T Score to P Value Calculator

Convert t-statistics to p-values for hypothesis testing. Supports one‑tailed and two‑tailed tests with interactive t‑distribution visualization.

Hypothesis testing: p-value = probability of observing a t-statistic as extreme as yours, under the null hypothesis.

t=1.96, df=30
t=2.576, df=30
t=-2.13, df=20
t=1.671, df=50
t=2.0, df=10
Computing...

Understanding the t‑distribution and p‑value

In statistics, the t‑distribution is used when the sample size is small or the population standard deviation is unknown. The p‑value tells you how likely it is to observe a t‑score as extreme as yours if the null hypothesis is true.

Mathematical Definition:

For a t‑statistic t with ν degrees of freedom, the p‑value depends on the alternative hypothesis:

  • Right‑tailed: p = P(Tν > t)
  • Left‑tailed: p = P(Tν < t)
  • Two‑tailed: p = 2·P(Tν > |t|)

where Tν follows a t‑distribution with ν degrees of freedom.

Common t‑distribution values (two‑tailed, α = 0.05)

df Critical t df Critical t
1 12.706 15 2.131
2 4.303 20 2.086
3 3.182 25 2.060
4 2.776 30 2.042
5 2.571 40 2.021
10 2.228 ∞ (z) 1.960

Interpretation Guidelines

1

Small p‑value (typically ≤ 0.05): Strong evidence against the null hypothesis. The result is statistically significant.

2

Large p‑value (> 0.05): Weak evidence against the null; fail to reject the null hypothesis.

3

One‑tailed vs two‑tailed: Two‑tailed tests are more conservative (p‑value ≈ double the one‑tailed p). Choose based on your research hypothesis.

Applications

  • Independent t‑test: Compare means of two groups.
  • Paired t‑test: Compare means of matched samples.
  • Regression coefficients: Test if a predictor is significant.
  • Quality control: Determine if a process deviates from target.

Calculator features:

  • Uses jStat library for precise t‑distribution CDF
  • Supports any degrees of freedom (≥1)
  • Interactive graph with shaded tail area corresponding to your test type
  • Displays both one‑tailed and two‑tailed p‑values for complete insight

Frequently Asked Questions

A p‑value is the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true. It quantifies the evidence against the null.

Use two‑tailed when you are testing for a difference in either direction (e.g., treatment effect could be positive or negative). Use one‑tailed only if you have a strong directional hypothesis and are willing to ignore the opposite direction.

It uses the jStat library, which implements high‑accuracy numerical approximations of the t‑distribution cumulative distribution function. Results match published t‑tables and standard statistical software.

The t‑distribution is defined for positive real degrees of freedom. You can enter non‑integer values (e.g., from Satterthwaite approximation), and the calculator will handle them correctly.

Yes, the calculator displays the critical t‑value for two‑tailed test at your chosen α. For one‑tailed, the critical t is also shown in the detailed results.