2. A paired-samples t-test compares the means of two related sets of data to see if they differ statistically.
IQ Example
We may want to compare the IQ scores of a group of students at the beginning and end of their degrees to see if there is any change.
We take IQ measurements of 10 students at the beginning of their degrees, and IQ measurements from the same 10 students at the end of their degrees.

3. Does the mean IQ of the students change over the course of their degrees?
Student Number
Start IQ
End IQ 1
111
110 2
116
117 3
114 4
109 5
115 6
124
122 7
106 8 9
113 10

4. STEP 1: Calculate the mean difference between pairs
Start IQ
End IQ D
111
110 1
116
117 -1
114 -3
109
115 4
124
122 2
106
113

5. STEP 1: Calculate the mean difference between pairs
Is this mean difference significantly different from zero?
(this is what a paired-samples t-test determines)

6. STEP 2: Calculate Standard Deviation of differences1 -1 -2 4 -3 -4 16 3 9 2

7. STEP 2: Calculate Standard Deviation of differences

8. STEP 3: Compute the t-statistic

9. STEP 4: Compute the Degrees of Freedom

10. Significance Level (α)
STEP 5: Look-up Critical t-value in Table of Values df
Significance Level (α)
0.1
0.05
0.01 1
6.314
12.706
63.657 2
2.920
4.303
9.925 3
2.353
3.182
5.841 4
2.132
2.776
4.604 5
2.015
2.571
4.032 6
1.943
2.447
3.707 7
1.895
2.365
3.499 8
1.860
2.306
3.355 9
1.833
2.262
3.250 10
1.812
2.228
3.169 11
1.796
2.201
3.106 12
1.782
2.179
3.055

11. STEP 6: Compare t-statistic with Critical t-value
Calculated t-statistic = 1.25
Critical t-value = 2.262
If calculated t-statistic is larger than critical t-value: SIGNIFICANT
If calculated t-statistic is smaller than critical t-value: NON-SIGNIFICANT
The calculated t-statistic is smaller than the critical t-value, therefore the mean IQ of students at the end of their degrees (113) was not significantly different from the mean IQ at the start of their degrees (114).

12. Summarise Findings
The IQ of students at the end of their degrees (M = 113) was not significantly different from the IQ at the start of their degrees
(M = 114), t(9) = 1.25, p > .05.