The document describes the paired samples t-test, which is used to compare means when observations are paired or related. It explains that the paired t-test accounts for natural variability between observational units by looking at the differences between each pair of observations, rather than comparing independent groups. An example compares the mileage from two types of gasoline in the same taxis to control for differences between vehicles. The test determines if the average difference between the paired observations is significantly different from zero.

1. Quantitative Methods
Compare Means Test (T-test)
Independent samples and
Paired samples
Keiko Ono, Ph.D.
(2005)

2. Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)
Compare X by (Group)
• Compare group 1 to group2
• Ha: X1 bar > X2 bar OR X1 bar < X2 bar
Ho: X1 bar = X2 bar
• Assumption: two subsamples were drawn
independently
• Sample size equality not necessary
• Test variable: must be interval
• Group variable: categorical, ordinal, interval

3. Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)
• Compare Democrats and Republicans
• Compare Men and Women
• Compare Experimental group and Control group
• Compare drug and placebo
• Compare Brand A and Brand B
• Compare and

4. Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)
• Compare between two groups when there are more than
two groups
→ e.g. Liberal (coded “1”), Moderate (coded “3”),
Conservative (coded “5”).
To compare liberals and conservatives, specify Group1 =
1, Group 2 = 5

5. Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)
• Compare between groups based on interval level
variable
e.g. Feeling thermometer score for feminists
Group 1: respondents 30-years or older
Group 2: under 30
(“cutoff” value would be 30).
Group Statistics
1207 53.62 22.060 .635
213 58.97 20.954 1.436
Respondent age
>= 30
< 30
Thermometer feminists
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.040 .842 -3.283 1418 .001 -5.342 1.627 -8.535 -2.150
-3.403 301.023 .001 -5.342 1.570 -8.432 -2.253
Equal variances
assumed
Equal variances
not assumed
Thermometer feminists
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means

6. PID Bush FT
1 D 20
2 R 70
3 D 15
4 D 35
5 R 85
6 R 70
7 D 50
8 R 90
9 R 65
n=9
Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)

7. PID Bush FT
1 D 20
2 R 70
3 D 15
4 D 35
5 R 85
6 R 70
7 D 50
8 R 90
9 R 65
n=9
Sample 1 (Democrats)
n1=4
X1 bar= 30
s1=15.8
Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)

8. PID Bush FT
1 D 20
2 R 70
3 D 15
4 D 35
5 R 85
6 R 70
7 D 50
8 R 90
9 R 65
n=9
Sample 2 (Republicans)
n2=5
X2 bar= 76
s2 = 10.8
Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)
Sample 1 (Democrats)
n1=4
X1 bar= 30
s1=15.8

9. Two-Samples Compare Means Test
(aka Independent-Samples Compare Means Test)
Equal variance assumed (Tomlinson)

10. Two-Samples Compare Means Test
Example 1. Do men and women feel differently about the military?
(NES 2000)
Group Statistics
665 73.25 20.860 .809
852 72.19 20.130 .690
Gender
1. Male
2. Female
Post:Thermometer
military
N Mean Std. Deviation
Std. Error
Mean

11. Two-Samples Compare Means Test
Example 1. Do men and women feel differently about the military?
(NES 2000)
Group Statistics
665 73.25 20.860 .809
852 72.19 20.130 .690
Gender
1. Male
2. Female
Post:Thermometer
military
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.272 .602 1.001 1515 .317 1.060 1.058 -1.016 3.136
.997 1402.109 .319 1.060 1.063 -1.025 3.145
Equal variances
assumed
Equal variances
not assumed
Post:Thermometer
military
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means

12. Two-Samples Compare Means Test
Example 1. Do men and women feel differently about the military?
(NES 2000)
Group Statistics
665 73.25 20.860 .809
852 72.19 20.130 .690
Gender
1. Male
2. Female
Post:Thermometer
military
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.272 .602 1.001 1515 .317 1.060 1.058 -1.016 3.136
.997 1402.109 .319 1.060 1.063 -1.025 3.145
Equal variances
assumed
Equal variances
not assumed
Post:Thermometer
military
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
Confidence Interval

13. Two-Samples Compare Means Test
Example 1. Do men and women feel differently about the military?
(NES 2000)
Group Statistics
665 73.25 20.860 .809
852 72.19 20.130 .690
Gender
1. Male
2. Female
Post:Thermometer
military
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.272 .602 1.001 1515 .317 1.060 1.058 -1.016 3.136
.997 1402.109 .319 1.060 1.063 -1.025 3.145
Equal variances
assumed
Equal variances
not assumed
Post:Thermometer
military
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
Critical value of t = 1.96
t = Mean difference / S.E. of Difference

14. Two-Samples Compare Means Test
Example 1. Do men and women feel differently about the military?
(NES 2000)
Group Statistics
665 73.25 20.860 .809
852 72.19 20.130 .690
Gender
1. Male
2. Female
Post:Thermometer
military
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.272 .602 1.001 1515 .317 1.060 1.058 -1.016 3.136
.997 1402.109 .319 1.060 1.063 -1.025 3.145
Equal variances
assumed
Equal variances
not assumed
Post:Thermometer
military
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
P-value
At 95% confidence level, critical value of p (2-tailed) is .05. If one-tailed, divide by half (.025).

15. Two-Samples Compare Means Test
Example 1. Do men and women feel differently about the military?
(NES 2000)
Group Statistics
665 73.25 20.860 .809
852 72.19 20.130 .690
Gender
1. Male
2. Female
Post:Thermometer
military
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
.272 .602 1.001 1515 .317 1.060 1.058 -1.016 3.136
.997 1402.109 .319 1.060 1.063 -1.025 3.145
Equal variances
assumed
Equal variances
not assumed
Post:Thermometer
military
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
P-value

16. Group Statistics
659 61.93 22.127 .862
825 66.20 20.482 .713
Gender
1. Male
2. Female
D2t. Thermometer
environmentalists
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
1.430 .232 -3.852 1482 .000 -4.272 1.109 -6.447 -2.096
-3.819 1358.693 .000 -4.272 1.119 -6.466 -2.077
Equal variances
assumed
Equal variances
not assumed
D2t. Thermometer
environmentalists
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
Two-Samples Compare Means Test
Example 2. Do men and women feel differently about
environmentalists? (NES 2000)

17. Group Statistics
659 61.93 22.127 .862
825 66.20 20.482 .713
Gender
1. Male
2. Female
D2t. Thermometer
environmentalists
N Mean Std. Deviation
Std. Error
Mean
Independent Samples Test
1.430 .232 -3.852 1482 .000 -4.272 1.109 -6.447 -2.096
-3.819 1358.693 .000 -4.272 1.119 -6.466 -2.077
Equal variances
assumed
Equal variances
not assumed
D2t. Thermometer
environmentalists
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
Two-Samples Compare Means Test
Example 2. Do men and women feel differently about
environmentalists? (NES 2000)
P-value

18. Compare Means test – Paired Samples
Independent Samples Compare Means Test dealt
with comparing two independent groups (men vs.
women, Democrat vs. Republican, etc.)
Paired Samples test involves comparing
traits/characteristics of matched observations.
What does this mean?

19. • Compare before and after new treatment, new
drug, new law, etc.
• Compare t1 and t2 (e.g. GDP in 2005 and GDP
in 2000 across countries, the unit of analysis is
the country)
• Compare average male and female test scores
across schools (the unit of analysis: school)
• Compare opinions and evaluations of different
objects (the unit of analysis: survey
respondent)
Paired-Samples Compare Means Test

20. Independent Samples Compare Means Test
On average, are men more (or less) favorable toward Clinton than
women?
Respondent Sex Clinton Gore
Andy M 70 50
Matt M 60 45
Richard M 30 50
Dave M 25 40
Elaine F 50 60
Julia F 70 65
Rachel F 50 40
Margaret F 35 55
Male
Mean
Female
Mean

21. What’s wrong with this picture?
Respondent Clinton Rating Gore Rating
Andy 70
Elaine 60
Can we conclude Clinton is more popular (even with
increased n) based on this set of data?
No…because of Natural Variability

22. Paired Samples Compare Means Test
On average, is Clinton rating different from Gore rating?
Respondent Sex Clinton Gore
Andy M 70 50
Matt M 60 45
Richard M 30 50
Dave M 25 40
Elaine F 50 60
Julia F 70 65
Rachel F 50 40
Margaret F 35 55

23. To test the hypothesis, we use the difference
between the two original variables.
Respondent Sex Clinton Gore C-G
Andy M 70 50 20
Matt M 60 45 15
Richard M 30 50 -20
Dave M 25 40 -15
Elaine F 50 60 -10
Julia F 70 65 5
Rachel F 50 40 10
Margaret F 35 55 -20

24. To test the hypothesis, we use the difference
between the two original variables.
Respondent Sex Clinton Gore C-G
Andy M 70 50 20
Matt M 60 45 15
Richard M 30 50 -20
Dave M 25 40 -15
Elaine F 50 60 -10
Julia F 70 65 5
Rachel F 50 40 10
Margaret F 35 55 -20
Ho:
X bar = 0
Ha:
X bar<0
or
Xbar >0

25. The Logic of paired samples test
Say someone has developed a new kind of hand
cream. She claims the new cream is far superior to
the conventional one. How do we test this
proposition?
OLD NEW

26. One way is to assign the old cream to one group of
experimental subjects and give the new one to
another group.
However, there is natural variability due to skin
differences among research subjects.

27. In other words, our hands are different from our
neighbors’.
So, a better way to test the difference
between the two brands is…

28. Randomly assign the two brands to each subject’s
right or left hands! This eliminates variability due
to skin differences.

29. Randomly assign the two brands to each subject’s
right or left hands! This eliminates variability due
to skin differences.

30. Paired-Samples Compare Means Test
Example
Which gives better mileage, Gasoline A
or Gasoline B?

31. Paired-Samples Compare Means Test
Better mileage, Gasoline A or
Gasoline B?
Taxi # Gasoline mileage
1 A 25.6
2 A 32.4
3 A 28.6
4 A 31.2
5 A 29.8
6 A 27.9
7 B 24.9
8 B 26.7
9 B 30.6
10 B 29.8
11 B 30.7
12 B 28.4
One method is to randomly assign
gasoline A or B to cars and compare the
means.
Gasoline A
Mean
Gasoline B
Mean

32. Paired-Samples Compare Means Test
Better mileage, Gasoline A or
Gasoline B?
Problem: Natural variability in driving habits
and conditions of the car

33. Paired-Samples Compare Means Test
A better method is to assign
gasoline A and B to the same
cars and compare the means.
Taxi # Gasoline A Gasoline B
1 25.6 24.9
2 32.4 26.7
3 28.6 31.2
4 31.2 30.7
5 29.8 29.5
6 27.9 28.7
7 25.9 30.6
8 26.5 28.4
9 31.3 25.7
10 29.5 29.4
11 31.2 32.8
12 28.8 25.2

34. Paired-Samples Compare Means Test
We hypothesize if there were no difference between Gasoline A and Gasoline B,
on average, the difference would be zero (this is the null hypothesis).
Taxi # Gasoline A Gasoline B Difference (A-B)
1 25.6 24.9 0.7
2 32.4 26.7 5.7
3 28.6 31.2 -2.6
4 31.2 30.7 0.5
5 29.8 29.5 0.3
6 27.9 28.7 -0.8
7 25.9 30.6 -4.7
8 26.5 28.4 -1.9
9 31.3 25.7 5.6
10 29.5 29.4 0.1
11 31.2 32.8 -1.6
12 28.8 25.2 3.6

35. Paired-Samples Compare Means Test
We hypothesize if there were no difference between Gasoline A and Gasoline B,
on average, the difference would be zero (this is the null hypothesis).
Taxi # Gasoline A Gasoline B Difference (A-B)
1 25.6 24.9 0.7
2 32.4 26.7 5.7
3 28.6 31.2 -2.6
4 31.2 30.7 0.5
5 29.8 29.5 0.3
6 27.9 28.7 -0.8
7 25.9 30.6 -4.7
8 26.5 28.4 -1.9
9 31.3 25.7 5.6
10 29.5 29.4 0.1
11 31.2 32.8 -1.6
12 28.8 25.2 3.6
Ho:
X bar = 0
Ha:
X bar<0
or
Xbar >0

36. Paired-Samples Compare Means Test
Example
Paired Samples Statistics
55.43 1771 29.675 .705
57.57 1771 25.663 .610
Pre:Thermometer Bill
Clinton
Pre:Thermometer Al Gore
Pair
1
Mean N Std. Deviation
Std. Error
Mean
Paired Samples Correlations
1771 .720 .000
Pre:Thermometer Bill
Clinton &
Pre:Thermometer Al Gore
Pair
1
N Correlation Sig.
Paired Samples Test
-2.141 21.042 .500 -3.121 -1.160 -4.281 1770 .000
Pre:Thermometer Bill
Clinton -
Pre:Thermometer Al Gore
Pair
1
Mean Std. Deviation
Std. Error
Mean Lower Upper
95% Confidence
Interval of the
Difference
Paired Differences
t df Sig. (2-tailed)

37. Paired-Samples Compare Means Test
Example
Paired Samples Statistics
55.43 1771 29.675 .705
57.57 1771 25.663 .610
Pre:Thermometer Bill
Clinton
Pre:Thermometer Al Gore
Pair
1
Mean N Std. Deviation
Std. Error
Mean
Paired Samples Correlations
1771 .720 .000
Pre:Thermometer Bill
Clinton &
Pre:Thermometer Al Gore
Pair
1
N Correlation Sig.
Paired Samples Test
-2.141 21.042 .500 -3.121 -1.160 -4.281 1770 .000
Pre:Thermometer Bill
Clinton -
Pre:Thermometer Al Gore
Pair
1
Mean Std. Deviation
Std. Error
Mean Lower Upper
95% Confidence
Interval of the
Difference
Paired Differences
t df Sig. (2-tailed)
Confidence Interval
P-value
t = (mean of difference / (S.D. of difference/√n))

38. • Compare before and after new treatment, new
drug, new law, etc.
• Compare t1 and t2 (e.g. GDP in 2005 and GDP
in 2000 across countries, the unit of analysis is
the country)
• Compare average male and female test scores
across schools (the unit of analysis: school)
• Compare opinions and evaluations of different
objects (the unit of analysis: survey
respondent)
Paired-Samples Compare Means Test

39. GDP 2000 GDP 2005 D
Chile
Mexico
Argentina
Ecuador
Peru
Columbia
Venezuela
Cholesterol Level
Before drug After drug D
Patient A
Patient B
Patient C
Patient D
Patient E
Patient F
Patient G
Patient H
Paired-Samples Compare Means Test
Average SAT scores
Female Male D
OU
Texas
Kansas
Missouri
Nebraska
Texas A&M
Rice
Arkansas