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Independent Samples t-test
Research Design
• An independent samples t-test is used to analyze data collected when two independent groups
(e.g., created through random assignment) are compared to determine whether their means are
significantly different from one another.
• The design involves separate and independent samples and makes a comparison between two
groups of individuals. These are between-participants designs.
o Experimental designs
▪ One sample may be drawn from a population and randomly assigned to an
experimental group (to which treatment is applied) or to a control group (which
remains representative of the underlying population).
▪ These two groups may then be compared to determine whether the treatment
‘made’ the experimental group significantly different from the control group, and
therefore representative of some other population – no longer representative of the
population from which it was drawn.
o Quasi-experimental designs
▪ Two pre-formed samples from two presumably different populations may be
compared to determine whether the means of the samples are different enough to
conclude that they did, in fact, came from populations that are significantly
different from one another.
The Independent Samples t-test
• As with other statistics, the independent-samples t-statistic forms a ratio of “good” variability
(the difference or effect) to “bad” variability (the error).
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• The numerator compares the difference between the means of the samples with the difference in
means of the populations from which those samples were drawn, under the null hypothesis.
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o If the null hypothesis is true – and the samples were drawn from the same population, or
from populations with the same mean – then the difference between population means
would be zero.
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o So, in effect, the numerator simply compares the two sample means (as estimates of the
means of the populations from which they were drawn) to determine if they are
significantly different from one another, thus indicating a significant difference between
their respective populations.
• The estimated standard error in the denominator establishes the expected average difference
between a sample statistic (the difference between the sample means) and the corresponding
population parameter (the difference between the population means) – in other words, how large
(on average) the difference is between the sample means (since the difference between the
population means would be zero under the null hypothesis).
• The degrees of freedom for the independent samples t-statistic are determined by the df values
for the two separate samples.
o The degrees of freedom for the t-statistic are computed by summing the degrees of
freedom for each of the two groups.
o The degrees of freedom for the t-statistic equal the total of the sample sizes minus two,
since one degree of freedom is lost for each of the two sample means calculated.
= 1 + 2 = (1 − 1) + (2 − 1) = 1 + 2 − 2
• Note that the variability of the scores within each group (or sample) reflects the consistency of
any treatment effect.
o If the ‘treatment’ impacts each participant similarly, then the scores within each group
will be clustered together, with relatively little variability.
o But if the ‘treatment’ has inconsistent effects for individuals within the same group
(under the same treatment condition), then variability within the groups will be large.
o The larger the variability within each group (or treatment condition), the larger the
estimated error (denominator), and the smaller the observed value of t as a ratio of the
difference between the means (numerator) relative to the variability in the denominator.
• In a non-directional hypothesis using a two-tailed test, the hypothesis tested (the null
hypothesis, or 0) is that there is no treatment effect – that is, no significant difference between
the means of the populations ( = 1 − 2 = 0) from which the samples were drawn.
o When the difference between the sample means (numerator) is much greater than the
expected variability or consistency of the differences between the means (denominator),
we obtain an extreme value for t – large positive and/or large negative, beyond the critical
value(s) of t – and conclude that the data are not consistent with the null hypothesis, and
our decision is to “reject 0.”
o But when the difference between the sample means is not much more than the expected
variability of the difference between the means, we obtain a t-statistic closer to zero, and
our decision is to “retain 0” and find support for the alternative hypothesis
(: ≠ 1: 1 − 2 ≠ 0).
• In a directional hypothesis, we use the predicted direction of the difference between the means
under the research hypothesis to formulate the alternative hypothesis and to determine the sign of
the critical value of t.
o If the difference between the means would be negative ( 1 − 2 < 0) under the research
hypothesis, then : < , with 0: 1 ≥ 2.
o If the difference between the means would be positive ( 1 − 2 > 0) under the research
hypothesis, then : > , with 0: 1 ≤ .
• To test hypotheses with the t-statistic, some basic assumptions must be met:
o The observations within each sample must be independent from one another.
o The populations from which the samples were selected must be normal.
o The populations from which the samples were selected must have equal variances.