Pearson chi-square test for normality

Performs the Pearson chi-square test for the composite hypothesis of normality, see e.g. Thode (2002, Sec. 5.2).

pearson.test(x, n.classes = ceiling(2 * (n^(2/5))), adjust = TRUE)
a numeric vector of data values. Missing values are allowed.
The number of classes. The default is due to Moore (1986).
logical; if TRUE (default), the p-value is computed from a chi-square distribution with n.classes-3 degrees of freedom, otherwise from a chi-square distribution with n.classes-1 degrees of freedom.

The Pearson test statistic is $P=\sum (C_{i} - E_{i})^{2}/E_{i}$, where $C_{i}$ is the number of counted and $E_{i}$ is the number of expected observations (under the hypothesis) in class $i$. The classes are build is such a way that they are equiprobable under the hypothesis of normality. The p-value is computed from a chi-square distribution with n.classes-3 degrees of freedom if adjust is TRUE and from a chi-square distribution with n.classes-1 degrees of freedom otherwise. In both cases this is not (!) the correct p-value, lying somewhere between the two, see also Moore (1986).


A list with class “htest” containing the following components:
the value of the Pearson chi-square statistic.
the p-value for the test.
the character string “Pearson chi-square normality test”.
a character string giving the name(s) of the data.
the number of classes used for the test.
the degress of freedom of the chi-square distribution used to compute the p-value.


The Pearson chi-square test is usually not recommended for testing the composite hypothesis of normality due to its inferior power properties compared to other tests. It is common practice to compute the p-value from the chi-square distribution with n.classes - 3 degrees of freedom, in order to adjust for the additional estimation of two parameters. (For the simple hypothesis of normality (mean and variance known) the test statistic is asymptotically chi-square distributed with n.classes - 1 degrees of freedom.) This is, however, not correct as long as the parameters are estimated by mean(x) and var(x) (or sd(x)), as it is usually done, see Moore (1986) for details. Since the true p-value is somewhere between the two, it is suggested to run pearson.test twice, with adjust = TRUE (default) and with adjust = FALSE. It is also suggested to slightly change the default number of classes, in order to see the effect on the p-value. Eventually, it is suggested not to rely upon the result of the test.

The function call pearson.test(x) essentially produces the same result as the S-PLUS function call chisq.gof((x-mean(x))/sqrt(var(x)), n.param.est=2).


Moore, D.S. (1986): Tests of the chi-squared type. In: D'Agostino, R.B. and Stephens, M.A., eds.: Goodness-of-Fit Techniques. Marcel Dekker, New York.

Thode Jr., H.C. (2002): Testing for Normality. Marcel Dekker, New York.

See Also

shapiro.test for performing the Shapiro-Wilk test for normality. ad.test, cvm.test, lillie.test, sf.test for performing further tests for normality. qqnorm for producing a normal quantile-quantile plot.

  • pearson.test
pearson.test(rnorm(100, mean = 5, sd = 3))
pearson.test(runif(100, min = 2, max = 4))

Documentation reproduced from package nortest, version 1.0-4, License: GPL (>= 2)

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