Lilliefors/Van Soest’s
test of normality
Hervé Abdi1
& Paul Molin
1 Overview
The normality assumption is at the core of a majority of standard
statistical procedures, and it is important to be able to test this
assumption. In addition, showing that a sample does not come
from a normally distributed population is sometimes of importance
per se. Among the many procedures used to test this assumption,
one of the most well-known is a modification of the
Kolomogorov-Smirnov test of goodness of fit, generally referred to
as the Lilliefors test for normality (or Lilliefors test, for short). This
test was developed independently by Lilliefors (1967) and by Van
Soest (1967). The null hypothesis for this test is that the error is
normally distributed (i.e., there is no difference between the observed
distribution of the error and a normal distribution). The
alternative hypothesis is that the error is not normally distributed.
Like most statistical tests, this test of normality defines a criterion
and gives its sampling distribution. When the probability
associated with the criterion is smaller than a given α-level, the
1
In: Neil Salkind (Ed.) (2007). Encyclopedia of Measurement and Statistics.
Thousand Oaks (CA): Sage.
Address correspondence to: Hervé Abdi
Program in Cognition and Neurosciences, MS: Gr.4.1,
The University of Texas at Dallas,
Richardson, TX 75083–0688, USA
E-mail: herve@utdallas.edu http://www.utd.edu/∼herve
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H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
alternative hypothesis is accepted (i.e., we conclude that the sample
does not come from a normal distribution). An interesting peculiarity
of the Lilliefors’ test is the technique used to derive the
sampling distribution of the criterion. In general, mathematical
statisticians derive the sampling distribution of the criterion using
analytical techniques. However in this case, this approach fails
and consequently, Lilliefors decided to calculate an approximation
of the sampling distribution by using the Monte-Carlo technique.
Essentially, the procedure consists of extracting a large number of
samples from a Normal Population and computing the value of the
criterion for each of these samples. The empirical distribution of
the values of the criterion gives an approximation of the sampling
distribution of the criterion under the null hypothesis.
Specifically, both Lilliefors and Van Soest used, for each sample
size chosen, 1000 random samples derived from a standardized
normal distribution to approximate the sampling distribution of a
Kolmogorov-Smirnov criterion of goodness of fit. The critical values
given by Lilliefors and Van Soest are quite similar, the relative
error being of the order of 10−2
.
According to Lilliefors (1967) this test of normality is more powerful
than others procedures for a wide range of nonnormal conditions.
Dagnelie (1968) indicated, in addition, that the critical
values reported by Lilliefors can be approximated by an analytical
formula. Such a formula facilitates writing computer routines
because it eliminates the risk of creating errors when keying in the
values of the table. Recently, Molin and Abdi (1998), refined the
approximation given by Dagnelie and computed new tables using
a larger number of runs (i.e., K = 100,000) in their simulations.
2 Notation
The sample for the test is made of N scores, each of them denoted
Xi . The sample mean is denoted MX is computed as
MX =
1
N
N
i
Xi , (1)
2
H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
the sample variance is denoted
S2
X =
N
i
(Xi − MX )2
N −1
, (2)
and the standard deviation of the sample denoted SX is equal to
the square root of the sample variance.
The first step of the test is to transform each of the Xi scores
into Z-scores as follows:
Zi =
Xi − MX
SX
. (3)
For each Zi -score we compute the proportion of score smaller
or equal to its value: This is called the frequency associated with
this score and it is denoted S (Zi ). For each Zi -score we also compute
the probability associated with this score if is comes from a
“standard” normal distribution with a mean of 0 and a standard
deviation of 1. We denote this probability by N (Zi ), and it is equal
to
N (Zi ) =
Zi
−∞
1
2π
exp −
1
2
Z2
i . (4)
The criterion for the Lilliefors’ test is denoted L. It is calculated
from the Z-scores, and it is equal to
L = max
i
{|S (Zi )−N (Zi )|,|S (Zi )−N (Zi−1)|} . (5)
So L is the absolute value of the biggest split between the probability
associated to Zi when Zi is normally distributed, and the frequencies
actually observed. The term |S (Zi )−N (Zi−1)| is needed
to take into account that, because the empirical distribution is discrete,
the maximum absolute difference can occur at either endpoints
of the empirical distribution.
The critical values are given by Table 2. Lcritical is the critical
value. The Null hypothesis is rejected when the L criterion is greater
than or equal to the critical value Lcritical.
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H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
3 Numerical example
As an illustration, we will look at an analysis of variance example
for which we want to test the so-called “normality assumption”
that states that the within group deviations (i.e., the “residuals”)
are normally distributed. The data are from Abdi (1987, p.
93ff.) and correspond to memory scores obtained by 20 subjects
who were assigned to one of 4 experimental groups (hence 5 subjects
per group). The score of the sth subject in the ath group is
denoted Ya,s, and the mean of each group is denoted Ma.. The
within-group mean square MSS(A) is equal to 2.35, it correspond
to the best estimation of the population error variance.
G. 1 G. 2 G. 3 G. 4
3 5 2 5
3 9 4 4
2 8 5 3
4 4 4 5
3 9 1 4
Ya. 15 35 16 21
Ma. 3 7 3.2 4.2
The Normality assumption states that the error is normally distributed.
In the analysis of variance framework, the error corresponds
to the residuals which are equal to the deviations of the
scores to the mean of their group. So in order to test the normality
assumption for the analysis of variance, the first step is to compute
the residuals from the scores. We denote Xi the residual corresponding
to the ith observation (with i going from 1 to 20). The
residuals are given in the following table:
Yas 3 3 2 4 3 5 9 8 4 9
Xi 0 0 −1 1 0 −2 2 1 −3 2
Yas 2 4 5 4 1 5 4 3 5 4
Xi −1.2 .8 1.8 .8 −2.2 .8 −.2 −1.2 .8 −.2
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H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
Next we transform the Xi values into Zi values using the following
formula:
Zi =
Xi
MSS(A)
(6)
because MSS(A) is the best estimate of the population variance,
and the mean of Xi is zero. Then, for each Zi value, the frequency
associated with S (Zi ) and the probability associated with Zi under
the Normality condition N (Zi ) are computed [we use a table
of the Normal Distribution to obtain N (Zi )]. The results are presented
in Table 1.
The value of the criterion is (see Table 1)
L = max
i
{|S (Zi )−N (Zi )|,|S (Zi )−N (Zi−1)|} = .250 . (7)
Taking an α level of α = .05, with N = 20, we find (from Table 2)
that the critical value is equal Lcritical = .192. Because L is larger
than Lcritical, the null hypothesis is rejected and we conclude that
the residuals in our experiment are not distributed normally.
4 Numerical approximation
The available tables for the Lilliefors’ test of normality typically report
the critical values for a small set of alpha values. For example,
the present table reports the critical values for
α = [.20, .15, .10, .05, .01].
These values correspond to the alpha values used for most tests
involving only one null hypothesis, as this was the standard procedure
in the late sixties. The current statistical practice, however,
favors multiple tests (maybe as a consequence of the availability
of statistical packages). Because using multiple tests increases the
overall Type I error (i.e., the Familywise Type I error or αPF ), it has
become customary to recommend testing each hypothesis with a
corrected α level (i.e., the Type I error per comparison, or αPC )
such as the Bonferonni or ˘Sidák corrections. For example, using
a Bonferonni approach with a familywise value of αPF = .05, and
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H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
Table 1: How to compute the criterion for the Lilliefors’ test of
normality. Ni stands for the absolute frequency of a given value
of Xi , Fi stands for the absolute frequency associated with a given
value of Xi (i.e., the number of scores smaller or equal to Xi ),
Zi is the Z-score corresponding to Xi , S (Zi ) is the proportion of
scores smaller than Zi , N (Zi ) is the probability associated with
Zi for the standard normal distribution, D0 =| S (Zi ) − N (Zi ) |,
D−1 =| S (Zi )−N (Zi−1) |, and max is the maximum of {D0,D−1}.
The value of the criterion is L = .250.
Xi Ni Fi Zi S (Zi ) N (Zi ) D0 D−1 max
−3.0 1 1 −1.96 .05 .025 .025 .050 .050
−2.2 1 2 −1.44 .10 .075 .025 .075 .075
−2.0 1 3 −1.30 .15 .097 .053 .074 .074
−1.2 2 5 −.78 .25 .218 .032 .154 .154
−1.0 1 6 −.65 .30 .258 .052 .083 .083
−.2 2 8 −.13 .40 .449 .049 .143 .143
.0 3 11 .00 .55 .500 .050 .102 .102
.8 4 15 .52 .75 .699 .051 .250 .250
1.0 2 17 .65 .85 .742 .108 .151 .151
1.8 1 18 1.17 .90 .879 .021 .157 .157
2.0 2 20 1.30 1.00 .903 .097 .120 .120
testing J = 3 hypotheses requires that each hypothesis is tested at
the level of
αPC = 1
J
αPF = 1
3
×.05 = .0167 . (8)
With a ˘Sidák approach, each hypothesis will be tested at the level
of
αPC = 1−(1−αPF )
1
J = 1−(1−.05)
1
3 = .0170 . (9)
As this example illustrates, both procedures are likely to require using
different α levels than the ones given by the tables. In fact, it is
rather unlikely that a table could be precise enough to provide the
wide range of alpha values needed for multiple testing purposes. A
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H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
more practical solution is to generate the critical values for any alpha
value, or, alternatively, to obtain the probability associated to
any value of the Kolmogorov-Smirnov criterion. Such an approach
can be implemented by approximating the sampling distribution
“on the fly" for each specific problem and deriving the critical values
for unusual values of α.
Another approach to finding critical values for unusual values
of α, is to find a numerical approximation for the sampling distributions.
Molin and Abdi (1998) proposed such an approximation
and showed that it was accurate for at least the first two significant
digits. Their procedure, somewhat complex, is better implemented
with a computer and comprises two steps.
The first step is to compute a quantity called A obtained from
the following formula:
A =
−(b1 + N)+ (b1 + N)2 −4b2 b0 −L−2
2b2
, (10)
with
b2 = 0.08861783849346
b1 = 1.30748185078790
b0 = 0.37872256037043 . (11)
The second step implements a polynomial approximation and
estimates the probability associated to a given value L as:
Pr(L) ≈ −.37782822932809+1.67819837908004A
−3.02959249450445A2
+2.80015798142101A3
−1.39874347510845A4
+0.40466213484419A5
−0.06353440854207A6
+0.00287462087623A7
+0.00069650013110A8
−0.00011872227037A9
+0.00000575586834A10
. (12)
For example, suppose that we have obtained a value of L =
.1030 from a sample of size N = 50. (Table 2 shows that Pr(L) = .20.)
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H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
To estimate Pr(L) we need first to compute A, and then use this
value in Equation 12. From Equation 10, we compute the estimate
of A as:
A =
−(b1 + N)+ (b1 + N)2 −4b2 b0 −L−2
2b2
=
−(b1 +50)+ (b1 +50)2 −4b2 b0 −.1030−2
2b2
= 1.82402308769590 . (13)
Plugging in this value of A in Equation 12 gives
Pr(L) = .19840103775379 ≈ .20 . (14)
As illustrated by this example, the approximated value of Pr(L) is
correct for the first two decimal values.
References
[1] Abdi, H. (1987). Introduction au traitement statistique des
données expérimentales. Grenoble: Presses Universitaires de
Grenoble.
[2] Dagnelie, P. (1968). A propos de l’emploi du test de
Kolmogorov-Smirnov comme test de normalité, Biométrie et
Praximétrie 9, 3–13.
[3] Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov test for
normality with mean and variance unknown, Journal of the
American Statistical Association, 62, 399–402.
[4] Molin, P., Abdi H. (1998). New Tables and numerical
approximation for the KolmogorovSmirnov/Lillierfors/Van
Soest test of normality. Technical
report, University of Bourgogne. Available from
www.utd.edu/∼herve/MA_Lilliefors98.pdf.
[5] Van Soest, J. (1967). Some experimental results concerning
tests of normality. Statistica. Neerlandica, 21, 91–97, 1967.
8
Table 2: Table of the critical values for the Kolmogorov-Smirnov/Lillefors
test of normality obtained with K = 100,000 samples
for each sample size. The intersection of a given row and column
shows the critical value Lcritical for the sample size labelling the row
and the alpha level labelling the column. For N > 50 the critical
value can be found by using fN =
.83+ N
N
−.01.
N α = .20 α = .15 α = .10 α = .05 α = .01
4 .3027 .3216 .3456 .3754 .4129
5 .2893 .3027 .3188 .3427 .3959
6 .2694 .2816 .2982 .3245 .3728
7 .2521 .2641 .2802 .3041 .3504
8 .2387 .2502 .2649 .2875 .3331
9 .2273 .2382 .2522 .2744 .3162
10 .2171 .2273 .2410 .2616 .3037
11 .2080 .2179 .2306 .2506 .2905
12 .2004 .2101 .2228 .2426 .2812
13 .1932 .2025 .2147 .2337 .2714
14 .1869 .1959 .2077 .2257 .2627
15 .1811 .1899 .2016 .2196 .2545
16 .1758 .1843 .1956 .2128 .2477
17 .1711 .1794 .1902 .2071 .2408
18 .1666 .1747 .1852 .2018 .2345
19 .1624 .1700 .1803 .1965 .2285
20 .1589 .1666 .1764 .1920 .2226
21 .1553 .1629 .1726 .1881 .2190
22 .1517 .1592 .1690 .1840 .2141
23 .1484 .1555 .1650 .1798 .2090
24 .1458 .1527 .1619 .1766 .2053
25 .1429 .1498 .1589 .1726 .2010
26 .1406 .1472 .1562 .1699 .1985
27 .1381 .1448 .1533 .1665 .1941
28 .1358 .1423 .1509 .1641 .1911
Table continues on the following page ...
H. Abdi & P. Molin: Lilliefors / Van Soest Normality Test
Table 3: . . . Continued. Table of the critical values for the Kolmogorov-Smirnov/Lillefors
test of normality obtained with K = 100,000
samples for each sample size. The intersection of a given row and
column shows the critical value Lcritical for the sample size labelling
the row and the alpha level labelling the column. For N > 50 the
critical value can be found by using fN =
.83+ N
N
−.01.
N α = .20 α = .15 α = .10 α = .05 α = .01
29 .1334 .1398 .1483 .1614 .1886
30 .1315 .1378 .1460 .1590 .1848
31 .1291 .1353 .1432 .1559 .1820
32 .1274 .1336 .1415 .1542 .1798
33 .1254 .1314 .1392 .1518 .1770
34 .1236 .1295 .1373 .1497 .1747
35 .1220 .1278 .1356 .1478 .1720
36 .1203 .1260 .1336 .1454 .1695
37 .1188 .1245 .1320 .1436 .1677
38 .1174 .1230 .1303 .1421 .1653
39 .1159 .1214 .1288 .1402 .1634
40 .1147 .1204 .1275 .1386 .1616
41 .1131 .1186 .1258 .1373 .1599
42 .1119 .1172 .1244 .1353 .1573
43 .1106 .1159 .1228 .1339 .1556
44 .1095 .1148 .1216 .1322 .1542
45 .1083 .1134 .1204 .1309 .1525
46 .1071 .1123 .1189 .1293 .1512
47 .1062 .1113 .1180 .1282 .1499
48 .1047 .1098 .1165 .1269 .1476
49 .1040 .1089 .1153 .1256 .1463
50 .1030 .1079 .1142 .1246 .1457
> 50
0.741
fN
0.775
fN
0.819
fN
0.895
fN
1.035
fN
10