# D’Agostino-Pearson Test

In Skewness and Kurtosis Analysis we showed how to use the skewness and kurtosis to determine whether a data set is normally distributed. In particular, we demonstrated the Jarque-Barre test. We now described a more powerful test which is also based on skewness and kurtosis.

We first describe Skewness and Kurtosis tests, and then we describe the D’Agostino-Pearson Test, which is an integration of these two tests.

Skewness Test

The normal distribution has skewness equal to zero. This test determines whether the skewness of the data is statistically different from zero. The test is based on the fact that when the data is normally distributed the test statistic zs = skew/s.e. has a standard normal distribution, where skew = the skewness of the sample data and the standard error is given by the following formulas where n = the sample size.

Example 1: Conduct the skewness test for the data in range B4:C15 of Figure 1.

Figure 1 – Sample data

The test results are shown in Figure 2.

Figure 2 – Skewness Test

We see from Figure 2 that the skewness is not significantly different from zero and in fact the 95% confidence interval is (-.72991, 1.21315).

Real Statistics Function: The Real Statistics Resource Pack provides the following array function.

SKEWTEST(R1, lab, alpha) – array function which tests whether the skewness of the sample data in range R1 is zero (consistent with a normal distribution). The output consists of a 6 × 1 range containing the sample skewness, standard error, test statistic zs, p-value and 1–alpha confidence interval.

If lab = TRUE then the output contains a column of labels (default = FALSE). The default value for alpha is .05.

The output in range X7:Y12 of Figure 2 can be obtained using the array formula

=SKEWTEST(B4:C15,TRUE).

Kurtosis Test

The normal distribution has kurtosis equal to zero. This test determines whether the kurtosis of the data is statistically different from zero. The test is based on the fact that when the data is normally distributed the test statistic zk = kurt/s.e. has a standard normal distribution, where kurt = the kurtosis of the sample data and the standard error is given by the following formulas where n = the sample size.

Example 2: Conduct the kurtosis test for the data in range B4:C15 of Figure 1.

The test results are shown in Figure 3.

Figure 3 – Kurtosis Test

Real Statistics Function: The Real Statistics Resource Pack provides the following array function.

KURTTEST(R1, lab, alpha) – array function which tests whether the kurtosis of the sample data in range R1 is zero (consistent with a normal distribution). The output consists of a 6 × 1 range containing the sample kurtosis, standard error, test statistic zk, p-value and 1–alpha confidence interval.

If lab = TRUE then the output contains a column of labels (default = FALSE). The default value for alpha is .05.

The output in range AC7:AD12 of Figure 3 can be obtained using the array formula =KURTTEST(B4:C15,TRUE).

D’Agostino-Pearson Omnibus Test

The D’Agostino-Pearson test is based on the fact that when the data is normally distributed the test statistic $z^2_k+z^2_s$ has a chi-square distribution with 2 degrees of freedom, i.e.

This test should generally not be used for data sets with less than 20 elements

Real Statistics Functions: The Real Statistics Resource Pack contains the following functions.

DAGOSTINO(R1) = the D’Agostino-Pearson test statistic for the data in the range R1

DPTEST(R1) = p-value of the D’Agostino-Pearson test on the data in R1

Example 3: Use the D’Agostino-Pearson Test to determine whether the data in range B4:C15 of Figure 1 is normally distributed

From Figure 4, we see that p-value = .63673 > .05 = α, and so conclude that there are no grounds to reject the null hypothesis that the data is normally distributed, a conclusion which agrees with that obtained using the Shapiro-Wilk test.

Figure 4 – D’Agostino-Pearson Test for Normality