PSYB06H3 Chap.7 Independent and Paired Samples t Tests
Independent and Paired Samples t Tests
Two samples, no known population
Most psychology studies know nothing about the population beyond the samples collected. The two-sample t tests handle that situation by comparing groups with each other rather than with a known mean. The first decision is structural: did different people supply the two sets of scores, or did the same people supply both?
Separate people call for an independent samples t test, while repeated or matched measurements call for a paired samples t test.
Independent samples
The independent test divides the difference between the two sample means by an error term built from both sample variances, each divided by its own group size.
Degrees of freedom equal the two sample sizes added together minus two, and groups of different sizes are acceptable.
Before testing, compare the variances with Hartley's F max, the larger variance over the smaller, and check the ratio against the tabled threshold for two conditions.
Paired samples
The paired test subtracts each participant's first score from the second, then asks whether the mean of those differences departs from zero.
Because the analysis runs on differences, any participant missing one score must be removed, and degrees of freedom count the remaining pairs minus one.
Pairing removes stable individual differences from the error term, which is why a modest average change can still be significant.
Beyond the decision
A complete answer states the assumptions, checks homogeneity of variance, screens for impossible or extreme scores and, where helpful, draws the means with standard error bars.
Effect size d expresses the mean difference in pooled standard deviation units, with 0.20, 0.50 and 0.80 marking small, medium and large effects. When the test fails to reject the null, the next step is a power analysis rather than a claim that the groups are the same.
What this chapter covers
- 01
Choosing Independent or Paired Designs
- 02
Hartley's F Max Check
- 03
Independent Samples t Statistic
- 04
Pooled Variance and Effect Size d
- 05
Paired Difference Scores
- 06
Deleting Incomplete or Impossible Data
Worked example · free
A paired test from summary values
- 1Hypotheses: the population mean difference equals zero versus does not equal zero.
- 1Standard error of the differences: 2.4 / √10 = 0.7589.
- 1Statistic: t = (1.8 − 0) / 0.7589 = 2.3717 with df = 10 − 1 = 9.
- 1Compare with the two-tailed critical value 2.262 for df = 9 and reject the null.
Key terms
- Independent Samples
- Two groups made of different participants, so each person contributes a score to only one condition.
- Paired Samples
- Two sets of scores from the same or matched participants, analysed as one column of difference scores.
- Homogeneity of Variance
- The assumption that the compared groups have similar variances, checked by hand with Hartley's F max.
- Hartley's F Max
- The ratio of the largest group variance to the smallest, compared with a tabled critical threshold.
- Difference Score
- One participant's second measurement minus the first, the unit a paired t test analyses.
- Cohen's d
- The mean difference divided by the pooled standard deviation, with 0.80 or more classed as a large effect.
Independent and Paired Samples t Tests FAQ
How do I decide between an independent and a paired t test?
Ask who produced each score. If every participant appears in only one condition, the groups are independent. If each participant, or each matched pair, contributes one score to both conditions, the design is paired and the analysis uses difference scores. Group sizes are a clue too: paired data always have equal counts once incomplete pairs are removed.
What happens to participants with a missing score?
In a paired design, a participant missing either measurement cannot produce a difference score, so the whole pair is deleted before analysis. In an independent design the remaining scores are kept and the groups simply end up with unequal sizes. Scores that are impossible, such as recall above the number of items shown, are removed in either design.
Why check the variances before an independent t test?
The comparison assumes the two populations spread similarly. Hartley's F max divides the larger sample variance by the smaller; a ratio below the tabled value for two conditions supports that assumption. Reporting the check shows the marker the test was appropriate, which the practice answers treat as part of a full solution.
Is a large t value the same as a large effect?
No. The t statistic grows as samples grow, so a tiny difference can become significant with enough participants. Effect size d divides the mean difference by the pooled standard deviation instead, which keeps it on the same scale whatever the sample size. Report both so a reader knows whether the difference is reliable and whether it is substantial.
Exam move
Build a two-column habit for every problem: first decide the design and write the matching formula, then run the variance check before the main test. Practise one raw-data and one summary-statistic problem for each test, and time yourself on reading the critical t for the right degrees of freedom.
Finish each solution with direction, significance and effect size in one sentence, and mark any nonsignificant result as needing a power analysis.
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