What is a paired samples t test

Another important test of differences is the t-test for
paired samples. This is also known as a t-test for
repeated measures or a t-test for matched samples.

Whenever two distributions of a dependent variable are highly
correlated, either because they are distributions of pre and post
tests from the same people,

tests from the same people,
one month later

tests from the same people, or from two samples that are matched
such that there is a one-to-one correspondence between each
subject in group one and its matched pair in group two,

one month later
Brown
Haired
Matched
Subjects

one month later
one month later
Brown
Haired
Matched
Subjects
Red
Haired
Matched
Subjects

subject in group one and its matched pair in group two, the
appropriate test of differences is the paired samples t-test.

If we have a single dependent variable that exists on an
interval or ratio scale such as scores on a test,

and is reasonably normally distributed;

and a single independent variable (e.g., when you took
the test)

the test) which has two levels

the test) which has two levels which are repeated or
matched,

the test) which has two levels which are repeated or
matched, then we use the pair wise t-test to test for
differences between the two samples of the dependent
variable.

First we begin with the null hypothesis:

There are no significant difference in pre and post scores
(dependent variable).

Or: There are no significant difference between group one
and its matched group in terms of the dependent variable.

Or: There are no significant difference between group one
and its matched group in terms of the dependent variable.
no difference

Another way to state the null-hypothesis for a paired-sample
t-test is by hypothesizing that the difference
between the pre-post test or matched pairs is ZERO.

Name _________
Name _________
1. The word of the day is ______
2. The State bird of Oklahoma is ______
3. The favorite dessert of Queen Elizabeth is
2. The State ________
bird of Oklahoma is ______
4. The capital of Siam is ___________
5. ________
The best movie of 1972 was ___________
6. The worst spelling bee in the world is _______
7. What is the capital of Mars?
8. How many people fit in a subway car in Tokyo?
9. The best episode of the Doctor Who reboot is
5. The best movie of 1972 was ___________
8. How many ___________________________
people fit in a subway car in Tokyo?
10. The worst thing you can do with silly putty is
___________________________
________________
11. How heavy is a 15 pound bowling ball?
12. ________________
What is the best flavor of Mamba fruit chews?
13. The weather in Alaska is __________________
14. Bugs Bunny is funniest dressed up as a
12. What is the best flavor of Mamba fruit chews?
13. The weather _____________
in Alaska is __________________
15. How much does a taxi cost in London?
16. _____________
What is the best flavor Jello Pudding pop?
17. How far is Wichita from Baghdad?
18. What is the most important thing to do on Arbor
16. What is the best flavor Jello Pudding pop?
18. 19. What What is the goes most best important with vanilla thing to custard?
do on Arbor
Day?
Day?
19. What goes best with vanilla custard?
Name _________
________
___________________________
________________
_____________
Day?

Name _________
Name _________
2. The State ________
bird of Oklahoma is ______
5. ________
The best movie of 1972 was ___________
8. How many ___________________________
people fit in a subway car in Tokyo?
___________________________
________________
12. ________________
What is the best flavor of Mamba fruit chews?
13. The weather _____________
in Alaska is __________________
16. _____________
What is the best flavor Jello Pudding pop?
18. 19. What What is the goes most best important with vanilla thing to custard?
do on Arbor
Day?
Day?
Name _________
________
− ___________________________
= 0
________________
_____________
Day?

Conceptually, aggregating the exact differences
between each pair makes most sense and leads to the
correct degrees of freedom, sampling distribution and
critical value.

Conceptually, aggregating the exact differences
between each pair makes most sense and leads to the
correct degrees of freedom, sampling distribution and
critical value.
Correct Degrees of Freedom
– 50 people take a pre-test
– Same 50 take a post test
– 50 – 1 = 49 degrees of freedom

Correct Sampling Distribution
− =
Sample Mean Distribution of
difference between the pre and
post samples test scores
pre-test scores.
post-test scores.

The formula for the paired samples t-test is:

Σ(X1-X2)
SEdiff
t =

Σ(X1-X2)
SEdiff
t =
Paired t-test value or
the number of
standard error
values that separate
these two means

Pre Post
1 7
2 6
1 8
Σ(Xpre-Xpost)
SEdiff
t =

Pre Post
1 7
2 6
1 8
Σ(Xpre-Xpost)
SEdiff
t =
Subtract each pretest
score from each
posttest score and
the sum them up

Pre Post
1 7
2 6
1 8
Σ(Xpre-Xpost)
SEdiff
t =
Difference
6
4
7

Pre Post
1 7
2 6
1 8
Σ(Xpre-Xpost)
SEdiff
t =
Difference
6
4
7
Sum of all
Differences
6 + 4 + 7 = 17

Pre Post
1 7
2 6
1 8
Σ(Xpre-Xpost)
SEdiff
t =
Difference
6
4
7
Sum of all
Differences
6 + 4 + 7 = 17
17
SEdiff
t =

Σ(Xpre-Xpost)
SEdiff
t =

Σ(Xpre-Xpost)
SEdiff
t =
Now let’s calculate
the Standard Error of
the difference

Σ(Xpre-Xpost)
SEdiff
t =
Now let’s calculate
the Standard Error of
It is this statistic (Standard Error of
the Difference) that makes it possible
the difference
to determine if this difference occurred only by chance
or if there is a strong probability that they are actually
different!

One thing to note: If the Standard Error of the
Difference is large (say 170) then the t value will be
small.

small.
For example:

small.
For example:
Σ(Xpre-Xpost)
SEdiff
t =

small.
For example:
Σ(Xpre-Xpost)
SEdiff
t =
17
170
t =

small.
For example:
Σ(Xpre-Xpost)
SEdiff
t =
17
170
t =
t = .01

However, if the Standard Error of the Difference is
small (say 1.7) then the t value will be larger.

For example:

For example:
Σ(Xpre-Xpost)
SEdiff
t =

For example:
Σ(Xpre-Xpost)
SEdiff
t =
17
1.7
t =

For example:
Σ(Xpre-Xpost)
SEdiff
t =
17
1.7
t =
t = 10

So what effect will a smaller or larger estimated
standard error have on whether a result is statistically
significantly different or not?

Well let’s say that our null hypothesis (Ho) is that there
is no statistically significant difference between the pre
and post test scores below:

Well let’s say that our null hypothesis (Ho) is that there
is no statistically significant difference between the pre
and post test scores below:
Students Pre Post
1 1 7
2 2 6
3 1 8
mean: 1.3 6.0

Of course, the alternative hypothesis would be that
there is no statistically significant difference between
the two.

the two.
With a t value of .01

the two.
With a t value of .01
Σ(Xpre-Xpost)
SEdiff
t =
17
170
t =
t = .01

the two.
With a t value of .01, and a sample of 3, and therefore
degrees of freedom of 2, we would look up the t critical
value at a .05 alpha level (a .05 alpha level means that
we are willing to consider an outcome to be significant
if it were likely to happen 5 times out of 100 (.05), in
other words, if it were a very rare occurrence.)

So we go to our table of t-Distribution Critical Values
to find the critical value that needs to be exceeded by
our result in order be considered statistically
significantly different.

So, the critical value we need to
exceed in order to reject the null
hypothesis is 2.920.

However, our t-value is

However, our t-value is
t = .01

However, our t-value is .01

Our t value of 0.1 does not exceed the t critical of
2.920, therefore we will fail to reject the null-hypothesis
(which essentially means to accept the null-hypothesis).

On the other hand, when our t value is 10, under the
same conditions,

same conditions,
Σ(Xpre-Xpost)
SEdiff
t =
17
1.7
t =
t = 10

same conditions, our t value of 10 does exceed the t-critical
of 2.920, therefore we will reject the null-hypothesis
(which essentially means to accept the
alternative hypothesis).

So when it comes to inferential statistics (inferring
meaning to a larger population from a smaller sample),
the size of the standard error determines everything.

So when it comes to inferential statistics (inferring
meaning to a larger population from a smaller sample),
the size of the standard error determines everything.
Understanding the standard error theoretically may or
may not be important for your learning purposes. If it
is not, you may want to click quickly through the next
10 slides. If it is important then consider what follows:

If we took 1000 samples of pre- and post-tests and
subtracted each pre-test sample from each post-test
sample we would have a sampling distribution called
the sampling distribution of differences between pre-and
post-test samples.

post-test samples.
− =
pre-test scores.
post-test scores.

post-test samples.
pre-test scores.
post-test scores.

post-test samples.
pre-test scores.
post-test scores.
−

post-test samples.
pre-test scores.
− =
post-test scores.

post-test samples.
pre-test scores.
− =
post-test scores.
etc …

If you calculated the standard deviation of this new
subtracted sampling distribution you would have the
actual standard error we are looking for for this
equation.

If you calculated the standard deviation of this new
subtracted sampling distribution you would have the
actual standard error we are looking for for this
equation.
Since this is almost impossible to do, the standard error
will be estimated.

Let’s see how this is done with a very simple data set.
Let’s begin with our null-hypothesis: “Post-test scores
are not statistically significantly higher than pre-test
scores”.

Let’s see how this is done with a very simple data set.
Let’s begin with our null-hypothesis: “Post-test scores
are not statistically significantly higher than pre-test
scores”.
Students Pre Post
1 1 7
2 2 6
3 1 8
mean: 1.3 7.0

We will calculate each element of the equation below:

Let’s begin with the sum of the differences between
the pre and post tests:

What is a paired samples t test

What is a paired samples t test

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