MATH2209 Chap.7 Independent and Paired Sample Tests
Independent and Paired Sample Tests
Two sets of measurements can arise from unrelated groups or from linked observations on the same units. That design distinction controls the analysis. This chapter reduces paired data to within-pair differences, keeps subtraction direction explicit, and routes independent samples through normality and variance checks.
Levene's test is treated as a variance-row selector rather than evidence about population means, preventing two p-values from being confused.
What this chapter covers
- 01
Independent and paired designs
- 02
Within-pair differences
- 03
Paired t test
- 04
Independent-samples t test
- 05
Normality checks
- 06
Levene's equality-of-variance test
Worked example · free
Test a paired mean change
- 1The six linked differences form one sample.
- 1SE = 1.414 / sqrt(6) = 0.577.
- 1t = -2.0 / 0.577 = -3.47 with df = 5.
- 1The negative sign means the after value is lower under the chosen subtraction.
Key terms
- Paired Difference
- The consistently ordered subtraction between two linked measurements.
- Independent Samples
- Samples whose observational units do not belong to one another in pairs.
- Levene's Test
- A test used to assess equality of variances for independent-group analysis.
Independent and Paired Sample Tests FAQ
How do I recognise paired data?
Look for repeated measurements on the same unit or deliberate matching between units. Before-and-after readings, left and right measurements, and matched case-control pairs are linked. Two columns alone do not establish pairing.
Which normality assumption applies to a paired test?
Inspect the distribution of within-pair differences, because those differences are the observations analysed by the paired t test. The two raw measurement columns can each look acceptable while their differences contain a severe outlier, or the reverse.
What does Levene's p-value decide?
It decides which independent-samples variance row is appropriate. A small value points to unequal variances; a larger one allows the equal-variance row. It does not test the equality of group means, which is addressed by the t-test p-value.
How should I audit a two-sample analysis?
Begin with the unit that produced each value. Repeated measurements or deliberately matched units require paired differences; unrelated participants require an independent comparison. For paired work, verify the subtraction direction, number of complete pairs, distribution of differences and n minus one degrees of freedom.
For independent work, inspect each group, record sample sizes and distinguish the Levene p-value from the mean-test p-value. Use the variance row supported by the assumption evidence without changing the design label. Report the estimated difference in the same group order used in the calculation, with a confidence interval and units.
Finish by separating evidence of a population difference from any causal claim, which depends on allocation and the wider study design.
Assessment move
For every two-group question, draw lines between values that belong together. Repeated measures on the same person, matched units and naturally linked measurements are paired; separate participants in two groups are independent. Do not infer pairing merely because data appear in two columns. Write the design decision before choosing a formula, because the units of analysis differ.
For paired data, select a consistent subtraction direction and calculate one difference for each pair. Plot or list those differences, then find their mean, standard deviation and standard error. The one-sample t procedure is applied to the differences, with degrees of freedom equal to the number of pairs minus one. Check the distribution of differences rather than checking the two raw columns separately.
Interpret the population mean difference with the subtraction direction intact. For independent groups, record both sample sizes, means and standard deviations. Distinguish the Levene p-value, which informs the variance row, from the t-test p-value, which addresses the mean comparison. Label them before comparing either with alpha.
If unequal variances are indicated, use the corresponding output and degrees of freedom rather than changing the research design. Practise calculating a difference in sample means and describing its sign relative to the group order.
Build mixed scenarios and classify them without numbers: two treatments assigned to different participants, before-and-after scores, left and right measurements, siblings deliberately matched, and two unrelated clinics. Then add numbers and solve only after the design is settled. For every conclusion, name both populations, the parameter being compared and the direction of the estimate.
Pair the p-value with a confidence interval and avoid converting association into causation unless allocation and the wider design support it. End revision with an error audit: wrong design, reversed difference, incorrect degrees of freedom, variance-row confusion and a conclusion about sample means instead of population means. Explain how each error changes the analysis.
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