Direct answer
Paired comparisons analyze within-pair differences; independent comparisons analyze separate sampling units, and Welch’s method estimates the independent mean difference without assuming equal population variances.
Visual explanation
Pairing changes the unit of variation
What this calculation tells you
Dependence determines the relevant unit of variation. Pairing can remove stable between-unit differences, while treating paired observations as independent misstates uncertainty.
Identify the sampling unit and why each observation is or is not linked to another. For independent means, use an interval and test that accommodate unequal spread unless a stronger model is justified.
Where it is used
Experiments
Analyze separate randomized groups or repeated outcomes according to design.
Quality
Compare matched units, instruments, or process states transparently.
Clinical research education
Separate repeated-patient data from independent cohorts without offering treatment conclusions.
Business analysis
Distinguish repeated-account comparisons from separate customer groups.
Common situations
- Analyzing before-and-after data.
- Comparing two independent samples with unequal spread.
- Evaluating matched pairs.
- Explaining why the sampling unit controls n.
Start with the statistical question
Identify the sampling unit and why each observation is or is not linked to another. For independent means, use an interval and test that accommodate unequal spread unless a stronger model is justified.
Paired points are connected across conditions, showing individual differences; independent groups are drawn as separate clouds. A Welch standard-error diagram retains both group variances.
Worked example
Before-and-after values on the same 20 people yield 20 differences. They do not form two independent samples of size 20, because each person contributes to both conditions.
Assumptions that carry the result
Paired inference assumes independent pairs and an appropriate model for their differences. Welch inference assumes independent observations within and between groups and a suitable mean-based approximation.
Interpret the result without overreaching
Neither method repairs confounding, regression to the mean, attrition, baseline imbalance, nonrandom assignment, or poorly measured outcomes.
- Pairing observations only because sample sizes match.
- Ignoring genuine within-person or matched dependence.
- Using a preliminary equal-variance test to decide mechanically between pooled and Welch tests.
Practical questions
Frequently asked questions
Can I pair observations after seeing the data?
Only with a defensible preexisting match or design; arbitrary post-hoc pairing can distort inference.
Why use Welch instead of pooled t?
Welch does not require equal variances and performs well across many independent-group settings.
What if paired differences are strongly skewed?
Inspect the differences, sample size, outliers, estimand, and robust or rank-based alternatives rather than the two marginal distributions alone.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.