Tdistribution Calculator
Our free statistics calculator solves tdistribution problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
f(t) = [Gamma((df+1)/2) / (sqrt(df*pi) * Gamma(df/2))] * (1 + t^2/df)^(-(df+1)/2)
Where t is the test statistic, df is degrees of freedom, and Gamma is the gamma function. The CDF is computed using the regularized incomplete beta function. P-values represent tail probabilities under this distribution.
Worked Examples
Example 1: One-Sample T-Test
Problem:A sample of 11 measurements has mean 5.3 and standard deviation 1.2. Test whether the population mean differs from 5.0 at alpha = 0.05.
Solution:t = (5.3 - 5.0) / (1.2 / sqrt(11)) = 0.3 / 0.3617 = 0.8294\ndf = 11 - 1 = 10\nTwo-tailed p-value: P(|T| > 0.8294) with df = 10\nCritical value at alpha = 0.05 (two-tailed): 2.2281\nSince |0.8294| < 2.2281, fail to reject H0.
Result:t = 0.8294 | p-value = 0.4263 | Not significant at alpha = 0.05
Example 2: Confidence Interval Width
Problem:With df = 25, find the critical t-value for a 95% confidence interval and compare to the z-value of 1.96.
Solution:For 95% CI with df = 25:\nt-critical (two-tailed, alpha = 0.05) = 2.0595\nNormal z-critical = 1.96\nDifference = 2.0595 - 1.96 = 0.0995\nThe t-based interval is about 5.1% wider than the z-based interval.\nAt df = 100, t-critical = 1.984, only 1.2% wider.
Result:t-critical(25) = 2.0595 vs z = 1.96 | T-interval is 5.1% wider
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy