t test calculator

Which t test

All three tests compare a mean with something when the population standard deviation is unknown and has to be estimated from the sample, which is nearly always the case in practice.

• One-sample t: is the process mean different from a target or a claimed value μ₀? t = (x̄ − μ₀) / (s / √n), with n − 1 degrees of freedom.
• Two-sample t: do two independent groups (two machines, two suppliers, before and after on different parts) have different means? The Welch version does not assume equal variances and uses the Welch–Satterthwaite degrees of freedom, which are usually not a whole number. The pooled version assumes equal variances and uses n₁ + n₂ − 2 degrees of freedom.
• Paired t: the same unit is measured twice (before and after, gauge A and gauge B on the same part). Work with the differences d and run a one-sample test on them with n − 1 df, where n is the number of pairs. Pairing removes the unit-to-unit variation, which is why it detects smaller changes than a two-sample test on the same data.

Reading the result

The p value is the probability of a result at least as extreme as the one observed if the null hypothesis were true. If p is less than α, reject H₀. Failing to reject is not the same as proving H₀; it means the sample did not carry enough evidence. The confidence interval gives the same decision for a two-sided test: if the interval for the mean or the difference excludes the null value, the test rejects at α = 1 − confidence. A one-sided test gives a one-sided bound instead.

The tool also runs an F test on the ratio of the two sample variances. Many analysts use the Welch test by default: it loses little when the variances are equal and protects you when they are not.

Assumptions and common traps

• The data, or the paired differences, should come from an approximately normal population. Larger samples tolerate moderate skew.
• Running a two-sample test on paired data throws away the pairing and usually hides a real effect.
• Choose the alternative and α before seeing the data. Switching to a one-sided test after looking halves the p value and inflates the Type I error rate.

On the Green Belt exam

Expect to pick the right test from a short scenario, compute a t statistic by hand from summary statistics, find degrees of freedom, read a critical value from the t table and state the conclusion in terms of H₀. Paired versus two-sample is a favorite distinction, as is the link between a confidence interval and a two-sided test.

Common questions

How do I run a t test online?

Paste one sample (or two), pick one-sample, two-sample or paired, set the alternative and the confidence, and the calculator returns the t statistic, degrees of freedom, the exact p value and the confidence interval. Everything is computed in your browser.

What is the difference between Student's t test and Welch's t test?

Student's two-sample t test pools the two standard deviations and assumes the populations have equal variances. Welch's version does not assume that and adjusts the degrees of freedom. When the variances or sample sizes differ, Welch is the safer default; this calculator does both and shows an F test for equal variances.

When should I use a paired t test?

When each value in one sample has a natural partner in the other: the same part measured before and after, or the same unit measured by two gauges. Pairing removes the unit-to-unit variation, so it finds a real difference with far fewer measurements than a two-sample test.

What does the t statistic tell me?

How many standard errors the observed difference is from the hypothesized one. Compared with the t distribution at the right degrees of freedom it gives the p value: the chance of a difference at least this large if there were really no difference.

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