T Test Calculator

An independent two-sample t test evaluates whether the means of two unrelated groups differ beyond what random sampling variation would commonly produce. This calculator uses Welch's t test, which does not require the two populations to have equal variances. It works from summary statistics: each group's mean, sample standard deviation, and sample size. That makes it useful when raw observations are unavailable but reliable summaries are known. The test statistic is the difference between group means divided by its standard error. For Welch's method, the standard error is √(s₁²/n₁ + s₂²/n₂). A positive statistic means the first mean is larger; a negative statistic means the second is larger. Reversing the group order changes the sign but not the two-tailed p-value. The statistic's absolute magnitude expresses the observed mean difference in estimated standard-error units. Welch–Satterthwaite degrees of freedom account for both group variances and sample sizes. They are often fractional, which is expected. When group sizes and standard deviations match, the degrees of freedom reduce to n₁ + n₂ − 2, the same value produced by the classical pooled test. When variability or sample sizes differ, Welch's adjusted degrees of freedom generally provide more dependable Type I error control than automatically pooling variances. The reported p-value is two-tailed. It is the probability, under a null hypothesis of equal population means and the model assumptions, of obtaining a t statistic at least as extreme in either direction. A small p-value indicates incompatibility with the null model; it does not measure the probability that the null hypothesis is true. Compare it with a significance level chosen before analysis, and avoid treating an arbitrary threshold as a measure of practical importance. The procedure assumes independent observations within and between groups, a quantitative outcome, and samples that appropriately represent their target populations. With small samples, severe skew or influential outliers can undermine the test. Larger samples make inference more robust to moderate non-normality, but independence and sound sampling remain essential. If observations are paired, such as before-and-after measurements on the same people, an independent test is inappropriate; use a paired t test on within-pair differences instead. Statistical significance should be reported alongside the actual mean difference, uncertainty, and context. A confidence interval communicates the range of plausible population differences, while an effect size such as Cohen's d helps express magnitude. This calculator provides an accurate Welch statistic and p-value for quick analysis or arithmetic checking, but decisions should also consider study design, multiple comparisons, measurement quality, and whether the difference is meaningful in the subject area.