Calculate t-statistic, p-value, and Cohen's d for one-sample, two-sample (Welch's), and paired t-tests. Includes critical value table and distribution visual.
The T-Statistic Calculator performs complete t-test analysis for one-sample, independent two-sample, and paired designs. Enter your summary statistics or paired differences time slots calculator computes the t-statistic, exact p-values for two-tailed and one-tailed tests, Cohen's d effect size, and critical values at multiple significance levels.
The t-test is the workhorse of statistical hypothesis testing, used when comparing means with unknown population standard deviations and relatively small sample sizes. Unlike the z-test, the t-distribution accounts for the extra uncertainty from estimating σ, producing wider confidence intervals and more conservative p-values, especially for small n.
For two-sample comparisons, the calculator provides both Welch's t-test (which doesn't assume equal variances) and the pooled t-test. Welch's test is preferred as the default because it's valid whether or not variances are equal, and it has been shown to maintain better Type I error control across a wide range of conditions. Check the example with realistic values before reporting.
The t-test is the most commonly used statistical test in research, quality control, and data analysis. This calculator supports all three major variants — one-sample, independent, and paired — with proper Welch correction for unequal variances. Cohen's d provides the effect size that p-values alone cannot convey.
The critical value table lets you quickly see whether your result would be significant at various α levels, and the visual t-distribution display helps students and non-statisticians understand where the test statistic falls relative to the null distribution.
One-sample: t = (x̄ − μ₀) / (s/√n), df = n−1. Two-sample (Welch): t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂), df via Welch-Satterthwaite. Paired: t = d̄ / (s_d/√n), df = n−1.
Result: t = 1.558, df = 29, p = 0.130 (two-tailed), Cohen's d = 0.284
SE = 12.3/√30 = 2.246. t = (78.5-75)/2.246 = 1.558. With df=29, the two-tailed p-value is 0.130. Since p > 0.05, we fail to reject H₀. Cohen's d = 3.5/12.3 = 0.284 (small effect).
The one-sample t-test compares a mean to a known value (μ₀). The two-sample test compares means from two independent groups. The paired test compares means from matched or repeated measurements on the same subjects. Choosing the right design is often more important than the statistical test itself.
When group variances are unequal, the standard pooled t-test can produce misleading results. Welch's modification adjusts both the standard error formula and the degrees of freedom. The Satterthwaite approximation for df produces a non-integer value that better reflects the actual sampling distribution.
Statistical significance depends on sample size: large samples detect trivial effects. Cohen's d standardizes the effect, making it comparable across studies. Power analysis uses d, α, and n to determine the probability of detecting a real effect. Planning sample size around desired power (typically 0.80) produces more efficient studies.
Use the t-test when the population standard deviation is unknown (nearly always in practice) and you're estimating it from the sample. The z-test is only appropriate when σ is known. For large n, the t and z distributions converge.
Welch's t-test is a modified two-sample t-test that doesn't assume equal variances. It uses the Welch-Satterthwaite equation for degrees of freedom. It's the recommended default because it controls Type I error better than the pooled test.
Cohen's d measures effect size in standard deviation units. Guidelines: |d| < 0.2 is negligible, 0.2-0.5 is small, 0.5-0.8 is medium, > 0.8 is large. Always report effect size alongside p-values to convey practical significance.
The paired t-test assumes: (1) differences are normally distributed (or n is large), (2) pairs are independent, and (3) observations within pairs are dependent. It's more powerful than the two-sample test when pairing is appropriate.
The t-test is robust to moderate non-normality, especially for n > 30. For severely skewed data or small n, consider the Wilcoxon signed-rank test (paired) or Mann-Whitney U test (independent).
Degrees of freedom (df) represent the number of independent pieces of information. For one-sample and paired tests, df = n−1. For Welch's test, df is calculated from sample sizes and variances, and may not be an integer.