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The following resources are associated:
Normality checking
Statistical hypothesis testing worksheet
Checking normality – Example Solutions
Example 1: The following standardised residuals were obtained following regression analysis:
-2.12
0.34
1.33
0.97
1.31
1.71
2.64
-0.68
-0.48
-0.75
0.50
1.02
-1.63
1.71
Check whether they are approximately normally distributed.
-0.55
-0.46
0.47
-0.16
-0.63
-1.21
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
www.statstutor.ac.uk
© Ellen Marshall and Peter Samuels
Reviewer: Cheryl Voake-Jones
University of Sheffield / Birmingham City University
University of Bath
Checking normality for parametric tests
Page 2 of 4
assumption of normality cannot be rejected. As the assumption of normality has been met, the
standard regression can be used.
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
Before
85.1
108.4
79
109.1
97.3
96
102.1
After
86
70
78.9
69.7
59.4
55
65.6
Carry out the relevant normality checks and carry out an appropriate test.
8
91.2
50.2
9
89.2
60.5
10
100
82
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.
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© Ellen Marshall and Peter Samuels
Reviewer: Cheryl Voake-Jones
University of Sheffield / Birmingham City University
University of Bath
Checking normality for parametric tests
Page 3 of 4
Tests of Normality
Kolmogorov-Smirnova
Statistic
Diff
.296
df
Shapiro-Wilk
Sig.
10
.013
Statistic
.763
df
Sig.
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
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© Ellen Marshall and Peter Samuels
Reviewer: Cheryl Voake-Jones
University of Sheffield / Birmingham City University
University of Bath
Checking normality for parametric tests
Page 4 of 4
There was only one
negative change which was
ranked 2nd.
As p < 0.05, there is evidence of a change in blood pressure.
Calculate the median blood pressure before and after the
exercise.
before
after
Median
Median
96.7
67.7
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.
www.statstutor.ac.uk
© Ellen Marshall and Peter Samuels
Reviewer: Cheryl Voake-Jones
University of Sheffield / Birmingham City University
University of Bath