal population being greater than that for non-normal populations.
Note that the coefficients {ai} are just the normalized 'best linear unbiased' coefficients
tabulated in Sarhan & Greenberg (1956).
2 3. Some analytical properties of W LEMMA 1. W is scale and origin invariant Proof. This follows from the fact that for normal (more generally symmetric) distribu-
tions, -ai = ani+l
COROLLARY 1. W has a distribution which depends only on the sanmple size n, for samples from a normal distribution.
COROLLARY 2. W is statistically independent of S2 and of y, for samples from a normal distribution.
Proof. This follows from the fact that - and S2 are sufficient for It and o.2 (Hogg & Craig,
1956). COROLLARY 3. EWr = Eb2r/ES2r, for any r. LEMMA 2. The maximum value of W is 1. Proof. Assume yO0 since W is origin invariant by Lemma 1. Hence
W = [E aiy.]2/E Y2.
i i
Since ( ai y)2 < >a4 E yi - yi i i i i
because , a4 = a'a - 1, by definition, then W is bounded by 1. This maximum is in fact
achieved when yi = yai, for arbitrary y. LEMMA 3. The minimum value of W is na2/(n-1).
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594 S. S. SHAPIRO AND M. B. WILK Proof.t (Due to C. L. Mallows.) Since W is scale and origin invariant, it suffices to con-
n
sider the maximization of y2 subject to the constraints Eyi = 0, 2aiyi = 1. Since this *=1
is a convex region and Zy4 is a convex function, the maximum of the latter must o one of the (n - 1) vertices of the region. These are
(n-1 -1 1 nal nal na1
( n-2 (n-2) -2 -2 i n(a1+ a2)n(a1+a2)' n(a1+a2)'* n(al +a2)
(n(al+i.+an-1)'n(al+..'.+an- -'""n(al+...+an-)
It can now be checked numerically, for the values of the specific coefficients {ai}, that the
n
maximum of E y2 occurs at the first of these points and the corresponding minimum value
i=1
of W is as given in the Lemma. LEMMA 4. The half and first moments of W are given by
B 2rP{l(n - 1)} EWQ = RcrF(-n) <2
and EW= R2(R2 +
where R2 = m' V-1m, and c2 = m' V7- V-1m. Proof. Using Corollary 3 of Lemma 1, EWi = Eb/ES and EW = Eb2/ES2.
Now, ES = o2r ()/(n2 ) and ES2 = (n -) C.
From the general least squares theorem (see e.g. Kendall & Stuart, vol. ii (1961)), B2 B2
Eb = - Ea= C C B4 B4
and Eb2- = C -2 C2 {var (a) + (E)2}
= ,,2R2 (R2 + 1)/C2,
since var (a) - 2/M' V-1M = o-2/B2, and hence the results of t Values of these moments are shown in Fig. 1 for sample size LEMMA 5. A joint distribution involving W is defined by
h .** 6n-2) = KW4(1 - W)l(n-) cos2-4 . COS Onco over a region T on which the 6 's and W are not independent, and w
t Lemma 3 was conjectured intuitively and verified by certain numerical studies. Subsequently the above proof was given by C. L. Mallows.
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An analysis of variance test for normality 595

Proof. Consider an orthogonal transformation B such that y = Bu, where

n

n

l= yi/Vn and u2 =J aiyi = b

i=

i-l

The

ordered

n! ( 2exp (_1SCi))l (a < yl < ..< yn < 00) After integrating out, u1, the joint density for u2, ...,n iS

K*exP(212 exp ui

over the appropriate region T*. Changing to polar co-ordinates such that

M2= p sin 01, etc, and then integrating over p, yields the joint density of 01, Oan-2 as

over some region T**.

K** coSn-3 6l cos n-4 02... cos On-3

From these various transformations

W=-b2- Up2 _sins2i1n2=0s1ini22l 6o
S2 np 2
i=l1

from which the lemma follows. The 6i's an
in the sample space T.

098

E(WI)
096

0O*94 - _ _

0-92__ _
0^90 3 5 7 9 11 13 15 17 19 21 Sample size, n
Fig. 1. Moments of W, E(Wr), n = 3(1)20, r = , 1. COROLLARY 4. For n = 3, the density of W is
-(T-W)-I W-1 3 <
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596 S. S. SHAPIRO AND M. B. WILK
Note that for n = 3, the W statistic is equivalent (up to a constant multiplier) to the statistic (range/standard deviation) advanced by David, Hartley & Pearson (1954) and the result of the corollary is essentially given by Pearson & Stephens (1964).
It has not been possible, for general n, to integrate out of the 6i's of Lemma 5 to obtai
an explicit form for the distribution of W. However, explicit results have also been given for n = 4, Shapiro (1964).
2*4. Approximations associated with the W test
The {ai} used in the W statistic are defined by
n
a- Em:,viiIC (j = 1,2, ...,n),
j=1
where mj, v j and C have been defined in ? 2 2. To determ to know both the vector of means m and the covariance matrix V. However, to date, the elements of V are known only up to samples of size 20 (Sarhan & Greenberg, 1956). Various approximations are presented in the remainder of this section to enable the use of W for samples larger than 20.
By definition, (M'V-1 MVV-
a= 7-1 T7_1,b)-! C?
is such that a'a = 1. Let a* = m'V-1, then C2 = a*'a*. Suggested approximations are
ai 1 = 2m (i = 2, 3, ..., n-
( (ln)
and t = = I (l,{ n+1)} (n 20),
and al aln.- - ~P{l(n+l)}
A comparison of a* (the exact values) and a* for various values of i * 1 and n = 5, 10,
15, 20 is given in Table 1. (Note ai =-an-i+l.) It will be seen that the approximation is
generally in error by less than 1 %, particularly as n increases. This encourages one to trust the use of this approximation for n > 20. Necessary values of the mi for this approximation are available in Harter (1961).
Table 1. Comparison of Ia* l and I a* I = 1 2mni , for selected values of i(t+ 1) and n
nX 2 3 4 5 8 10
5 Exact 1.014 0.0 - Approx. 0.990 0 0
10 Exact 2 035 1.324 0 757 0 247 Approx. 2 003 1 312 0752 0 245 - -
1.5 Exact 2 530 1 909 1 437 1.036 0.0 Approx. 2 496 1 895 1 430 1 031 0.0
20 Exact 2 849 2 277 1 850 1 496 0 631 0-124 Approx. 2 815 2 262 1*842 1 491 0630 0 124
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An analys8is of variance test for normality 597
A comparison of a2 anda 2 for n = 6(1) 20 is given in Table 2. While the errors of this approximation are quite small for n < 20, the approximation and true values appear to cross over at n = 19. Further comparisons with other approximations, discussed below, suggested the changed formulation of al for n > 20 given above.
Table 2. Comparison of a2 and a2
n Exact Approximate n Exact Approximate 6 0*414 0*426 13 0*287 0-283 7 *388 *392 14 *276 *272 8 *366 *365 15 *265 *261 9 *347 *343 16 *256 *254 10 *329 *324 17 *247 *245 11 *314 *308 18 *239 *237 12 *300 *295 19 *231 *231 20 *224 *226
C2 R2

70

35

-3--

60 3C0 50 25 40 20
30 1 5~~~~~~~

C2=-72 +4.08n
4R2 -241+1 98n

20 1 0

1 0 5

0 0

0 2 4 6 8 10 12 14 16 18 20

Sample size, n

Fig. 2. Plot of C2 = m'V-1 V-lm and 12 = m'V-1m as functions of the sample size n.

What is required for the W test are the normalized coefficients {ai}. Thus &2 is direct usable but the * (i = 2, ..., n-1), must be normalized by division by C = (in'V-VmV-m).
A plot of the values of C2 and of R2 = m' V-m as a function of n is given in Fig. 2. The linearity of these may be summarized by the following least-squares equations:
C2 = - 2.722 + 4*083n,

which gave a regression mean square of 7331-6 and a residual mean square of 0-0186, and R2 = -2-411+1*981n,

with a regression mean square of 1725-7 and a residual mean square of 0-0016.

38

Biom.

52

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598 S. S. SHAPIRO AND M. B. WILK
These results encourage the use of the extrapolated equations to estimate c2 and R2 for higher values of n.
A comparison can now be made between values of c2 from the extrapolation equation
n
and from a a2, using
a2 n-I
a*2= 2 a, a2.
For the case n = 30, these give values of 119-77 and 120-47, respectively. This concordance of the independent approximations increases faith in both.
Plackett (1958) has suggested approximations for the elements of the vector a and R2. While his approximations are valid for a wide range of distributions and can be used with censored samples, they are more complex, for the normal case, than those suggested above. For the normal case his approximations are
* =nm [F(m.j.)-F(m._1)] (j=2,3,...,n-1),
= {M(f(Mj)2 + mjf(mj) -f(mj) + m,[F(m,+?) - F(m1)]} (j = 1),
where F(m1) = cumulative distribution evaluated at mj,
f(mj) = density function evaluated at m,
and a* = -4* Plackett's approximation to R2 is
R2= 2 1 +m3f(ml) +mlf(ml) - 2F(m) + I}
Plackett's da approximations and the present a' approximations are compared with the exact values, for sample size 20, in Table 3. In addition a consistency comparison of the two approximations is given for sample size 30. Plackett's result for a1 (n = 20) was the only case where his approximation was closer to the true value than the simpler approximations suggested above. The differences in the two approximations for a, were negligible, being less than 0-5 %. Both methods give good approximations, being off no more than three units in the second decimal place. The comparison of the two methods for n = 30 shows good agreement, most of the differences being in the third decimal place. The largest discrepancy occurred for i = 2; the estimates differed by six units in the second decimal place, an error of less than 2 %.
The two methods of approximating R2 were compared for n = 20. Plackett's method gave a value of 36-09, the method suggested above gave a value of 37-21 and the true
value was 37-26.
The good practical agreement of these two approximations encourages the belief that there is little risk in reasonable extrapolations for n > 20. The values of constants, for n > 20, given in ? 3 below, were estimated from the simple approximations and extrapolations described above.
As a further internal check the values of an, an-1 and an-4 were plotted as a function n for n = 3(1) 50. The plots are shown in Fig. 3 which is seen to be quite smooth for each of the three curves at the value n = 20. Since values for n < 20 are 'exact' the smooth transition lends credence to the approximations for n > 20.
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An analysis of variance test for normality 599

Table 3. Comparison of approximate values of a* m m'V-1

n i Present approx. Exact Plackett

20 1 - 4-223 - 4-2013 -4-215 2 -2x815 - 2x8494 - 2x764 3 - 2x262 - 2x2765 -2x237 4 - 1x842 -1x8502 -1-820 5 - 1*491 -1*4960 -1-476 6 -1*181 -1*1841 -1P169 7 -0*897 - 08990 - 0-887 8 -0*630 - 0*6314 - 0-622 9 -0374 -0-3784 - 0 370
10 -0*124 - 0*1243 - 04123

30

1 -4*655 - -4671 2 -3*231 - 3-170 3 -2*730 - -2768 4 2-357 - 2-369 5 -2*052 - -2013 6 -1*789 - -1760 7 -1553 - 1-528 8 -1-338 -- -1*334 9 - 1*137 - -1132
10 -0*947 - -0-941 11 - 0-765 - -0759 12 -0-589 - -0582 13 - 0*418 - 0*413 14 - 0*249 - -0*249 15 - 0*083 - -0082

0.5 0~3 ______

0 5 10 15 20 25 30 35 40 45 50 Sample size, n
Fig. 3. ai plotted as a function of sample size, n = 2(1) 50, for iz_ = ,n-1,n-4(n> 8).
38-
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600 S. S. SHAPIRO AND M. B. WILK
1.0
0*8 0-7 0*6 p 05
0 70 0 75 0-80 0-85 0.90 0 95 1 00
W
Fig. 4. Empirical O.D.F. of W for n = 5, 10, 15, 20, 35, 50. 090
081-00 _____n 1 1 21____ w~~~~~
1080 0.95
995 %f 95X9 0*70~5
0685
0 5 10 15 20 25 30 35 40 45 50 Sample size, n
Fig. 5. Selected empirical percentage points of W, n = 3(1)50.
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An analysis of variance test for normality 601
Table 4. Some theoretical moments (jti) and Monte Carlo m
n 6~i /t l #3 2 /3//2 4//2
3 0 9549 0 9547 0-9135 0-9130 0-005698 - 0 5930 2-3748
4 *9486 *9489 *9012 *9019 *005166 - *8944 3*7231 5 *9494 *9491 *9026 *9021 *004491 - *8176 7-8126
6 0-9521 0-9525 0-9072 0-9082 0 003390 - 1-1790 5-4295 7 *9547 *9545 *9123 *9120 *002995 - 1-3229 6-4104
8 *9574 *9575 *9174 *9175 *002470 - 1-3841 7-1092 9 *9600 *9596 *9221 *9215 *002293 - 1-5987 8-4482 10 *9622 *9620 *9264 *9260 *001972 1-6655 9-2812
11 0-9643 0-9639 0 9303 0 9295 0-001717 -1o7494 11-0547 12 *9661 *9661 *9337 *9338 *001483 -1o7744 11-9185 13 *9678 *9678 -9369 *9369 *001316 1-7581 13-0769 14 *9692 *9693 *9398 *9399 *001168 -1o9025 14-0568 15 *9706 -9705 *9424 *9422 *001023 - 1-8876 16-7383
16 0-9718 0-9717 0-9447 0.9445 0*000964 - 1-7968 17-6669 17 *9730 *9730 -9470 -9470 *000823 - 1-9468 22-1972 18 *9741 *9741 *9491 *9492 *000810 - 2-1391 24-7776 19 *9750 *9750 -9508 -9509 *000711 - 2-1305 29-7333 20 *9757 *9760 *9523 *9527 *000651 - 2-2761 32-5906
21 0 9771 - 0 9549 0 000594 -2-2827 36-0382 22 *9776 *9558 *000568 - 2-3984 44-5617 23 - *9782 -9570 *000504 -2-1862 40 7507 24 -9787 -- .9579 *000504 - 2-3517 43-4926 25 *9789 - 9584 *000458 - 2-3448 46-3318
26 - 0-9796 - 0-9598 0 000421 - 2-4978 58-9446 27 - *9801 *9607 *000404 - 2-5903 60-5200 28 - *9805 -9615 *000382 -2-6964 64 1702 29 - *9810 - *9624 *000369 - 2-6090 68-9591 30 - *9811 *9626 *000344 - 2-7288 71-7714
31 - 0-9816 0-9636 0*000336 -2-7997 77.4744 32 *9819 -9642 *000326 - 2-6900 76-8384
33 *9823 *9650 *000308 - 3-0181 93-2496 34 *9825 - *9654 *000293 - 3.0166 100-4419 35 *9827 - 9658 *000268 - 2-8574 108-5077
36 - 0-9829 - 0-9662 0 000264 - 2-7965 91-7985 37 *9833 - *9670 *000253 -3-1566 120 0005 38 *9837 *9677 *000235 - 3-0679 118-2513
39 *9837 -9678 *000239 - 3-3283 134 3110
40 *9839 *9682 *000229 -3-1719 136 4787
41 0-9840 - 0 9684 0-000227 - 3 0740 129 9604 42 *9844 *9691 *000212 -3-2885 136 3814 43 - *9846 - *9694 *000196 - 3-2646 151-7350 44 - *9846 - 9695 *000193 - 3 0803 140 2724 45 *9849 - *9701 *000192 - 3 1645 137-2297
46 - 0-9850 0 9703 0-000184 -3-3742 176-0635
47 - *9854 - *9710 *000170 - 3.3353 179 2792
48 *9853 *9708 *000179 -3*2972 173-6601
49 *9855 *9712 *000165 -3 2810 183-9433
50 *9855 - *9714 *000154 - 3-3240 212-4279
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602 S. S. SHAPIRO AND M. B. WILK
2-5. Approximation to the distribution of W
The complexity in the domain of the joint distribution of W and the angles {Oi} in Lemma 5
necessitates consideration of an approximation to the null distribution of W. Since only the first and second moments of normal order statistics are, practically, available, it follows that only the one-half and first moments of W are known. Hence a technique such as the Cornish-Fisher expansion cannot be used.
In the circumstance it seemed both appropriate and efficient to employ empirical sampling to obtain an approximation for the null distribution.
Accordingly, normal random samples were obtained from the Rand Tables (Rand Corp. (1955)). Repeated values of W were computed for n = 3(1) 50 and the empirical percentage points determined for each value of n. The number of samples, m, employed was as follows:
for n = 3(1)20, M= 5000,
n = 21(1) 50, m= [1nO0001
Fig. 4 gives the empirical C.D.F.'S for values of n = 5, 10, 15, 20, 35, 50. Fig. 5 gives a plot of the 1, 5, 10, 50, 90, 95, and 99 empirical percentage points of W for n = 3(1)50.
A check on the adequacy of the sampling study is given by comparing the empirical one-half and the first moments of the sample with the corresponding theoretical moments of W for n = 3(1) 20. This comparison is given in Table 4, which provides additional assurance of the adequacy of the sampling study. Also in Table 4 are given the sample variance and the standardized third and fourth moments for n = 3(1) 50.
After some preliminary investigation, the SB system of curves suggested by Johnson (1949) was selected as a basis for smoothing the empirical null W distribution. Details of this procedure and its results are given in Shapiro & Wilk (1965 a). The tables of percentage points of W given in ? 3 are based on these smoothed sampling results.

3. SUMMARY OF OPERATIONAL INFORMATION

The objective of this section is to bring together all the tables and descriptions needed to execute the W test for normality. This section may be employed independently of notational or other information from other sections.
The object of the W test is to provide an index or test statistic to evaluate the supposed normality of a complete sample. The statistic has been shown to be an effective measure of normality even for small samples (n < 20) against a wide spectrum of non-normal alternatives (see ?5 below and Shapiro & Wilk (1964a)).
The W statistic is scale and origin invariant and hence supplies a test of the composite null hypothesis of normality.

To compute the value of W, given a complete random sample of size n, X1, X2, ...,X
one proceeds as follows:

(i) Order the observations to obtain an ordered sample yl , Y2 < ... < Yn.

(ii) Compute

n n
S2 = ,(Xy~)2 EX(Xi-X)2.

1

1

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An analysis of variance test for normality 603
(iii) (a) If n is even, n = 2k, compute
b = E an-i+1(yn-+l-Yi) i= 1
where the values of an_i 1 are given in Table 5. (b) If n is odd, n = 2k + 1, the computation is just as in (iii) (a), since ak+1 = 0 when
n = 2k + 1. Thus one finds
b = an(yn-Y1)? +* + ak+2(yk+2- Y),
where the value of Yk+1' the sample median, does not enter the computation of b. (iv) Compute W = b2/S2. (v) 1, 2, 5, 10, 50, 90, 95, 98 and 99 % points of the distribution of W are given in Table
Small values of W are significant, i.e. indicate non-normality. (vi) A more precise significance level may be associated with an observed W value by
using the approximation detailed in Shapiro & Wilk (1965a).
Table 5. Coefficients {a-i+1} for the W test for normality,
forn = 2(1)50.
n 2 3 4 5 6 7 8 9 10
i\ 1 07071 0-7071 06872 06646 06431 06233 0.6052 0.5888 05739 2 - *0000 *1677 *2413 *2806 *3031 *3164 *3244 *3291 3 - *0000 *0875 *1401 *1743 *1976 *2141
4 - - 0000 *0561 *0947 *1224 5 - - - 0000 *0399
n 11 12 13 14 15 16 17 18 19 20
i\ 1 0-5601 0*5475 0*5359 0*5251 0*5150 0-5056 0.4968 0.4886 0.4808 0.4734 2 *3315 *3325 *3325 *3318 *3306 *3290 *3273 *3253 *3232 *3211 3 *2260 *2347 *2412 *2460 *2495 *2521 *2540 *2553 *2561 *2565 4 *1429 *1586 *1707 *1802 *1878 *1939 *1988 *2027 *2059 *2085 5 *0695 *0922 *1099 *1240 *1353 *1447 *1524 *1587 *1641 *1686
6 0.0000 0*0303 0.0539 0.0727 0-0880 0*1005 0-1109 0.1197 0*1271 0*1334 7 - *0000 *0240 *0433 *0593 *0725 *0837 *0932 *1013
8 - - - *0000 *0196 *0359 *0496 *0612 *0711
9 - - - - 0000 *0163 *0303 *0422
10 -- - - - - 0000 *0140
n
i\\ 21 22 23 24 25 26 27 28 29 30
1 04643 0X4590 0X4542 0X4493 0X4450 0X4407 0X4366 0X4328 0X4291 0X4254 2 *3185 *3156 *3126 *3098 *3069 *3043 *3018 *2992 *2968 *2944 3 *2578 *2571 *2563 *2554 *2543 *2533 *2522 *2510 *2499 *2487 4 *2119 *2131 *2139 *2145 *2148 *2151 *2152 *2151 *2150 *2148 5 *1736 *1764 *1787 *1807 *1822 *1836 *1848 *1857 *1864 *1870
6 0X1399 0X1443 0X1480 0X1512 0X1539 0X1563 0X1584 0X1601 0X1616 0X1630 7 *1092 *1150 *1201 *1245 *1283 *1316 *1346 *1372 *1395 *1415 8 *0804 *0878 *0941 *0997 *1046 *1089 *1128 *1162 *1192 *1219 9 *0530 *0618 *0696 *0764 *0823 *0876 *0923 *0965 *1002 *1036 10 *0263 *0368 *0459 *0539 *0610 *0672 *0728 *0778 *0822 *0862
11 0X0000 0X0122 0X0228 0X0321 0X0403 0X0476 0-0540 0X0598 0-0650 0-0697 12 - *0000 *0107 *0200 *0284 *0358 *0424 *0483 *0537 13 - - - *0000 *0094 *0178 *0253 *0320 *0381 14 - - - - - *0000 *0084 *0159 *0227
15 - - - - - - 0000 *0076
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604 S. S. SHAPIRO AND M. B. WILK

Table 5. Coefficients {an-i+l} for the W test
for n = 2(1) 50 (cont.)
n
iX 31 32 33 34 35 36 37 38 39 40
1 0X4220 0X4188 0X4156 0X4127 0-4096 0X4068 0X4040 0X4015 0X3989 0X3964 2 *2921 *2898 *2876 *2854 *2834 *2813 *2794 *2774 *2755 *2737 3 *2475 *2463 *2451 *2439 *2427 *2415 *2403 *2391 *2380 *2368 4 *2145 *2141 *2137 *2132 *2127 *2121 *2116 *2110 *2104 *2098 5 *1874 *1878 *1880 *1882 *1883 *1883 *1883 *1881 *1880 *1878
6 0X1641 0-1651 0X1660 0X1667 0X1673 0X1678 0-1683 0X1686 0*1689 0-1691 7 *1433 *1449 *1463 *1475 *1487 *1496 *1505 *1513 *1520 *1526 8 *1243 *1265 *1284 *1301 *1317 *1331 *1344 *1356 *1366 *1376 9 *1066 *1093 *1118 *1140 *1160 *1179 *1196 *1211 *1225 *1237 10 *0899 *0931 *0961 *0988 *1013 *1036 *1056 *1075 *1092 *1108
11 0-0739 0-0777 0*0812 0-0844 0-0873 0*0900 0-0924 0*0947 0-0967 0*0986 12 *0585 *0629 *0669 *0706 *0739 *0770 *0798 *0824 *0848 *0870 13 *0435 *0485 *0530 *0572 *0610 *0645 *0677 *0706 *0733 *0759 14 *0289 *0344 *0395 *0441 *0484 *0523 *0559 *0592 *0622 *0651 15 *0144 *0206 *0262 *0314 *0361 *0404 *0444 *0481 *0515 *0546
16 0*0000 0*0068 0-0131 0*0187 0*0239 0-0287 0*0331 0-0372 0-0409 0-0444 17 - *0000 *0062 *0119 *0172 *0220 *0264 *0305 *0343 18 - - - *0000 *0057 *0110 *0158 *0203 *0244
19 - - - *0000 *0053 *0101 *0146 20 - - - - *0000 *0049

n
41 42 43 44 45 46 47 48 49 50

1 0X3940 0X3917 0X3894 0X3872 0X3850 0X3830 0-3808 0X3789 0X3770 0X3751 2 *2719 *2701 *2684 *2667 *2651 *2635 *2620 *2604 *2589 *2574 3 *2357 *2345 *2334 *2323 *2313 *2302 *2291 *2281 *2271 *2260 4 *2091 *2085 *2078 *2072 *2065 *2058 *2052 *2045 *2038 *2032 5 *1876 *1874 *1871 *1868 *1865 *1862 *1859 *1855 *1851 *1847

6 0X1693 0X1694 0X1695 0X1695 0X1695 0-1695 0-1695 0*1693 0-1692 0-1691 7 *1531 *1535 *1539 *1542 *1545 *1548 *1550 *1551 *1553 *1554 8 *1384 *1392 *1398 *1405 *1410 *1415 *1420 *1423 *1427 *1430 9 *1249 *1259 *1269 *1278 *1286 *1293 *1300 *1306 *1312 *1317 10 *1123 *1136 *1149 *1160 *1170 *1180 *1189 *1197 *1205 *1212

11 0*1004 0*1020 0*1035 0*1049 0*1062 0*1073 0*1085 0*1095 0*1105 0-1113 12 *0891 *0909 *0927 *0943 *0959 *0972 *0986 *0998 *1010 *1020 13 *0782 *0804 *0824 *0842 *0860 *0876 *0892 *0906 *0919 *0932 14 *0677 *0701 *0724 *0745 *0765 *0783 *0801 *0817 *0832 *0846 15 *0575 *0602 *0628 *0651 *0673 *0694 *0713 *0731 *0748 *0764

16 0*0476 0-0506 0*0534 0-0560 000584 0*0607 0-0628 0-0648 0-0667 0-0685 17 *0379 *0411 *0442 *0471 *0497 *0522 *0546 *0568 *0588 *0608 18 *0283 *0318 *0352 *0383 *0412 *0439 *0465 *0489 *0511 *0532
19 *0188 *0227 *0263 *0296 *0328 *0357 *0385 *0411 *0436 *0459 20 *0094 *0136 *0175 *0211 *0245 *0277 *0307 *0335 *0361 *0386

21 0.0000 0*0045 0*0087 0*0126 0*0163 0*0197 0-0229 0-0259 0-0288 0-0314

22 - 0000 *0042 *0081 *0118 *0153 *0185 *0215 *0244
23 - - *0000 *0039 *0076 *0111 *0143 *0174

24 - -- - *0000 '0037 *0071 *0104

25

-

*0000

*0035

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An analysis of variance test for normality 605
Table 6. Percentage points of the W test* for n = 3(1) 50
Level
n 0*01 0'02 0*05 0'10 0'50 0 90 0.95 0-98 0.99
3 0 753 0-756 0-767 0-789 0.959 0.998 0.999 1 000 1 000 4 *687 707 *748 *792 *935 *987 *992 *996 *997 5 *686 *715 *762 *806 *927 *979 *986 *991 *993
6 0-713 0*743 0-788 0.826 0-927 0974 0-981 0-986 0-989 7 *730 *760 *803 *838 *928 *972 *979 *985 *988 8 *749 *778 *818 *851 *932 *972 *978 *984 *987 9 *764 *791 *829 *859 *935 *972 *978 *984 *986 10 *781 *806 *842 *869 *938 *972 *978 *983 *986
11 0-792 0.817 0.850 0-876 0940 0973 0-979 0.984 0-986 12 *805 *828 *859 *883 *943 *973 *979 *984 *986 13 *814 *837 *866 *889 *945 *974 *979 *984 *986 14 *825 *846 *874 *895 *947 *975 *980 *984 *986 15 *835 *855 *881 *901 -950 *975 *980 *984 *987
16 0-844 0.863 0.887 0.906 0-952 0-976 0.981 0.985 0-987 17 *851 *869 *892 *910 *954 *977 *981 *985 *987 18 *858 *874 *897 *914 *956 *978 *982 *986 *988 19 *863 *879 *901 *917 .957 *978 *982 *986 *988 20 *868 *884 -905 *920 *959 *979 *983 *986 *988
21 0.873 0-888 0-908 0-923 0.960 0.980 0-983 0.987 0.989 22 *878 *892 *911 *926 *961 *980 *984 *987 *989 23 *881 *895 *914 *928 *962 *981 *984 *987 *989 24 *884 *898 *916 -930 *963 *981 *984 *987 *989 25 *888 *901 *918 *931 *964 *981 *985 *988 *989
26 0891 0 904 0-920 0 933 0-965 0-982 0-985 0.988 0-989 27 *894 *906 *923 *935 *965 *982 *985 *988 -990 28 *896 *908 *924 *936 *966 *982 *985 *988 -990 29 *898 *910 *926 *937 *966 *982 *985 *988 990 30 *900 *912 *927 .939 *967 *983 *985 *988 *900
31 0-902 0-914 0-929 0 940 0.967 0.983 0-986 0.988 0.990 32 *904 *915 *930 *941 -968 *983 *986 *988 -990
33 *906 *917 *931 *942 *968 *983 *986 *989 *990 34 *908 *919 *933 *943 *969 *983 *986 *989 *990 35 *910 *920 *934 *944 *969 *984 *986 *989 *990
36 0912 0-922 0*935 0945 0970 0.984 0.986 0-989 0990 37 *914 *924 *936 *946 -970 *984 *987 *989 -990 38 *916 *925 *938 *947 *971 *984 *987 *989 *990 39 *917 *927 *939 *948 *971 *984 *987 *989 *991 40 *919 *928 *940 *949 *972 *985 *987 *989 *991
41 0-920 0.929 0-941 0-950 0-972 0.985 0-987 0.989 0.991 42 *922 *930 *942 *951 *972 *985 *987 *989 *991
43 *923 *932 *943 *951 *973 *985 *987 990 *991 44 *924 *933 *944 *952 *973 *985 *987 *990 *991
45 *926 *934 *945 *953 *973 *985 *988 -990 *991
46 0927 0935 0945 0953 0974 0.985 0-988 0.990 0991
47 *928 *936 *946 *954 *974 *985 *988 *990 *991
48 *929 *937 *947 *954 *974 *985 *988 *990 *991
49 *929 .937 *947 *955 *974 *985 *988 *990 *991
50 *930 *938 *947 *955 *974 *985 *988 *990 *991
e Based on fitted Johnson (1949) SB approximation, see Shapiro & Wilk (1965a) for details.
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606 S. S. SHAPIRO AND M. B. WILK
To illustrate the process, suppose a sample of 7 observations were obtained, namely
xi = 6, x2 = 1,x3 = -4, x4 = 8, x5 = -2, x6 = 5, x7 = 0.
(i) Ordering, one obtains
Yi = -4, Y2 =-2, y3 = 0,y4 = 1, y5 = 5, Y6 = 6~ Y7 = 8.
(ii) S2 = Jy4 - 17 (Eyi)2 =146 -28 = 118.
(iii) From Table 5, under n = 7, one obtains
a7 = 0-6233, a6 = 0-3031, a5 = 01401, a4 = 0 0000.
Thus b = 0.6233(8 + 4) + 0*3031(6 + 2) + 0*1401(5-0) = 10*6049.
(iv) W = (10.6049)2/118 = 0 9530.
(v) Referring to Table 6, one finds the value of W to be substantially larger than the tabulated 50 % point, which is 0-928. Thus there is no evidence, from the W test, of non normality of this sample.
4. EXAMPLES
Example 1. Snedecor (1946, p. 175), makes a test of normality for the following sample of weights in pounds of 11 men: 148, 154, 158, 160, 161, 162, 166, 170, 182, 195, 236.
The W statistic is found to be 079 which is just below the 1 % point of the null distrib tion. This agrees with Snedecor's approximate application of the Vb1 statistic test.
Example, 2. Kendall (1948, p. 194) gives an extract of 200 'random sampling numbers' from the Kendall-Babington Smith, Tracts for Computers No. 24. These were totalled, as number pairs, in groups of 10 to give the following sample of size 10: 303, 338, 406, 457, 461, 469, 474, 489, 515, 583.
The W statistic in this case has the value 0 9430, which is just above the 50 % point of t
null distribution. Example 3. Davies et al. (1956) give an example of a 25 experiment on effects of five
factors on yields of penicillin. The 5-factor interaction is confounded between 2 blocks. Omitting the confounded effect the ordered effects are:
C 0-0958 ABC 0 0002 BC *0333 CD - 0-0026
ACDE *0293 B - 0*0036
BCE *0246 BD - 0-0042 ACD *0206 BCD -0-0113 ABCE *0194 ABE - 0-0139 DE *0191 ABD -0-0211 BE *0182 AC - 0-0333 BDE *0173 AD - 0-0341 ADE *0132 ACE - 0.0363 BCDE *0102 ABCD - 0-0363 ABDE *0084 AB - 0-0402 CDE *0077 CE - 0-0582 D *0058 A -0-1184
AE *0016 E -0-1398
In their analysis of variance, Davies et al. pool the 3- and 4-factor interactions for an error term. They do not find the pooled 2-factor interaction mean square to be significant but note that CE is significant at the 5 % point on a standard F-test. However, on the basis of a Bartlett test, they find that the significance of CE does not reach the 5 % level.
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An analysis of variance test for normality 607
The overall statistical configuration of the 30 unconfounded effects may be evaluated against a background of a null hypothesis that these are a sample of size 30 from a normal population. Computing the W statistic for this hypothesis one finds a value of 0-8812,
which is substantially below the tabulated 1 % point for the null distribution.
One may now ask whether the sample of size 25 remaining after removal of the 5 main effects terms has a normal configuration. The corresponding value of W is 0-9326, which is
above the 10 % point of the null distribution.
To investigate further whether the 2-factor interactions taken alone may have a nonnormal configuration due to one or more 2-factor interactions which are statistically 'too large', the W statistic may be computed for the ten 2-factor effects. This gives
W= 0-9465,
which is well above the 50 % point, for n = 10.
Similarly, the 15 combined 3 and 4-factor interactions may be examined from the same
point of view. The W value is 0-9088, which is just above the 10 % value of the null distr
tion. Thus this analysis, combined with an inspection of the ordered contrasts, would suggest
that the A, C and E main effects are real, while the remaining effects may be regarded as a random normal sample. This analysis does not indicate any reason to suspect a real CE effect based only on the statistical evidence.
The partitioning employed in this latter analysis is of course valid since the criteria employed are independent of the observations per se.
In the situation of this example, the sign of the contrasts is of course arbitrary and hence their distributional configuration should be evaluated on the basis of the absolute values, as in half-normal plotting (see Daniel, 1959). Thus, the above procedure had better be carried out using a half-normal version of the W test if that were available.
5. COMPARISON WITH OTHER TESTS FOR NORMALITY
To evaluate the W procedure relative to other tests for normality an empirical sampling investigation of comparative properties was conducted, using a range of populations and sample sizes. The results of this study are given in Shapiro & Wilk (1964a), only a brief extract is included in the present paper.
The null distribution used for the study of the W test was determined as described above. For all other statistics, except the %2 goodness of fit, the null distribution employed was determined empirically from 500 samples. For the x2 test, standard %2 table values were used. The power results for all procedures and alternate distributions were derived from 200 samples.
Empirical sampling results were used to define null distribution percentage points for a combination of convenience and extensiveness in the more exhaustive study of which the results quoted here are an extract. More exact values have been published by various authors for some of these null percentage points. Clearly one employing the KolmogorovSmirnov procedure, for example, as a statistical method would be well advised to employ the most accurate null distribution information available. However, the present power results are intended only for indicative interest rather than as a definitive description of a procedure, and uncertainties or errors of several percent do not materially influence the comparative assessment.
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608 S. S. SHAPIRO AND M. B. WILK
Table 7 gives results on the power of a 5 % test for sam procedures and for fifteen non-normal populations. The tests shown in Table 7 are: W; chi-squared goodness of fit (x2); standardized 3rd and 4th moments, Vb. and b2; Kolmogoro
Smirnov (KS) (Kolmogorov, 1933); Cramer-Von Mises (CVM) (Cramer, 1928); a weighted, by F/(1-F), Cramer-Von Mises (WCVM), where F is the cumulative distribution function (Anderson & Darling, 1954); Durbin's version of the Kolmogorov-Smirnov procedure (D) (Durbin, 1961); range/standard deviation (u) (David et al. 1954).
Table 7. Empirical power for 5 0% tests for selected alternative distributions; samples all of size 20
Population
title V/h 32 W x2 Vbl b2 KS CVM WCVM D u x2(l) 2-83 15-0 0X98 0*94 0X89 0 53 0 44 044 0*54 0X87 0.10 X2 (2) 2-00 9*0 *84 *33 *74 *34 *27 *23 *27 *42 *08
X2 (4) 141 6.0 *50 13 *49 *27 *18 *13 *16 *15 *06 X2 (10) 089 4.2 *29 07 *29 19 *11 *10 *11 *07 *06 Non-cent. x2 0X73 3X7 *59 *10 50 *20 *19 *16 *18 *20 *10 Log normal 6419 113X9 *93 *95 *89 *58 *44 *48 *62 *82 *06 Cauchy - - 88 *41 *77 *81 *45 *55 *98 *85 *56 Uniform 0 1-8 *23 *11 00 *29 *13 09 *10 08 *38 Logistic 0 4.2 *08 -06 *12 *06 *06 -03 *05 -05 -07 Beta (2, 1) -0-57 2-4 *35 *08 *08 *13 *08 *10 -12 *12 *23 La Place 0 6-0 *25 *17 *25 *27 -07 -07 *29 *16 *19 Poisson (1) 1900 4 0 *99 1900 *26 *11 *55 *22 *31 1 00 *35 Binomial, 0 2-5 *71 1900 *02 -03 *38 *15 *17 1.00 *20 (4, 0.5)
*T(5, 2.4) 0 79 2-2 *55 *14 *24 *20 *23 *20 *22 *T(10, 3.1) 0*97 2-8 *89 *32 *51 *24 *32 *30 *30 - -
* Variates from this distribution T(a, A) are defined by y = aRl -_(1- R)A, where R is uniform (0, 1) (Hastings, Mosteller, Tukey & Winsor, 1947). Also note that (a) the non-central x2 distribution
has degrees of freedom 16, non-centrality parameter 1; (b) the beta distribution has p = 2, q = 1 in standard notation; (c) the Poisson distribution has expectation 1.
In using the non-scale and non-origin invariant tests the mean and variance of the hypothesized normal was taken to agree with the known mean and variance of the alternative distribution. For the Cauchy the mode and intrinsic accuracy were used.
The results of Table 7 indicate that the W test is comparatively quite sensitive to a wide range of non-normality, even with samples as small as n = 20. It seems to be especially sensitive to asymmetry, long-tailedness and to some degree to short-tailedness.
The %2 procedure shows good power against the highly skewed distributions and reasonable sensitivity to very long-tailedness.
The Vbl test is quite sensitive to most forms of skewness. The b2 statistic can usefully augment Vb, in certain circumstances. The high power of Vbl for the Cauchy alternative i
probably due to the fact that, though the Cauchy is symmetric, small samples from it will often be asymmetric because of the very long-tailedness of the distribution.
The KS test has similar properties to that of the CVM procedure, with a few exceptions. In general the WCVM test has higher power than KS or CVM, especially in the case of longtailed alternatives, such as the Cauchy, for which WCVM had the highest power of all the statistics examined.
The use of Durbin's procedure improves the KS sensitivity only in the case of highly
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An analysis of variance test for normality 609
skewed and discrete alternatives. Against the Cauchy, the D test responds, like Vb1,
to the asymmetry of small samples. The u test gives good results against the uniform alternative and this is representative of
its properties for short-tailed symmetric alternatives. The %2 test has the disadvantages that the number and character of class intervals used
is arbitrary, that all information concerning sign and trend of discrepancies is ignored and that, for small samples, the number of cells must be very small. These factors might explain some of the lapses of power for x2 indicated in Table 7. Note that for almost all cases the power of W is higher than that of x2.
As expected, the Ibj6 test is in general insensitive in the case of symmetric alternatives as illustrated by the uniform distribution. Note that for all cases, except the logistic, Vb, power is dominated by that of the W test.

Table 8. The effect of mis-specification of parameters

(n = 20, 5 % test, assumed parameters are ,c = 0, o = 1)

Actual parameters Tests Sample
,u a'r ,u/ size KS CM WCVM

D

x2

0 00 1-2 0*00 20 0*06 0-08 0*18 0 09 0*07 *00 1-3 *00 20 *12 *12 *29 *10 *09 *15 1.0 *15 20 *05 *08 *10 *03 *04 *18 1.2 *15 20 *08 *16 *24 *11 *12 *195 1*3 *15 20 -07 *12 *31 *12 *10 30 1.0 *30 20 *14 *26 *31 *07 *11 *36 1.2 *30 20 *21 *34 *46 *16 *21 *39 1F3 30 20 *21 *38 *55 *19 *26

The b2 test is not sensitive to asymmetry. Its performance was inferior to that of W except in the cases of the Cauchy, uniform, logistic and Laplace for which its performance was equivalent to that of W.
Both the KS and CVM tests have quite inferior power properties. With sporadic exception in the case of very long-tailedness this is true also of the WCVM procedure. The D procedure does improve on the KS test but still ends up with power properties which are not as good as other test statistics, with the exceptions of the discrete alternatives. (In addition, the D test is laborious for hand computation.)
The u statistic shows very poor sensitivity against even highly skewed and very longtailed distributions. For example, in the case of the x2(1) alternative, the u test has power of 10 % while even the KS test has a power of 44 % and that for W is 98 %. While the u te shows interesting sensitivity for uniform-like departures from normality, it would seem that the types of non-normality that it is usually important to identify are those of asymmetry and of long-tailedness and outliers.
The reader is referred to David et al. (1954, pp. 488-90) for a comparison of the power of the b2, u and Geary's (1935) 'a' (mean deviation/standard deviation) tests in detecting departure fromi normality in symmetrical populations. Using a Monte Carlo technique, they found that Geary's statistic (which was not considered here) was possibly more effective than either b2 or u in detecting long-tailedness.
The test statistics considered above can be put into two classes. Those which are valid

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610 S. S. SHAPIRO AND M. B. WILK
for composite hypotheses and those which are valid for simple hypotheses. For the simple hypotheses procedures, such as x2, KS, CVM, WCVM and D, the parameters of the null distribution must be pre-specified. A study was made of the effect of small errors of specification on the test performance. Some of the results of this study are given in Table 8. The apparent power in the cases of mis-specification is comparable to that attained for these
procedures against non-normal alternatives. For example, for lu/o- = 03, WC
apparent power of between 0-31 and 0 55 while its power against %2 (2) is only 0-27.
6. DisCUSSION AND CONCLUDING REMARKS 6- 1. Evaluation of test
As a test for the normality of complete samples, the W statistic has several good featuresnamely, that it may be used as a test of the composite hypothesis, that is very simple to compute once the table of linear coefficients is available and that the test is quite sensitive against a wide range of alternatives even for small samples (n < 20). The statistic is responsive to the nature of the overall configuration of the sample as compared with the configuration of expected values of normal order statistics.
A drawback of the W test is that for large sample sizes it may prove awkward to tabulate or approximate the necessary values of the multipliers in the numerator of the statistic. Also, it may be difficult for large sample sizes to determine percentage points of its distribution.
The W test had its inception in the framework of probability plotting. The formal use of the (one-dimensional) test statistic as a methodological tool in evaluating the normality of a sample is visualized by the authors as a supplement to normal probability plotting and not as a substitute for it.
6 2. Extensions It has been remarked earlier in the paper that a modification of the present W statistic may be defined so as to be usable with incomplete samples. Work on this modified W* statistic will be reported elsewhere (Shapiro & Wilk, 1965b). The general viewpoint which underlies the construction of the W and W* tests for normality can be applied to derive tests for other distributional assumptions, e.g. that a sample is uniform or exponential. Research on the construction of such statistics, including necessary tables of constants and percentage points of null distributions, and on their statistical value against various alternative distributions is in process (Shapiro & Wilk, 1964b). These statistics may be constructed so as to be scale and origin invariant and thus can be used for tests of composite hypothesis. It may be noted that many of the results of ? 2-3 apply to any symmetric distribution. The W statistic for normality is sensitive to outliers, either one-sided or two-sided. Hence it may be employed as part of an inferential procedure in the analysis of experimental data as suggested in Example 3 of ?4.
The authors are indebted to Mrs M. H. Becker and Mrs H. Chen for their assistance in various phases of the computational aspects of the paper. Thanks are due to the editor and referees for various editorial and other suggestions.
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