The Paired t-Test
(A.K.A. Dependent Samples t-Test,
or t-Test for Correlated Groups)
Advanced Research Methods in Psychology
- lecture –
Matthew Rockloff
1
When to use the paired t-test
• In many research designs, it is helpful to measure
the same people more than once.
• A common example is testing for performance
improvements (or decrements) over time.
• However, in any circumstance where multiple
measurements are made on the same person (or
“experimental unit”), it may be useful to observe if
there are mean differences between these
measurements.
• The paired t-test will show whether the
differences observed in the 2 measures will
be found reliably in repeated samples.
2
Example 4.1
• In this example, we will look at the throwing
distance for junior varsity javelin toss (in
meters).
• Five players are selected at random from
the entire league.
• We are interested in the following research
question:
• Do players improve on their distance
between the pre and post season?
• The average throwing distance, in both pre
and post season, is recorded in columns
1 and 2 (see next slide) for each of 5
people (P1-P5):
3
Example 4.1 (cont.)
Column 1
Column 2
Column 3
Column 4
X1 :
X2 :
Pre-season
Post-season
P1:
1
2
1
4
P2:
2
4.5
0
0.25
P3:
2
3
0
1
P4:
2
4.5
0
0.25
P5:
3
6
1
4
1  2
2  4
s x2 
2
(



)

n
( 1  1 ) 2 ( 2   2 ) 2
 0.4
1.9
4
Example 4.1 (cont.)
• Unlike the independent samples t-test, on
each row the numbers in columns 2 and 3
come from the same people.
• Person 1, for example, threw an average of
1 meter pre-season, but improved to an
average of 2 meters in the post-season
(after all competition was completed).
• It appears that this player may have
improved through practice.
• How can we find if the league has improved
overall from the pre to the post season?
5
Example 4.1 (cont.)
• The paired t-test will allow us to see if the
improvement that we see in this sample is
reliable.
• If we selected another 5 players
at random from the league,
would we still see an improvement?
• Without having to go through the trouble
and expense of repeated sampling (called
replication), we can estimate whether the
difference in the 2 means is so large in
magnitude that we would likely find the
same result if we chose another 5 persons.
6
Example 4.1 (cont.)
t
1   2
S  S  2rx1x2 S x1 S x2
2
x1
, df = n-1
2
x2
n 1
7
Example 4.1 (cont.)
• This paired “t” needs a couple more
values that we have not yet computed.
• First, we need to find the Standard
Deviation of X1 and X2, called Sx1 and
Sx2.
• These are simply the square-root of
the variances
( S x2  0.4  0.6325and S x2  1.9  1.3784 ).
1
2
8
Example 4.1 (cont.)
• Second, we need to find the correlation
between the pre and post-season distances
( rx x ), or likewise columns 2 and 3.
1 2
• Another section will illustrate how to
compute a correlation.
• This computation is somewhat long, so
we’ll avoid it for now.
• I’ll just tell you the correlation is:
rx1x2=0.9177.
• Any scientific or statistical calculator can
get you this answer.
9
Example 4.1 (cont.)
t
24
.4  1.9  2(.9177)(.6325)(1.3784)
5 1
 -4.78, df = 4
10
Example 4.1 (cont.)
• Finally, this computed “t” statistic must be
compared with the critical value of the tdistribution.
• The critical value of the “t” is the highest
magnitude we should expect to find if there
is really no difference between the
population means of X1 and X2, or in other
words, no difference between performance
in the pre and post season in the league.
• Since we expect there should be
improvement in throwing distance,
this is a 1-tailed test.
11
Example 4.1 (cont.)
• The C.V. t(4), α=.05 = 2.132, therefore we reject
the null hypothesis because the absolute
value of our “t” at 4.78 is greater than the
critical value.
• This is a 1-tailed t-test, so we must verify
this conclusion by noting that the mean of
the post season at 4 meters, is greater than
the mean of the pre-season throw average
of 2 meters.
12
Example 4.1 (cont.)
Our research conclusion states the
facts in simple terms:
Throwing distances increased
significantly from the pre-season
(M = 2) to the post-season (M = 4),
t(4) = 4.78, p < .05 (one-tailed).
13
Example 4.1 Using SPSS
• First, we must setup the variables in SPSS.
• Although not strictly necessary, it is good
practice to give a unique code to each
participant (“personid”).
• Unlike the independent samples t-test, the
paired t-test has separate entries for 2
dependent variables, rather than an
independent and dependent:
– DependentVariable1 = preseas
(for Pre-season scores)
– DependentVariable2 = postseas
(for Post-season scores)
14
Example 4.1 Using SPSS (cont.)
• In our example, the variables are setup as
follows in the SPSS variable view:
15
Example 4.1 Using SPSS (cont.)
• It is no longer necessary to provide “codes” (or
values) for the independent variable, simply
because one does not exist! We can proceed to
typing in the data in the SPSS data view:
16
Example 4.1 Using SPSS (cont.)
• Notice, this is where the “personid” variables has
helped.
• If we had incorrectly tried to analyze this problem as
an independent samples t-test, then we would have
coded for 10 people under “personid.”
• Of course, since we have only 5 people in this
example, this would have been incorrect.
• The personid variable thus allows a simple check
for whether we have typed-in the data correctly.
• The number of “rows” in SPSS should always equal
the number of “subjects” (or likewise, experimental
units).
17
Example 4.1 Using SPSS (cont.)
• Next, we need the SPSS syntax to run
a paired t-test. The code is as follows:
t-test pairs = DependentVariable1 DependentVariable2.
• In our example, the following code is written:
18
Example 4.1 Using SPSS (cont.)
After running the syntax, the following
appears in the SPSS output viewer:
19
Example 4.1 Using SPSS (cont.)
• You should focus your attention first of the
mean values for the pre and the post
season performance.
• As before, the means (Pre-season=2 and
Post-season=4) give us our conclusion.
• Namely, we conclude that performance
increased from the pre to the post season.
• The statistics tell us that our conclusion is
true not only for this sample of 5 persons,
but also for other samples of 5 persons in
the league.
20
Example 4.1 Conclusion
• Our test is 1-tailed, so we must divide the
2-tailed probability provided by SPSS in
half (p=.009/2 = .0045).
• When expressed to 2 significant digits, this
value will round to “.00” and as a result the
lowest value that can be represented in APA
style is “p<.01.”
• In short, we can now write our conclusion
as follows:
Throwing distances increased
significantly from the pre-season
(M = 2) to the post-season (M = 4),
t(4) = 4.78, p < .01 (one-tailed).
21
Thus
concludes
The Paired t-Test
(A.K.A. Dependent Samples t-Test,
or t-Test for Correlated Groups)
Advanced Research Methods in Psychology
- Week 3 lecture –
Matthew Rockloff
22