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Statistical Analysis 3: Paired t-test
Research question type: Difference between (comparison of) two related ( paired , repeated
or matched ) variables
What kind of variables? Continuous ( scale/interval/ratio )
Common Applications: Comparing the means of data from two related samples; say,
observations before and after an intervention on the same participant; comparison of measurements from the same participant using 2 measurement techniques
Example 1:
Research question: Is there a difference in marks following a teaching intervention?
The marks for a group of students before (pre) and after (post) a teaching intervention are recorded below:
Marks are continuous (scale) data. Continuous data are often summarised by giving their average and standard deviation (SD), and the paired t-test is used to compare the means of the two samples of related data.
The paired t-test compares the mean difference of the values to zero. It depends on the mean difference, the variability of the differences and the number of data.
Various assumptions also need to hold – see validity section below.
You should practise entering the data into SPSS (PASW), but the data are available on W:\EC\STUDENT\ MATHS SUPPORT CENTRE STATS WORKSHEETS\marks.sav
[NB The Diff column is given here for illustration purposes; it does not have to be entered to SPSS]
Hypotheses: The 'null hypothesis' might be: H 0 : There is no difference in mean pre- and post-marks And an 'alternative hypothesis' might be: H 1 : There is a difference in mean pre- and post-marks
Steps in SPSS (PASW): The data need to be entered in SPSS in 2 columns, where one column indicates the pre-mark and the other has the post-mark – see over. [A third column could include participant numbers].
Loughborough University Mathematics Learning Support Centre
Student Before mark^ After mark Diff 1 18 22 4 2 21 25 4 3 16 17 1 4 22 24 2 5 19 16 - 3 6 24 29 5 7 17 20 3 8 21 23 2 9 23 19 - 4 10 18 20 2 11 14 15 1 12 16 15 - 1 13 16 18 2 14 19 26 7 15 18 18 0 16 20 24 4 17 12 18 6 18 22 25 3 19 15 19 4 20 17 16 - 1 Mean 18.40 20.45 2.
Coventry University Mathematics Support Centre Analyze > Compare Means > Paired Samples T-test Select the two paired variables as the Paired Variables, selecting the after variable first (post), followed by the before variable (pre) – see below Click OK
Output should look something like below:
[There is another table showing correlation – not needed for this purpose]
Paired Samples Statistics
Mean N
Std. Deviation
Std. Error Mean
Pair 1
Mark after 20.45 20 4.058. Mark before 18.40 20 3.
.
Paired Samples Test Paired Differences
Mean Std.^ t^ df^ Sig. (2^ tailed)- Deviation
Std. Error Mean
95% Confidence Interval of the Difference Lower Upper Mark after - Mark before 2.050^ 2.837^ .634^ .722^ 3.378^ 3.231^19.
Results: Notice that this option automatically gives you the sample summary data.
The relevant results for the paired t-test are in bold. From this row observe the t statistic, t = 3.231, and p = 0.004; ie, a very small probability of this result occurring by chance, under the null hypothesis of no difference. The null hypothesis is rejected, since p < 0.05 (in fact p = 0.004).
Conclusion: There is strong evidence (t = 3.23, p = 0.004) that the teaching intervention improves marks. In this data set, it improved marks, on average, by approximately 2 points. Of course, if we were to take other samples of marks, we could get a 'mean paired difference' in marks different from 2.05. This is why it is important to look at the 95% Confidence Interval (95% CI).
p-value
Example 2: In an experiment to compare anxiety levels induced between looking at real spiders and pictures of spiders, the following data was collected from 12 people with a fear of spiders (arachnophobia):
Suitable null and alternate hypotheses could be: H 0 : There is no difference in mean anxiety scores between looking at real or pictures of spiders, and
H 1 : There is a difference in mean anxiety scores between looking at real or pictures of spiders
You should practise entering the data into SPSS (PASW), but the data are available on W:\EC\STUDENT\ MATHS SUPPORT CENTRE STATS WORKSHEETS\spiders.sav
Results: Following the steps in SPSS (PASW) outlined previously, you should get the following output:
Paired Samples Statistics Mean N Std. Deviation Std. Error Mean Pair 1 Anxiety score when looking at a spider picture
40.00 12 9.293 2.
Anxiety score when looking at a real spider
47.00 12 11.029 3.
Paired Samples Test Paired Differences
t df
Sig. (2- Mean SD tailed)
Std. Error Mean
95% CI of the Difference Lower Upper
Pair 1
Anxiety score when looking at a spider picture
- Anxiety score when looking at a real spider
- 7.000 9.807 2.831 - 13.231 - .769 (^) 2.473- 11.
Conclusion: You would report something along the lines that there is evidence to suggest that participants experienced statistically significantly greater anxiety (p = 0.031) when exposed to real spiders (mean = 47.0 units, SD = 9.3) than to pictures of spiders (mean = 40.0 units, SD = 11.0).`The 95% confidence interval for the difference is (-13.2,-0.77).
#Example taken from Field A (2009) Discovering statistics using SPSS 2 nd (^) ed. Sage Publications
Anxiety score Participant Picture Real Diff 1 30 40 10 2 35 35 0 3 45 50 5 4 40 55 15 5 50 65 15 6 35 55 20 7 55 50 - 5 8 25 35 10 9 30 30 0 10 45 50 5 11 40 60 20 12 50 39 11
