StatReason
When data is genuinely paired
Use a paired analysis when each observation in one condition has a natural partner in the other condition. The partner might be the same person measured before and after a program, a matched pair of twins, or two readings from the same device on the same item.
Why the within-pair difference is the key information
A paired t-test calculates one difference for every confirmed pair and then asks whether the average difference is compatible with the selected null value. A Wilcoxon signed-rank test similarly uses paired differences, although it ranks their magnitudes. If the pair relationship is real, using it can reduce irrelevant person-to-person variation and give a more precise estimate of change than treating the two conditions as independent groups would.
When data is independent instead
Independent groups have no such one-to-one relationship. Examples include a sample of customers assigned to one website version and a separate sample assigned to another version, or a group of plants receiving one treatment and a different group receiving another. An independent t-test compares group summaries and must not use row position as if it created a pair.
Worked example
Consider a simple before-and-after example. Five people record a baseline score and repeat the measurement after training. The relevant data are five changes, one for each person. Sorting the before column from low to high and the after column from high to low would preserve group means but destroy the actual pair information. A paired result could change dramatically even though the two column summaries did not.
What you must verify yourself
The calculator cannot discover valid pairing from a pasted layout. You must verify that every row represents the same unit measured twice or an intentional match. You must also decide how to handle a participant with one missing measurement. A paired test generally requires a complete pair; removing that pair changes the analysis population and should be visible in the receipt.
Pairing is not automatically better
Pairing is useful only when it reflects the way the data were collected. Treating unrelated observations as paired can create a misleading standard error. Treating true pairs as independent can throw away useful design information. Neither mistake is corrected by the arithmetic itself; both require you to know your own study design.
Source
This distinction follows the paired-observation guidance in the NIST/SEMATECH e-Handbook of Statistical Methods and the shared statistical reasoning contract every StatReason engine is built against.
Limitations
This page covers the two-group paired and independent cases. Repeated measures across three or more conditions, and matched designs with more than two conditions, use a related but distinct set of engines covered under ANOVA and Group Comparisons.