Difference of Means Test
- Classical design used
- N subjects are
randomly assigned to two groups.
assignment insures that individuals have an
equal chance of being in either treatment
- After treatment,
the individuals are measured on the dependent
- A test of
differences in means between groups provides
evidence for the treatment's effect.
- Features of good
- Approximation in
political and social research
- Little potential
for nor experience with experimental
- Instead, social
researchers tend to select out two populations (or
samples) on the basis of some control variable
between the two groups.
- This is not as
neat as psychologists or statisticians like, but
it's what social researchers usually have to work
- Related Samples
- The observations are
not made on independent samples but on matched
subjects or even the same subjects.
- Cases in both
groups chosen to be similar on certain factors
thought to be causes of the dependent
matched on sex, height, weight,
matched on GNP, size, type of
- Because some
sources of variation are controlled, a
matched-pairs test is designed to detect smaller
variations in the dependent variable -- such as
expenditures for defense, in the case of
design (paired data in SPSS)
- Uses the same
cases but takes observations before and
possible variation even more.
- In political
and social research, may be used with repeated
measures on the same states over time.
- If random factors
only make the determination, the test for differences
of means should be conducted as if the data were from
- If the groups are
matched before the experiment or if the same subjects
are used, the test of difference of means should be
conducted as if the data were from related
- In political and
social research, most tests for the difference between
means employ independent samples.
The appropriate test for
difference of means in SPSS: T-Test
- Available under the
Analyze Menu in SPSS 10
- Choose Compare
Independent-Samples T Test -- if your data are
divided into two unmatched groups, e.g.
- Democrats v.
- Men v.
- Undergraduates v.
Paired-Samples T Test -- if your data consists
of repeated measures on the same units, e.g.
- comparing same
students' performance on 1/3 and 2/3
- comparing same
states' voting turnout in 1996 and 2000
The conceptual model underlying both types of T-Test:
- compare the difference
of means for sample 1 with sample 2:
- Strictly speaking,
the test is for the difference computed between
the means of two samples, drawn and computed an
infinite number of times, compared with the
difference between the two means of their
- But since the two
population means, 1
are assumed to be equal, 1
= 0, so the second term drops out, and the test
becomes simply a test for the observed difference
between the sample means:
- As in the standard
"model" of hypothesis testing, this observed
difference in sample means can be evaluated by
dividing by some "standard error" allowable for sampling
- Assuming independent
samples, the STANDARD ERROR of the sampling
distribution of the DIFFERENCE IN MEANS is
- When 1
are known, the formula becomes
- When 1
are estimated by sample values and where: s = unbiased
estimate of ,
we can compute
- Thus, the formula
as an estimate of becomes
- Using estimated values
for the population variances produces the same formulate,
except that when SAMPLE SIZES are SMALL (30 cases or
less), the t-distribution must be used.
- For historical reasons,
the difference of means test is referred to as the
t-test, even thought the t distribution is really
necessary only for small samples.