Kolmogrorov-Smirnov (a)

Statistic df Sig.

.21 1598 .000*

* This is lower bound of the true signifcance

(a) Lilliefors Significance Correction

SCORE

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.

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.

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**

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.

- If the normal probability plot is approximately linear (the data follow a normal curve), the test statistic will be relatively high.

- If the normal probability plot has curvature that is evidence of non-normality in the tails of a distribution, the test statistic will be relatively low.

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.

**Reject the assumption of normality if the test of signficance reports a p-value of less (<) than 0.05**

**DO NOT REJECT the assumption of normality if the test of significance reports a p-value of more (>) than 0.05.**

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.

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.** **

- If the Kolmogorov-Smirnov Z test yields a significance level of
**less (<) than 0.05**, it means that the.**distribution is NOT normal**

- If the Kolmogorov-Smirnov Z test yields a significance level of
**more (>) than 0.05**, it means that thel.**distribution is norma**

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.

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.

You have TWO choices if the distribution is not normal and they are:

- Use a Nonparametric Statistic Instead

- Transform the Variable to Make ti Normal

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:

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

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*]

Examine the SPSS output and determine if the sample is normally distributed.