ASSESSING NORMALITY USING STATISTICAL TECHNIQUES
Kolmogrorov-Smirnov (a)
Statistic          df                    Sig.
    .21          1598                  .000*
* This is lower bound of the true signifcance 
(a) Lilliefors Significance Correction  
SCORE
KOLMOGOROV-SMIRNOV TEST
You could use the Kolmogorov-Smirnov statistic Z test evaluates statistically whether the difference between the observed distribtuion and a theoretical nornal distribtuion is small enough to be just due to chance. If it could be due to chance you would treat the distribution as being normal. If the distribution between the actual distribtuion and the theoretical normal distribution is larger than is likely to be due to chance (sampling error) then you would treat the actual distribution as not being normal.
Table showing the Kolmogorov-Smirnov statistic for assessing normality
The graphical methods discussed present qualitative information about the distribution of data that may not be apparent from statistical tests. Histograms, box plots and normal probability plots are graphical methods are useful for determining whether data follow a normal curve. Extreme deviations from normality are often readily identified from graphical methods. However, in many instances the decision is not straightforward. Using graphical methods to decide whether a data set is normally distributed involves making a subjective decision; formal test procedures are usually necessary to test the assumption of normality.
In general, both statistical tests and graphical plots should be used to determine normality. However, the assumption of normality should not be rejected on the basis of a statistical test alone. In particular, when the sample is large, available, statistical tests for normality can be sensitive to very small (i.e., negligible) deviations in normality. Therefore, if the sample is very large, a statistical test may reject the assumption of normality when the data set, as shown using graphical methods, is essentially normal and the deviation from normality too small to be of practical significance.
CHAPTER 4: THE NORMAL DISTRIBUTION
At the end of this topic, you should be able to:

  • explain what is the normal distribution
  • assessing normality using graphical  techniques - histogram
  • assessung normality using graphical techniques - box plots
  • assessing normality using graphical techniques - normality plot
  • assessing normality using statistical techniques

SHAPIRO - WILK TEST
Another powerful and most commonly employed tests for normality is the W test by Shapiro and Wilk, also called the Shapiro-Wilk test. It is an effective method for testing whether a data set has been drawn from a normal distribution.

In terms of hypothesis testing, the Shapiro-Wilk test is based on Ho: that the data are normally distributed. The test is used for samples which have less than 50 subjects.
      
                        Ho:  µ1 = µ2    OR      Ha:  µ1 ≠ µ2     

Tests of Normality

                                                        Shapiro-Wilk

Independent variable group   Statistic df Sig.
       Group 1                              .912     22        .055
       Group 2       .                             .166              14        .442
       Group 3                              .900     16        .084

The Shapiro-Wilk normality tests indicate that the scores are normally distributed in each of the three groups. All the p-values reported are more than 0.05 and hence you DO NOT REJECT the null hypothesis.
SPSS Output
In terms of hypothesis testing, the Kolmogorov-Smirnov test is based on Ho: that the data are normally distributed. The test is used for samples which have more than 50 subjects.   
        
                   Ho:  µ1 = µ2    OR      Ha:  µ1 ≠ µ2     


SPSS Output
Table showing the Kolmogorov-Smirnov statistic for assessing normality

Kolmogrorov-Smirnov (a)
Statistic          df                    Sig.
    .057          999                  .200*
* This is lower bound of the true signifcance 
(a) Lilliefors Significance Correction  
SCORE
The Kolmogorov-Smirnov Z test indicates that the p-value is less than 0.05 and hence you REJECT the null hypotheis. You conclude that for this particular distribution is NOT NORMAL.
NOTE:
It should be noted that with large samples even a very small deviation from normality can yield low significance levels, so a judgement still has to made as to whether the departure from nromality is large enough to matter.
WHAT TO DO IF THE DISTRIBUTION IS NOT NORMAL?
You have TWO choices if the distribution is not normal and they are:

a) Use a Nonparametric Statistic
In many cases, if the distribution is not normal an alternative statistic will be available, especially for bivariate analyses such as correlation or comparisons of means. These alternatives which do not require normal distributions are called nonparametric or distribution-free statistics. Some of these alternatives are shown below:
Purpose of                     Parametric          Non-Parametric
Analysis                          Statistics                   Statistics

Differences between                                  -  Mann-Whitney U test
two independent                    t-test            -  Kolmogorov-Smirnov
two means                                                          sample Z test

Differences between               t-test           - Wilcoxon's matched  
two dependent means                                  pairs test

Differences between more      Oneway        - Kurskal-Wallis analysis
than two means                     ANOVA          of ranks

Differences between more      Repeated      - Friedman's two-way
than two means that is           measures        analysis of variance
repeated                               ANOVA        - Cochran Q test

Relationship between            Pearson r      - Spearman Rho
variables                                                  -  Kendall's tau
                                                              - Chi-square

b) Transform the Variable to Make it Normal
The shape of a distribtuion can be changed by expressing it in different way statistically. This is referred to as transforming the distribution, Different types of transformations can ve applied to "normalise" the distribution. The type of transformation selected depends on the manner to which the distribution departs from normality. [We will not discuss transformation in this course]
LEARNING ACTIVITY
Examine the SPSS output and determine if the sample is normally distributed.
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