What this calculator does
Comparing Means is Module 3 of Analysis Studio, and it is a free module. It runs the four standard mean comparisons:
- One-sample t test against a stated value
- Independent two-sample t test, with Welch as the default and Student on request
- Paired t test for matched or repeated measurements
- One-way ANOVA across three or more groups, Welch or classic, with Games-Howell or Tukey-Kramer follow-ups
Every test reports its effect size with an interval: Hedges g and Cohen d for two-group tests, d for paired data, and eta squared and omega squared for ANOVA. Group descriptives come first on the results screen, so the test statistic is read next to the means it summarizes. Under a survey design, the module switches to design-based domain-mean differences and design-based Wald tests, matching the standard survey-package results, because a textbook t test on weighted clustered data understates its own uncertainty.
The method, in plain terms
The t statistic is a difference divided by the standard error of that difference. The Welch form, the default here, estimates each group variance separately and adjusts the degrees of freedom by the Welch-Satterthwaite formula, so unequal variances and unequal group sizes cost nothing and the equal-variance assumption is never needed. ANOVA generalizes the same idea: the F statistic compares the variance between group means with the variance within groups.
Worked examples
Two groups of five. Group A scores 22, 25, 28, 24, 26, a mean of 25.0. Group B scores 18, 20, 19, 22, 21, a mean of 20.0.
- Difference: 5.0 points
- Welch t: 4.082
- p: 0.0044
The difference is five points on a within-group spread of about two points, which is why the test is decisive at n of five per group: the signal is large relative to the noise.
Three groups of four, with means of 6.0, 8.0, and 11.5, give a one-way ANOVA of F = 31.000 with p = 0.0001. The follow-up comparisons then locate the separation: the third group stands apart from the first two, which sit closer together, and the post hoc table states which pairs differ with the family error rate held.
When the calculator fits, and when it does not
It fits when:
- The outcome is continuous, groups are independent or paired by design, and the question is about means.
- Variances may differ across groups. Welch handles that as the default rather than as a rescue.
- The data come from a survey design, in which case the design-based forms take over automatically once the design is declared.
It does not fit when:
- The outcome is ordinal or heavily skewed with small samples. The rank-based tests in Module 8 answer the same questions without leaning on the mean.
- Measurements repeat within the same cases across several conditions. That structure belongs to the mixed-effects models in Module 15.
- The real question is a proportion. Proportions belong to the crosstab module or the logistic module, where the intervals behave properly at the boundaries.
Run it
Module 3: Comparing MeansOpen in Analysis Studio →Load your data, pick the outcome and the grouping variable, and read the group descriptives before the test table.
Questions
Welch or Student: which should I use? Welch, unless a protocol fixed Student in advance. Welch keeps its stated error rate when variances differ and loses almost nothing when they match, which is why the module defaults to it and why the equal-variance test is a legacy habit rather than a safeguard.
What does the effect size add to the p value? The p value says whether the difference is distinguishable from noise at this sample size. The effect size says how large it is, on a standardized scale, with an interval. A significant result on ten thousand cases can be trivially small; the effect size is where that shows.
My paired data: which test? The paired t test, on the within-pair differences. Pairing removes the between-case variance from the comparison, which is the entire point of a paired design, and the effect size reported is the standardized mean of the differences.
ANOVA is significant. Which groups differ? That is what the post hoc table answers. Games-Howell pairs with the Welch ANOVA and tolerates unequal variances; Tukey-Kramer pairs with the classic form. Either holds the family error rate across the set of pairs, which unadjusted t tests across every pair do not.
Does it work on survey data? Yes. Declare the design and the module uses design-based differences and Wald tests, with the design degrees of freedom, so clustering and weighting are inside the standard errors rather than ignored by them.