Tdistribution Calculator
Our free statistics calculator solves tdistribution problems. Get worked examples, visual aids, and downloadable results.
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
Frequently Asked Questions
How accurate are the results from Tdistribution Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
What formula does Tdistribution Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.
Is Tdistribution Calculator free to use?
Yes, completely free with no sign-up required. All calculators on NovaCalculator are free to use without registration, subscription, or payment.