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Checking normality – Example SolutionsExample 1: The following standardised residuals were obtained following regression analysis:

-2.12 0.34 1.33 0.97 1.31 1.71 2.64 -0.55 0.47 -0.63-0.68 -0.48 -0.75 0.50 1.02 -1.63 1.71 -0.46 -0.16 -1.21

Check whether they are approximately normally distributed.

In SPSS, Analyze Descriptive Statistics Explore Plots, then select “Histogram” and also “Normality Plots with tests”.

Either a histogram or a QQplot should be used to graphically assess normality. The histogram shows that the residuals are approximately normally distributed. The points on the QQplot are all close to the line with no signs of skewness.

If a test has been used, the Shapiro-Wilk is more appropriate for this sample size. As p > 0.05, the assumption of normality cannot be rejected. As the assumption of normality has been met, the standard regression can be used.

© Ellen Marshall and Peter Samuels Reviewer: Cheryl Voake-Jones www.statstutor.ac.uk University of Sheffield / Birmingham City University University of Bath

The following resources are associated:Normality checking

Statistical hypothesis testing worksheet

Checking normality for parametric tests of 4

Example 2: A study was carried out to compare whether exercise has an effect on the blood pressure (measure in mm Hg). The blood pressure was measured on 15 people before and after exercising. The results were as follows:

Subject 1 2 3 4 5 6 7 8 9 10Before 85.1 108.4 79 109.1 97.3 96 102.1 91.2 89.2 100After 86 70 78.9 69.7 59.4 55 65.6 50.2 60.5 82

Carry out the relevant normality checks and carry out an appropriate test.

What analysis would you suggest for this study?

This is paired data so each person has two measurements ‘before’ and ‘after’. Therefore, if the differences between the before and after values are normally distributed a paired t-test should be used. If not, the Wilcoxon signed rank test is the non-parametric equivalent.

The first step is to calculate the differences in blood pressure for each subject:

Go to Transform Compute Variable

Name the new variable ‘Diff’ in the “Target Value” box and move the variables before and after to the “Numeric Expression” box, putting a – sign between them.

Then check the normality of the differences using:

Analyze Descriptive Statistics Explore > Plots, then select “Histogram” and also “Normality Plots with tests”.

Both plots suggest that the data is highly skewed. From the histogram, it’s clear that the data is negatively skewed.



Tests of Normality

Kolmogorov-Smirnova Shapiro-Wilk

Statistic df Sig. Statistic df Sig.

Diff .296 10 .013 .763 10 .005

a. Lilliefors Significance Correction

The Shapiro-Wilk test is also highly significant (p = 0.005) so the assumption of normality is not met. The non-parametric Wilcoxon signed rank should be used.

The Wilcoxon signed rank test ranks all the differences irrelevant of the sign. The sum of the positive ranks is compared to the sum of the negative ranks. If exercise has no effect on blood pressure, roughly the same number of positive and negative ranks would be expected.

To carry out the Wilcoxon test, go to

Analyse Non-parametric tests Legacy Dialogs 2 related samples and move ‘after’ and ‘before’ to the ‘Test pairs’ boxes



As p < 0.05, there is evidence of a change in blood pressure. Calculate the median blood pressure before and after the exercise.

Report:

A Wilcoxon-signed rank test showed a statistically significant reduction in blood pressure after exercise, Z = -2.601, p = 0.009. The median blood pressure decreased from 96.7 before the exercise to 67.7 after.


There was only one negative change which was ranked

2nd.

before after

Median Median

96.7 67.7

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