Purpose

    This procedure tests the hypothesis that the data come from the normal distribution. See. Why the "normal distribution" is important.

Preparations

    To run this procedure, select a range, and then run the Statistics→Basic Statistics and Tables→Normality Tests command.

Results

Count - analyzed sample size.

Mean - analyzed sample mean. See Elementary Concepts.
Standard Deviation, Median, Skewness, Kurtosis - See Elementary Concepts.
Kolmogorov-Smirnov/Lilliefor Test.
The Kolmogorov-Smirnov one-sample test for normality is based on the maximum difference between the sample cumulative distribution and the hypothesized cumulative distribution. If the D statistic is significant, then the hypothesis that the respective distribution is normal should be rejected. For many software programs, the probability values that are reported are based on those tabulated by Massey (1951); those probability values are valid when the mean and standard deviation of the normal distribution are known a-priori and not estimated from the data. However, usually those parameters are computed from the actual data. In that case, the test for normality involves a complex conditional hypothesis ("how likely is it to obtain a D statistic of this magnitude or greater, contingent upon the mean and standard deviation computed from the data"), and the Lilliefors probabilities should be interpreted (Lilliefors, 1967). Note that in recent years, the Shapiro-Wilks W test has become the preferred test of normality because of its good power properties as compared to a wide range of alternative tests.

Shapiro-Wilk W Test.
The Shapiro-Wilk W test is used in testing for normality. If the W statistic is significant, then the hypothesis that the respective distribution is normal should be rejected. The Shapiro-Wilk W test is the preferred test of normality because of its good power properties as compared to a wide range of alternative tests (Shapiro, Wilk, & Chen, 1968). W Statistics in computed as
    W = b² / S²

    where

        S² = (x_i-µ)²
        b = a_n-i+1(x_n-i+1-x_i)
            where
                µ    - mean
                a_n-i+1 contants

Hence, the closer W is to one, the more normal the sample is. The probability values for W are valid for samples in the range of 3 to 5000. W may not be as powerful as other tests when ties occur in your data.

D'Agostino Tests.
D'Agostino (1990) describes a normality test based on the skewness coefficient, . Recall that because the normal distribution is symmetrical, is equal to zero for normal data. Hence, a test can be developed to determine if the value of is significantly different from zero. If it is, the data are obviously nonnormal. The statistic, z²_s, is, under the null hypothesis of normality, approximately normally distributed.

Also D'Agostino (1990) describes a normality test based on the kurtosis coefficient. Recall that for the normal distribution, the theoretical value of kurtosis coefficient is 3. Hence, a test can be developed to determine if the value of kurtosis coefficient is significantly different from 3. If it is, the data are obviously nonnormal. The statistic, z²_k, is, under the null hypothesis of normality, approximately normally distributed for sample sizes n>20.
D'Agostino (1990) describes a normality test that combines the tests for skewness and kurtosis.  K2 The statistic, K² (K² = z²_s + z²_k) , is approximately distributed as a chi-square with two degrees of freedom.