T Test Two Samples Calculator

Student's t-test: comparing means

The Student's t-test compares means when you do not know the population standard deviation and the sample is small. The one-sample statistic is t = (x̄ − μ₀) / (s/√n), which you evaluate against the t distribution with df = n − 1. The two-sample version compares two means, using either pooled or Welch variance depending on whether the variances are equal. There is a nice backstory here. William Sealy Gosset published the method in 1908 under the pen name "Student" while working at the Guinness brewery in Dublin, because the company would not let employees publish under their own names. The t distribution carries heavier tails than the normal and approaches it as df → ∞. Worked example: two samples (50, s = 8, n = 30) and (55, s = 10, n = 28) give a pooled SE ≈ 2.39 and t ≈ −2.09 with df ≈ 56, so the two-tailed p ≈ 0.041, significant at the 5% level.

Applications

Checking a laboratory measurement against a certified reference value. Running A/B tests on website conversion rates. Weighing two teaching methods, the kind of comparison used in ENADE-style quality assessments. Two-arm clinical trials. Agronomy experiments that pit fertilizers against each other. And any quality-control workflow where you contrast two samples.

FAQ

When should I use t instead of z? Whenever σ is unknown, which is nearly always the case in practice, and n is small. Once n gets large, the two converge on similar results.

One-tailed or two-tailed? Go two-tailed when you just want to catch any difference at all. Go one-tailed when your hypothesis names a direction, as in "treatment is better than control". The one-tailed version roughly halves the p-value.

What if the variances are very different? Switch to Welch's t-test. It adjusts the degrees of freedom via Satterthwaite and makes no assumption that the variances are equal.

And if the data are not normal? For moderate n the t-test holds up well thanks to the CLT. When the data are badly skewed or n is small, reach for a nonparametric alternative such as Mann-Whitney.