T-Distribution Probability Calculator | t value to exact p-value

Enter a degree of freedom, a t value, and a test direction to get the Student's t-distribution PDF, CDF, and p-value at once. Supports any df from 1 to 1000 and flags whether the result clears the 0.001 / 0.01 / 0.05 significance thresholds.

💡 About this tool

Your stats software hands you a t-statistic and degrees of freedom, but the journal wants the exact p-value, and the printed t-table in the back of your textbook only lists round critical values for df = 10, 15, 20. When your real df is 37 or your t lands at 2.413, you end up eyeballing between two rows and hoping the reviewer agrees with your read.

This calculator removes that guesswork. Drop in df, the t value, and the tail you tested, and it returns the one- or two-tailed p-value to six decimal places, so you can report p = 0.018 instead of "p < 0.05". It is built for the small-sample situations the t-distribution was designed for, where a normal approximation would be wrong, but it scales cleanly up to df = 1000 when your sample is large.

🧐 Frequently Asked Questions

What do I put for degrees of freedom? For a one-sample t-test, use sample size minus one (n=20 gives df=19). For an independent two-sample test with equal variances, use (n₁-1)+(n₂-1). For a paired test, use the number of pairs minus one.

One-tailed or two-tailed? Use two-tailed when your alternative hypothesis is just "different (≠)". Use upper (>) or lower (<) when you are testing a specific direction. The two-tailed p-value is roughly double the one-tailed value because it counts both tails.

What is the difference between PDF, CDF, and p-value? PDF is the height of the density curve at your t value, CDF is the cumulative probability of being at or below it, and the p-value is the tail area that matches your chosen direction. You report the p-value for the decision.

Can I use a negative t value? Yes. The t-distribution is symmetric about zero, so a negative t with the lower (<) tail gives you the left-tail probability directly.

Why does a bigger df shrink my p-value? As degrees of freedom rise, the t-distribution loses its heavy tails and converges toward the standard normal, so the same t value sits further into the tail and yields a smaller p.

📚 Why "Student" and why it matters today

The distribution was published in 1908 by William Sealy Gosset, a chemist working on quality control at the Guinness brewery in Dublin. He needed to draw conclusions about beer ingredients from tiny batch samples, where the normal distribution overstated his confidence. Because Guinness barred staff from publishing, he wrote under the pen name "Student", and the heavy-tailed curve has carried that name ever since.

A practical rule that still circulates on Stack Overflow and stats forums is "switch from t to z once n passes about 30". The reason is visible right in this tool: push df up past 30 and the CDF you get for a fixed t value barely moves from the normal value, which is exactly why large-sample software often skips the t-table entirely.