**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.

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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.

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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,

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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,

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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.