How this calculator works
The one-sample relationship can also be inverted for a mean, standard deviation or sample size. Recovered sample sizes must be whole and consistent with the statistic sign. A t statistic divides the departure of a sample mean or mean difference from its null value by an estimated standard error. Welch keeps the two group variance estimates separate. Pooled inference assumes a common population variance. All means and standard deviations must share one unit. At least one contributing variance must be positive, and each summary sample size must be a whole number of at least two.
Formula
One sample: t = (x̄ - μ0)/(s/√n). Welch: t = (x̄1 - x̄2 - Δ0)/√(s1²/n1 + s2²/n2). Pooled: replace the denominator with sp√(1/n1 + 1/n2).
Worked example
A sample mean 2, null 0, sample standard deviation 2 and n = 4 give SE 1, t = 2 and df = 3. Means 12 and 10 with both SDs 2 and both n = 10 give t = √5 and pooled df = 18.
Advertisement
Frequently asked questions
Is t a probability?
No. It is a signed standardised statistic. A reference t distribution and alternative hypothesis convert it to a p-value.
Why choose Welch?
It allows unequal population variances and is often a reasonable default for independent groups.
What does negative t mean?
The sample mean or first-minus-second mean difference lies below its null value.