`
`Commentaries are informative essays dealing with viewpoints of statis-
`tical practice, statistical education, and other topics considered to be
`of general interest to the broad readership of The American Statistician.
`Commentaries are similar in spirit
`to Letters to the Editor, but
`they
`
`
`involve longer discussions of background, issues, and perspectives. All
`commentaries will be refereed for their merit and compatibility with these
`criteria.
`
`A Suggestion for Using Powerful and Informative
`
`Tests of Normality
`
`RALPH B. D’AGOSTINO, ALBERT BELANGER, AND RALPH B. D’AGOSTINO, JR.*
`
`
`
`tosis, respectively. The extensive power studies just men-4
`tioned have also demonstrated convincingly that
`the old
`warhorses,
`the chi—squared test and the Kolmogorov test
`(1933), have poor power properties and should not be used
`when testing for normality.
`Unfortunately, the preceding results have not been dis—
`seminated very well. The chi-squared and Kolmogorov tests
`are still suggested in textbooks for testing for normality.
`Major statistical packages such as SAS and SPSSX perform
`the excellent Shapiro—Wilk W test for sample sizes up to
`50. For larger samples, however,
`they supply the poor-
`power Kolmogorov test. These statistical packages do give
`skewness and kurtosis measures. They are not, however,
`the \/b_l and [)2 statistics. Rather they are functions of these,
`the so—called Fisher g statistics (Fisher 1973). The docu-
`mentation on this latter point is very incomplete.
`In our
`experience, many users are unaware of it, and descriptive
`evaluation of normality or nonnormality is confused because
`of it. Hypothesis testing using the powerful \/b_1 and 122 is
`not presented or even suggested.
`In this article, we discuss the skewness, \/b_, and kur-
`tosis, b2, statistics and indicate how they are excellent de-
`scriptive and inferential measures for evaluating normality.
`Further, we relate the Fisher g skewness and kurtosis mea-
`sures produced by the SAS and SPSSX software packages
`to V171 and b2 and show how a simple program (SAS macro)
`can be used to produce an excellent, informative analysis
`for investigating normality. This analysis contains separate
`tests based on \/b—] and b2 and the K2 test, which combines
`V5: and 192 for an omnibus test. Finally, we indicate how
`the preceding can be used in conjunction with normal prob-
`ability plotting. The latter gives an informative graphical
`component to an analysis for normality.
`
`2.
`
`POPULATION—MOMENTS DESCRIPTION OF
`NORMALITY AND NONNORMALITY
`
`For testing that an underlying population is normally dis-
`tributed the skewness and kurtosis statistics, \/b—, and b2,
`and the D’Agostino—Pearson K2 statistic that combines these
`two statistics have been shown to be powerful and infor-
`mative tests. Their use, however, has not been as prevalent
`as their usefulness. We review these tests and show how
`
`readily available and popular statistical software can be used
`to implement them. Their relationship to deviations from
`linearity in normal probability plotting is also presented.
`
`KEY WORDS: Via—1, b2; D’Agostino—Pcarson K2; Kur—
`tosis; Normal probability plot; Skewness.
`
`1.
`
`INTRODUCTION
`
`Tests of normality are statistical inference procedures de-
`signed to test that the underlying distribution of a random
`variable is normally distributed. There is a long history of
`these tests, and there are a plethora of them available for
`use (D’AgoStino 1971; D’Agostino and Stephens 1986, chap.
`9). Major studies investigating the statistical power of these
`over a wide range of alternative distributions have been
`undertaken, and a reasonably consistent picture has emerged
`as to which of these should be recommended for use. See
`
`D’Agostino and Stephens (1986, chap. 9) for a review of
`these power studies. The Shapiro—Wilk W test (Shapiro and
`Wilk 1965),
`the third sample moment (Vb—1) and fourth
`sample moment ([72) tests, and the D’Agostino—Pearson K2
`test combining these (D’Agostino and Pearson 1973) emerge
`as excellent tests. The W and K2 tests share the fine property
`of being omnibus tests, in that they have good power prop-
`erties over a broad range of nonnormal distributions. The
`\/b_l and b2 tests have excellent properties for detecting
`nonnormality associated with skewness and nonnormal kur-
`
`
`*Ralph B. D’Agostino is Chairman and Professor, Mathematics De-
`partment, Boston University, Boston, MA 02215. Albert Belanger is Stat-
`istician, Statistics and Consulting Unit, Boston University, Boston, MA
`02215. Ralph B. D’Agostino, Jr.,
`is a graduate student, Statistics De-
`partment, Harvard University, Cambridge, MA 02138. This work was
`supported in part by National Heart Lung and Blood Institute Grant 1-
`R01 «HL-40423-02.
`
`is said to be
`A population, or its random variable X,
`normally distributed if its density function is given by
`
`_l(-‘*#>Z
`2
`U
`
`f(X)=
`
` l
`V2770
`
`e
`
`‘°°<X<°°
`"0°<,u<°°
`0_>0.
`
`(1)
`
`316
`
`The American Statistician, November 1990, Vol. 44, N0. 4
`
`© 1 990 American Statistical Association
`
`This content downloaded from 128.255.6.125 on Mon, 29 May 2017 19:54:51 UTC
`All use subject to http://about.jstor.org/terms
`
`AstraZeneca Exhibit 2168 p. 1
`InnoPharma Licensing LLC V. AstraZeneca AB IPR2017-00904
`Fresenius-Kabi USA LLC V. AstraZeneca AB IPR2017-01910
`
`

`

`Here ,u. and 0' are the mean and standard deviation, respec-
`tively, of it. Of interest here are the third and fourth stan-
`dardized moments given by
`
`E<X — m3 M e m3
`VE _ [E(X — i021” "
`a3
`
`(2)
`
`and
`
`
`E<X — m4
`E<X — m4
`32 _ [E(X — “>212 _
`a4
`
`’
`
`(3)
`
`where E is the expected value operator. These moments
`measure skewness and kurtosis, respectively, and for the
`normal distribution they are equal to 0 and 3, respectively.
`The nonnormality of a population can be described by values
`of its central moments differing from the normal values.
`The normal distribution is symmetric, so VE = 0. A
`nonnormal distribution that is asymmetrical has a value of
`VB: 75 0 (see Fig.
`l); VE > 0 corresponds to skewness
`to the right and VB: < 0 corresponds to skewness to the
`left.
`
`The word kurtosis means “curvature,” and it has tradi-
`
`tionally been measured by the fourth standardized moment
`32. For the normal distribution,
`its value is 3. Figure 1
`displays two nonnormal distributions in which ,82 9e 3. Un-
`imodal distributions whose tails are heavier or thicker than
`
`the normal have ,82 > 3. These distributions also tend to
`have higher peaks in the center of the distribution, and in
`the past these distributions were often described in terms of
`the high peaks (leptokurtic). Unimodal distributions whose
`
`
`
`Illustration of Distributions With \/E 75 0 and [32 9e 3;
`Figure 1.
`Distributions Differing in Skewness and Differing in Kurtosis from
`the Normal Distribution; Top panel: A, \/E > 0; B, \/E = 0; C,
`\/E < 0; Bottom panel: A,
`[32 = 3; B, 32 < 3; C, 32 > 3.
`
`tails are lighter than the normal tend to have ,82 < 3. In
`terms of their peak, it tends to be broader than the normal
`(platykurtic). Readers are referred to D’Agostino and Ste-
`phens (1986) for further discussion of these and to Balanda
`and MacGillivray (1988) for a detailed discussion of kur-
`tosis. D’Agostino and Stephens (1986) also gave examples
`of well-known nonnormal distributions indexed by VE and
`32-
`
`3. SAMPLE MOMENTS AS INDICATORS OF
`NONNORMALITY
`
`Karl Pearson (1895) was the first to suggest that the
`sample estimates of \/[3_l and ,82 could be used to describe
`nonnormal distributions and used as the bases for tests of
`
`., X,, the sample
`.
`.
`normality. For a sample of size n, X1,
`estimates of VE and 32 are, respectively,
`
`x/b—l = Ina/mt”
`
`b2 = m4/mg,
`
`and
`
`where
`
`m, = 2(X, — Your;
`
`and )7 is the sample mean
`
`i = 2X,/n.
`
`(4)
`
`(5)
`
`(6)
`
`(7)
`
`As descriptive statistics, values of Via—1 and b2 close to 0
`and 3, respectively, indicate normality. To be more precise
`the expected values of these are 0 and 3(n — l)/(n + 1)
`under normality. Values differing from these are indicators
`of nonnormality. The signs and magnitudes of these give
`information about the type of nonnormality [e.g., \/b—l >
`0 corresponds to positive skewness and [22 > 3(n — 1)/ (n
`+ 1) relates to heavy tails in the population distribution].
`
`4. TESTS OF NORMALITY BASED ON SAMPLE
`MOMENTS
`
`The \/b_I and [)2 statistics are the bases for powerful tests
`of normality (D’Agostino and Stephens 1986, chap. 9).
`
`4.1 Tests of Skewness (Vb—1)
`
`Here the null hypothesis is H0: normality versus the al-
`ternative; H1: nonnormality due to skewness. For alterna-
`tives (VE 3:3 0), a two-sided test based on V2): is performed.
`For directional alternatives (VE > 0 or VF] < 0), one-
`sided tests are performed. Tables of critical values are avail-
`able (D’Agostino and Stephens 1986, chap. 9). For sample
`sizes n > 8, a normal approximation that is easily com-
`puterized is available. It is obtained as follows (D’Agostino
`1970):
`
`1. Compute \/b_1 from the sample data.
`2. Compute
`
`1/2
`
`,
`
`}
`
`(8)
`
`(9)
`
`_
`
`(n + 1)(n + 3)
`
`
`
`Y — \/b_I{———-——6(n_ 2)
`
`
`3(n2 + 2711 — 70)(n + l)(n + 3)
`Bzo/b—n —
`(n — 2)(n + 5)(n + 7)(n + 9)
`
`This content downloaded from 128.255.6.125 on Mon, 29 May 2017 19:54:51 UTC
`All use subject to http://about.jstor.org/terms
`
`The American Statistician, November 1990, V01. 44, N0. 4
`
`317
`
`AstraZeneca Exhibit 2168 p. 2
`
`

`

`W2 = —1 + {2(Bz(\/b_1) — 1)}“2,
`
`5 = MW,
`
`and
`
`3. Compute
`
`a = {2/(W2 — 1)}1/2.
`
`(10)
`
`(11)
`
`(12)
`
`Z(\/b_l) = 51n(Y/a + {(Y/a)2 + 1}“).
`
`(13)
`
`Z(\/b—1) is approximately normally distributed under the null
`hypothesis of population normality.
`
`4.2 Tests of Kurtosis (b2)
`
`Here the null hypothesis is HO: normality versus the al-
`temative; H1: nonnormality due to nonnormal kurtosis. Again
`a two-sided test (for [$2 75 3) or one-sided tests (for B2 >
`3 or [32 < 3) can be performed. Again elaborate tables are
`available (D’Agostino and Stephens 1986, chap. 9). More-
`over, a normal approximation due to Anscombe and Glynn
`(1983) is available. It is valid for n 2 20 and is as follows:
`
`4.3 Omnibus Test
`
`D’Agostino and Pearson (1973) presented a statistic that
`combines \/b_l and b2 to produce an omnibus test of nor-
`mality. By omnibus, we mean it is able to detect deviations
`from normality due to either skewness or kurtosis. The test
`statistic is
`
`K2 = 220/171) + 22(192),
`
`(20)
`
`where Z(\/b_1) and Z(b2) are the normal approximations to
`\/b_1 and b2 discussed in Sections 4.1 and 4.2. The K2
`statistic has approximately a chi-squared distribution, with
`2 df when the population is normally distributed.
`
`5. NUMERICAL EXAMPLE
`
`Table 1 contains a sample of cholesterol values from a
`sample of 62 subjects from the Framingham Heart Study.
`The data are presented as a stem-and-leaf plot. From these
`data we obtain
`
`V171 = 1.02, Z(\/b_1) = 3.14, p = .0017,
`II
`
`£22
`
`4.58, Z(b2) = 2.21, p = .0269,
`
`1. Compute [92 from the sample data.
`
`2. Compute the mean and variance of oz,
`
`and
`
`3
`
`- l
`
`E<b2> = 4H
`
`and
`
`var(b2) =
`
`24n(n — 2)(n - 3)
`(n + mm + 3)(n + 5) '
`
`3. Compute the standardized version of oz,
`
`K2 = 14.75, p = .0006.
`
`(14)
`
`(15)
`
`The preceding p values are the levels of significance for the
`corresponding two-sided tests. For the Kolmogorov test, p
`= .087. The data are clearly nonnormal. The \/b_1 and b2
`statistics quantify the nature of the nonnormality. The data
`distribution is skewed to the right and heavy in the tails.
`The Kolmogorov test gives no information about this non-
`normality and only indicates marginally nonnormality.
`
`x = (.172 — E(b2))/VVar(b2)-
`
`(l6)
`
`6. THE FISHER g STATISTICS
`
`4. Compute the third standardized moment of £72,
`
`~181(b2)— (n + 7)(n + 9) \/n(n — 2)(n — 3)
`
`_ 6(n2 — 5n + 2)
`
`6(n + 3)(n + 5).
`
`Both SAS and SPSSX routinely give skewness and kur-
`tosis statistics in their descriptive statistics output. Unfor-
`
`Tab/e 1. Cholesterol Data From the Framingham Heart Study
`
`5. Compute
`
`A = 6 + -—-—-— ——
`VB1(b2) VBsz)
`
`[
`
`2
`
`8
`
`+
`
`\/<
`
`1 +
`
`
`4
`
`131020)]
`
`-
`
`6. Compute
`
`Zb —
`(2)—
`
`l——2—>
`9A
`
`1/3
`
`(l7)
`
`(
`
`18
`
`)
`
`_
`
`l— 2/A
`1+ x\/2/(A — 4)
`
`]
`
`> /‘\/2/(9A).
`
`(19)
`
`Z(b2) is approximately normally distributed under the null
`hypothesis of population normality.
`
`Both Z(\/b—l) and Z ([72) can be used to test one-sided and
`two-sided alternative hypotheses.
`
`318
`
`The American Statistician, November 1990, Vol. 44, No. 4
`
`
`
`This content downloaded from 128.255.6.125 on Mon, 29 May 2017 19:54:51 UTC
`All use subject to http://ab0ut.jst0r.0rg/terms
`
`AstraZeneca Exhibit 2168 p. 3
`
` Stem-and-Ieaf plot Number
`
`39
`3
`1
`38
`37
`36
`35
`34
`33
`32
`31
`30
`29
`28
`27
`26
`25
`24
`23
`22
`21
`20
`1 9
`1 8
`1 7
`
`71 6 1
`NOTE: The descriptive statistics are sample size, n = 62; mean, )7 = 250.0; standard
`deviation, S = 41.4; skewness, Vb1 = 1.02; kurtosis, D2 = 4.58:
`
`3
`
`46
`7
`
`008
`
`35
`00288
`347778
`444668
`03678
`0000122244668
`0556
`0125678
`02
`28
`4
`
`1
`
`2
`1
`
`3
`
`2
`5
`6
`6
`5
`13
`4
`7
`2
`2
`1
`
`

`

`and Stephens (1986, chap. 2) for a detailed discussion of
`probability plotting.
`A normal probability plot is simply a plot of the inverse
`of the standard normal cumulative on the horizontal axis
`and the ordered observations on the vertical axis. The in-
`
`verse of the normal cumulative is usually defined in such a
`way to enhance the linearity of the plot, and one common
`procedure is to let the normal probability plot employ Blom’s
`(1958) approximation. In this case, the normal probability
`plot is a plot of
`
`=
`
`Z
`
`#1
`
`q)
`
`
`<11 + 1/4
`i —— 3/8
`
`0n
`
`(1):
`X-
`
`26
`
`(
`
`)
`
`where X“) is the ith ordered observation from the ordered
`sample X“) S X(2) S
`S X(,,) and
`
`i — 3/8
`
`Z = d)"
`
`<11 + 1/4)
`
`27
`
`(
`
`)
`
`tunately, neither give Vb: and b2. Rather, they give the
`Fisher g statistics defined as follows:
`
`skewness
`
`g,
`
`n2(X — 303
`: ——-——-———~——-
`(n _ 1)(n _ 2)S3
`
`1
`(2 )
`
`and
`
`where
`
`k rtosi
`u
`
`s
`
`n(n + 1)2(X — )_()4
`_
`g2 _ (n — 1)(n — 2)(n — 3)s4
`_
`2
`_ __3(n_2_~__ ,
`(n — 2)(n — 3)
`
`(22)
`
`52 =
`
`
`"
`n — l
`
`X — )7 2
`m2 2 L3— (23)
`n — l
`
`is the sample variance.
`These are related to \/b_, and b2 via the following:
`
`(n — 2)
`b : “-
`V—l m 81
`
`and
`
`(n — 2)(n — 3)
`b = ~—————————
`2
`(n + l)(n — 1) g2
`
`3(n — 1)
`+ — -
`(n + 1)
`
`(24)
`
`(25)
`
`software package does compute
`The BMDP statistics
`\/b_l and b2. All of the preceding software do not perform
`tests of normality based on skewness and kurtosis.
`
`7. RECOMMENDATIONS
`
`The tests just described based on \/b_l and b2 are excellent
`and powerful
`tests. We recommend that for all sample
`sizes VB: and b2 should be computed and examined as
`descriptive statistics. For all sample sizes 11 2 9, tests of
`hypotheses can be based on them. In particular, for n >
`50, where the Shapiro—Wilk test is no longer available, we
`recommend these tests and the D’Agostino—Pearson K2 test
`as the tests of choice. The justification for this is not only
`because of their fine power but also because of the infor-
`mation they supply on nonnormality. In conjunction with
`the use of standard statistical software, such as SAS, SPSSX,
`and BMDP, the skewness and kurtosis measures they pro-
`duce can be used as inputs to simple programs (macros) to
`perform these tests. In the appendix, we supply one such
`simple macro that can be used with SAS and that will pro-
`vide two-tailed tests.
`
`8. NORMAL PROBABILITY PLOT
`
`Another component in a good data analysis for investi-
`gating normality of data and an item again often not well
`handled routinely in computer packages is the normal prob-
`ability plot. This plot is a graphical presentation of the data
`that will be approximately a straight line if the underlying
`distribution is normal. Deviations from linearity correspond
`to various types of nonnormality. Some of these deviations
`reflect skewness and/or kurtosis. Others reflect features such
`
`as the presence of outliers, mixtures in the data, or truncation
`(censoring) in the data. Readers are referred to D’Agostino
`
`is the Z value such that
`
`
`i — 3/8
`l
`J2
`:
`n + 1/4
`~06 V27re
`
`2 dx
`
`#1-
`
`28
`
`(
`
`)
`
`fori=1,...,n.
`
`Figure 2 is a normal probability plot of the data of Section
`5. The expected straight line can be obtained by connecting
`the + ’s on the graph. Figure 3 contains a number of forms
`that the normal probability plot will produce in the presence
`of nonnormality. For the present data set, its skewness to
`the right is very evident in the plot.
`A program for the normal probability plot applicable to
`SAS is part of the macro given in the appendix.
`
`9. CONCLUSION
`
`We have discussed the uses of \/b—, and b2 as descriptive
`and inferential statistics for evaluating the normality of data.
`We have made specific recommendations for their uses.
`Further we have reviewed briefly the normal probability
`plot, which can be used in conjunction with \/b_1 and b2 for
`a graphical analysis. A good complete normality analysis
`would consist of the use of the plot plus the statistics. The
`use of these is superior to what is routinely given in standard
`computer software. Serious investigators should consider
`using the materials of this article in their data analysis.
`
`APPENDIX: A MACRO FOR USE WITH SAS
`STATISTICAL SOFTWARE
`
`The following macro takes as input a variable name and
`a data set name. It produces as output the results of a uni-
`variate descriptive analysis (PROC UNIVARIATE), skew-
`ness and kurtosis measures and test statistics, the D’Agostino—
`Pearson omnibus normality test statistic, p levels, and a
`normal probability plot.
`
`%MACRO NORMTEST(VAR,DATA);
`PROC UNIVARIATE NORMAL PLOT DATA = &DATA;
`VAR &VAR; OUTPUT OUT = XXSTAT N = N
`MEAN = XBAR STD = S SKEWNESS = G1
`
`KURTOSIS = G2;
`
`This content downloaded from 128.255.6.125 on Mon, 29 May 2017 19:54:51 UTC
`All use subject to http://about.jstor.org/terms
`
`The American Statistician, November 1990, Vol. 44, N0. 4
`
`319
`
`AstraZeneca Exhibit 2168 p. 4
`
`

`

`375 +
`
`350 +
`
`325 +
`
`300 +
`
`250 4-
`
`2 2 5 +
`
`200 +
`
`175 +
`
`
`
`
`
`
`275 +
`
`
`
`
`
`
`1
`
`1|!
`
`.
`
`+
`i 1.
`
`It It 1|! t
`
`It It
`
`It It
`
`+ 1*
`l .
`
`)t It It 10! 1k
`
`***:*
`
`it
`
`I! 1k
`
`150 +
`-+ ——————————— 4- ----------- + ——————————— + ----------- + ——————————— + ——————————— +
`- 3
`— 2
`- 1
`'
`o
`1
`2
`a
`NORMALIZED RANK
`
`Figure 2. Normal Probability Plot of Cholesterol Data.
`
`DATA;
`SET XXSTAT;
`DO _Z_= — 1,0,1; _X_= XBAR + _Z_*S; OUTPUT;
`END;
`KEEP _X_ _Z_;
`DATA; SET &DATA _LAST_;
`PROC RANK TIES = MEAN NORMAL = BLOM; VAR
`&VAR; RANKS BLOMRANK;
`OPTIONS LS = 80;
`PROC PLOT NOLEGEND;
`PLOT &VAR*BLOMRANK = ’*’ _X_*_Z_ = ’ + ’/
`OVERLAY HAXIS = — 3 TO 3 BY .5;
`LABEL BLOMRANK .= ”NORMALIZED RANK”
`&VAR = ”NORMAL PROBABILITY PLOT FOR
`&VAR”;
`
`DATA;
`SET XXSTAT;
`SQRTBl = (N — 2)/SQRT(N*(N — 1))*G1;
`Y = SQRTB1*SQRT((N + 1)*(N + 3)/(6*(N — 2D);
`BETA2 = 3*(N*N + 27*N — 70)*(N + 1)*(N + 3)/
`((N - 2)*(N + 5)*(N + 7)*(N + 9));
`W = SQRT( — l + SQRT(2*(BETA2 — 1)));
`DELTA = 1/SQRT(LOG(W));
`ALPHA = SQRT(Z/(W*W — 1));
`
`Z_Bl = DELTA*LOG(Y/ALPHA + SQRT((Y/
`ALPHA)**2 + 1));
`B2: 3*(N —1)/(N + 1) + (N — 2)*(N — 3)/
`((N+ 1)*(N — 1))*G2;
`MEANB2=3*(N—1)/(N+1);
`VARB2 = 24*N*(N — 2)*(N — 3)/
`((N+ 1)*(N + 1)*(N + 3)*(N + 5));
`X = (B2 — MEANB2)/SQRT(VARB2);
`MOMENT = 6*(N*N — 5*N + 2)/
`((N + 7)*(N + 9))*SQRT(6*(N + 3)*(N + 5)/
`(N*(N — 2)*(N — 3)));
`A = 6 + 8/MOMENT*(2/MOMENT + SQRT(I + 4/
`(MOMENT**2)));
`Z_B2 =(1 — 2/(9*A) — ((1 — 2/A)/(1 + X*SQRT(2/
`(A — 4))))**(1/3))/SQRT(2/(9*A));
`PRZBl = 2*(1 ~ PROBNORM(ABS(Z_B1)));
`PRZB2 = 2*(1 — PROBNORM(ABS(Z_BZ)));
`CHITEST = Z_B1*Z_B1+ Z_B2*Z_B2;
`PRCIH = 1 — PROBCHI(CHITEST,2);
`FILE PRINT;
`PUT “NORMALlTY TEST FOR VARIABLE &VAR "
`N = /
`@20 G1 = 8.5 @33 SQRTBI = 8.5 @50 ”2:” Z-
`B1 8.5 ” P = ” PRZBl @6.4/
`
`320
`
`The American Statistician, November 1990, Vol. 44, N0. 4
`
`This content downloaded from 128.255.6.125 on Mon, 29 May 2017 19:54:51 UTC
`All use subject to http://ab0ut.jst0r.0rg/terms
`
`AstraZeneca Exhibit 2168 p. 5
`
`C H
`
`O
`
`L
`E
`i
`E
`R
`0
`L
`
`

`

`LIL.
`
`Indication:
`
`W31 = o) l32< 3
`Symmetric
`Thin Tails
`
`W31 : 0,132) 3
`Symmetric
`Thick Tails
`
`X
`
`X
`
`.
`.
`Indication.
`
`Z
`
`Z
`
`W31 > 0
`Skewed to Right
`
`W31 < O
`Skewed to Left
`
`Z
`
`Z
`
`Z
`
`Z
`
`indication: Mixture
`of Normals
`
`Truncated
`at Left
`
`Truncated
`at Right
`
`Outlier
`at Right
`
`Figure 3.
`
`Indications of Deviations From Normality in a Normal Probability Plot.
`
`II
`CHI IEST 8.5 P
`
`:1;
`
`:1!
`
`-
`
`@20 G228'5 @33 32:85 @50 ”2:” Z_32 8.5 ”
`P=" PRZBZ 6.4//
`’
`*>l< =
`@33 'K 2 CHISQ (2 DP)
`PRCHI 6'4,
`%MEND NORMTEST;
`/*
`/* The SAS Options MACRO, DQUOTE and
`*
`INESIZE=
`;* :6 _
`ff
`t
`80 mUSt
`1n 6 0C
`/*
`/* Example of a statement to execute the macro above:
`/* %NORMTEST(CHOL,DATAl)
`/*
`
`*/
`*/
`*
`*;
`*/
`*/
`:1:/
`*/
`
`Balanda, K. 1),, and MacGillivray, H. L. (1988), “Kurtosis: A Critical
`Review,” The American Statistician, 42, 111—119.
`Blom, G. (1958), Statistical Estimates and Transformed Beta Variables,
`New York: John Wiley.
`D’Agostino, R. B. (1970), “Transformation to Normality of the Null Dis—
`
`tribution of g1
`Biometrika, 57, 679~681.
`(1971), “An Omnibus Test of Normality for Moderate and Large
`Dim? S‘zg'”BB"0’"fj”l§"“’ 58’ E‘s—fin)
`T .
`f D
`gostmo,
`.
`., an
`earson,
`.
`.
`, “ esting or
`epartures
`From Normality. I. Fuller Empirical Results for the Distribution of b2
`,,
`.
`.
`and \/b_, Biometrika, 60, 6134622.
`D’Agostino, R. B., and Stephens, M. A. ([986), Goodnessvof-fit Tech-
`"iquesa New York: Mam? Dekker-
`Fisher, R. A. (1973), Statistical Methodsfor Research Workers (14th ed.),
`New York: Hafner Publishing.
`Kolmogorov, A. (1933), “Sulla Determinazione Empirica di una Legge
`di Distribuzione,” Giornalle dell’lnstituto Italiano deg/i Attuari, 4, 83—
`91.
`Pearson, K. (1895), “Contributions to the Mathematical Theory of Evo—
`
`lution,” Philosophical Transactions of the Royal Society ofLondon, 91,
`343—358.
`
`[Received April 1989. Revised January [990.]
`
`REFERENCES
`
`Anscombe, F. J., and Glynn, W. J. (1983), “Distribution of the Kurtosis
`Statistic b2 for Normal Statistics,” Biometrika, 70, 227—234.
`
`Shapiro, S. S., and Wilk, M. B. (1965), “An Analysis of Variance Test
`for Normality (Complete Samples),” Biometrika, 52, 591—611.
`
`This content downloaded from 128.255.6.125 on Mon, 29 May 2017 19:54:51 UTC
`All use subject to http://ab0ut.jst0r.0rg/terms
`
`The American Statistician, November 1990, Vol. 44, No. 4
`
`32l
`
`AstraZeneca Exhibit 2168 p. 6
`
`