An experiment applying A and B to different batches, and one splitting each donor's cells into A and B, both produce two columns of numbers. Their error units are not the same.

Independence and pairing are generating relationships, not table shapes

For independent samples, an A observation has no natural match to a particular B observation: different donors, batches, animals, or independent cultures are examples. The SE of the mean difference reflects variation in both groups.

For paired samples, two values belong together—for example before/after values, left/right, or a matched pair from the same experimental unit. The analysis target is not two raw columns, but the pairwise difference dᵢ=Bᵢ−Aᵢ. Large baseline differences among units may cancel within pairs.

Do not create pairs after the fact by joining rows that look similar; pairing belongs to the design. Conversely, discarding genuine pairing and using an independent analysis can lose useful information and power.

Two independent means have at least two standard-error models

The mean difference x̄B−x̄A can be identical while its SE differs.

Pooled t combines the two sample variances under an equal-population-variance assumption. Welch t uses sA²/nA + sB²/nB, reflecting each group’s variance and n separately, then adjusts degrees of freedom for that uncertainty.

With similar group sizes and spread, results can be nearly the same. When n and variance differ substantially together, pooled results can be distorted. A plan may choose Welch as a robust default, but avoid an automatic procedure that switches after looking at a variance-test p-value.

A paired t-test is closer to a one-sample test of differences

With mean pairwise difference d̄, SD s_d, and number of pairs n, calculate t=d̄/(s_d/√n). Assumptions therefore concern the distribution of differences and independence of pairs, more than normality of each raw condition separately.

If one value is missing from a pair, the complete-pair count falls. When missingness may relate to treatment or response, keeping only complete cases can bias results; report the missing-data structure and plan an appropriate model.

Nonparametric does not mean “assumption-free mean test”

Mann–Whitney/Wilcoxon rank-sum uses ranks but does not generally test a mean difference. It is easier to interpret as a location difference when distribution shapes match and only location shifts. Paired Wilcoxon signed-rank likewise has conditions, including symmetry of differences, for some interpretations.

Do not switch automatically to a rank test because of an extreme value. First consider its cause, whether the question concerns a mean or central location, and whether data are censored or ordinal. Permutation, bootstrap, robust estimation, and explicit models are also candidates.

In-Silico Lab: analyze the same effect under two structures

1. Under independent, change the SD ratio and compare Welch and pooled SE.
2. Under paired, see how a shared baseline changes pairwise-difference SE.
3. At effect 0, change the seed and confirm false positives can occur in either structure.
4. Consider whether arbitrary rearrangement of pairs preserves the paired advantage.

The Lab’s z approximation makes relationships transparent. Real t procedures reflect sample SD, degrees of freedom, unequal n, and missingness.

JMP result names reveal model assumptions

Assuming equal variances and Assuming unequal variances are not decoration; they reveal which SE and degrees of freedom were used. Matched Pairs is meaningful only when row links were correctly defined beforehand. Software cannot know whether a well ID is a donor ID.

Order analysis choices by the question

1. Is response continuous and is a mean difference the research question?
2. What is the independent experimental unit, and does each row represent it?
3. Are the two values paired by design?
4. What are the distribution shapes, outliers, censoring, and spread imbalance?
5. What are the pre-specified analysis and sensitivity analyses?
6. How will effect, CI, and units be reported?

Example result statement

We compared A (n=18) and B (n=17) from different donors independently. The B−A mean difference was 1.7 percentage points; the Welch 95% CI was 0.2–3.2, t(29.4)=2.31, p=.028. One donor-level value was used per analysis unit, and the direction was the same in distribution plots and robust location estimates.

For a paired design, report number of complete pairs, mean pairwise difference, and SD of differences—not merely “two-group t-test.”

Takeaways

• Independence and pairing are relationships fixed by design.
• Welch separately reflects group variance and n.
• Do not automatically branch pooled/Welch on one preliminary spread test.
• Paired analysis uses pairwise differences.
• A rank test is not an assumption-free test of means.
• Do not count technical repeats as independent units.

The next unit extends two groups to three or more. ANOVA’s variation partitioning addresses the multiple-error problem created by repeated t-tests.

Official supplementary resources

• NIST/SEMATECH · Two-Sample t-Test for Equal Means
• NIST/SEMATECH · Wilcoxon Rank-Sum Test

The values, figures, and Lab in this article are educational synthetic material.