Independent and Paired Sample t-Tests
The reader will be able to:
n Understand the use of Dependent Samples
t Test as a test of the
difference between two related samples
n Interpret the results through the probability level that the null
hypothesis is true
Similar to Independent Samples t Test, Dependent Samples t Test
compares two means. However, these two means come from one sample
with repeated measures or from two matched samples. And since the
data is from one sample or related samples, we perform a Dependent
Samples t Test. Thus, the purpose of Dependent Samples t Test is to
compare the difference between two related or depended samples. For
example, if students learning English as a second language took a place-
ment test at the beginning of the school year, and took the test again at
the end of the school year, this would be an example of one sample
with two repeated measures, the same students measured twice over
time. This means that for each student there are two test scores. This
kind of before and after comparison is also known as a within-subject
design or repeated-measures design, which is very common in the edu-
cation-related contexts where knowledge gained over time is measured.
The advantage of this design is that since there is just one sample, this
controls for any pre-existing individual differences between samples.
In addition, having one sample is economical. However, participants’
repeated task such as repeated tests may show an improvement in scores
just from practice.
In another example, let us further explore Example 2 from Chapter
21. In this example, pairs of same-sex siblings (two brothers or two
134
The purpose of
Dependent Samples
t Test is to compare
the difference between
two related or
depended samples.
Pyrczak, Fred, and Deborah M. Oh. Making Sense of Statistics : A Conceptual Overview, Taylor & Francis Group, 2018. ProQuest Ebook Central,
http://ebookcentral.proquest.com/lib/capella/detail.action?docID=5425416.
Created from capella on 2022-07-31 15:40:47.
Copyright © 2018. Taylor & Francis Group. All rights reserved.
sisters) comprised the samples. They are pairs of siblings. Through a
coin toss, they were put into one of either group: one that received a
vitamin supplement and the other that received a placebo. For each
person in the experimental group, there was a same-sex sibling in the
control group. So, these samples are two matched samples. In measuring
the effects of the vitamin supplement intake, it makes sense to study
siblings of same-sex who are similar in their family diet and habits,
similar family setting, etc.
Different from independent samples in a Dependent Samples t Test,
the matching or paired participants in the depended groups have already
built in a similarity between the groups or repeated data points, making
it possible to reduce error.
In both examples, the Dependent Samples t Test procedure requires
one continuous data and one categorical data with two categories. This
is no different from Independent Samples t Test, as well as the assump-
tion of normality of data, but the Dependent Samples t Test assumes
that while the samples are related, the observations are independent. 1
(See the summary table, Table 21.1 in Chapter 21.)
Suppose when the children’s energy level was compared between
the experimental group and the control group, the experimental group
showed on average the energy level of 7.4 on a scale of 0 to 10, with
10 being the highest level of energy. The control group on the other
hand showed an average of 5.55. The null hypothesis states that the
difference between the two groups is equal to zero (0) or that there is
no significant difference between the two groups in their energy level.
Null Hypothesis: H 0:
1 –
2 = 0 or
1 =
2
Alternatively, the other possible explanation for this mean difference is
that there is a true difference in mean energy levels between the siblings
assigned to either one of the groups.
Alternative Hypothesis: H A:
1 –
2 0 or
1
2
The degree of freedom (df) in the Dependent Samples t Test is N,
which represents the total number of pairs, minus 1:
df = N – 1
In calculating for the t value, we take the difference between the two
related samples. In the case of one sample with repeated measures,
assuming that the “after” scores are higher than the “before” scores, we
Chapter 23: Dependent Samples
t Test
135
Dependent Samples t
Test compares means
just like the
Independent Samples
t Test.
The degrees of
freedom in the
Dependent Samples
t
Test is
N–1.
Pyrczak, Fred, and Deborah M. Oh. Making Sense of Statistics : A Conceptual Overview, Taylor & Francis Group, 2018. ProQuest Ebook Central,
http://ebookcentral.proquest.com/lib/capella/detail.action?docID=5425416.
Created from capella on 2022-07-31 15:40:47.
Copyright © 2018. Taylor & Francis Group. All rights reserved.
Part F: Means Comparison
136
take the difference between the two means and divide this difference
by the standard error of the difference between the means. See the
formula below:
In the Dependent Samples t Test, the standard error of the differ-
ence between means is calculated differently from the Independent
Samples t Test, however the same concept applies. The standard error
of the difference between means is the average expected difference
between dependent sample means. (See Appendix A for the computa-
tion formula.)
Table 23.1 shows the results of the comparison of the energy levels
between siblings.
t = Difference between two dependent sample meeans
Standard error of the difference betweeen means
Table 23.1 Results for Dependent Samples t Test Comparing the
Experimental and Control Groups
Group Number Cor. P Mean SD SE of
of Pairs Mean
Experimental 20 .201 .395 7.40 1.095 .245
Control 5.55 1.099 .246
Paired difference
Mean SD SE of T Value df P
Difference Mean
1.85 1.39 .310 5.965 19 .000
According to the results, the probability (P) that the t value of 5.965
occurred by chance or due to sampling error is .000. Therefore, it can
be concluded that the null hypothesis is rejected. There is a significant
difference between the siblings in their energy levels. In a journal, one
might state the following:
a paired samples analysis produced a significant t value [t (19) =
5.965, p < .001]. There was a significant difference in the energy
level for the experimental group (Mean = 7.40, SD = 1.095)
compared to the control group (Mean = 5.55, SD = 1.099). Siblings
in the experimental group that took the vitamin supplement showed
Pyrczak, Fred, and Deborah M. Oh. Making Sense of Statistics : A Conceptual Overview, Taylor & Francis Group, 2018. ProQuest Ebook Central,
http://ebookcentral.proquest.com/lib/capella/detail.action?docID=5425416.
Created from capella on 2022-07-31 15:40:47.
Copyright © 2018. Taylor & Francis Group. All rights reserved.
a significantly different energy level than the others in the control
group who did not take the vitamin supplements.
Notice that the statement says that there was a significant difference,
and not “significantly higher” energy level. This is because we performed
a nondirectional hypothesis test. But just the same, since the test estab-
lished a significant difference, we can safely assume that the experi-
mental group showed a significantly higher mean.
Chapter 23: Dependent Samples
t Test
137
Exercise for Chapter 23
Factual Questions
1. How is the Dependent Samples t Test different from Independent Samples t Test in terms
of samples?
2. What are the similarities in the assumptions behind both the Dependent and Independent
Samples t Test?
3. What is Dependent Samples t Test with one sample also known as?
4. In a Dependent Samples t Test what is the df for a sample of 100 participants in a particular
study?
5. The Dependent Samples t Test would have less sampling error in its design. Discuss the
reasons.
6. Given the results below what would you conclude about the mean difference between Group
A and Group B?
Group Number Cor. P Mean SD SE of
of Pairs Mean
Group A 50 .087 .549 7.64 1.03 .145
Group B 4.82 1.32 .187
Paired difference
Mean SD SE of T Value df P
Difference Mean
2.82 1.60 .226 12.46 49 .000
Pyrczak, Fred, and Deborah M. Oh. Making Sense of Statistics : A Conceptual Overview, Taylor & Francis Group, 2018. ProQuest Ebook Central,
http://ebookcentral.proquest.com/lib/capella/detail.action?docID=5425416.
Created from capella on 2022-07-31 15:40:47.
Copyright © 2018. Taylor & Francis Group. All rights reserved.
Question for Discussion
7. Give an example of one continuous data and one categorical data
with two categories that would require performing a Dependent
Samples t Test.
