T-Distribution Calculator

Student's t density, cumulative probability, and quantile for any degrees of freedom.

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About this tool

Compute the Student t density, cumulative probability or quantile for any degrees of freedom, with a plotted curve.

How to use

1
Step 1
Choose PDF, CDF or inverse CDF.
2
Step 2
Enter the degrees of freedom.
3
Step 3
Enter x (PDF or CDF) or a probability p between 0 and 1, then press Calculate.

How it works

The t distribution is bell-shaped like the normal but with heavier tails, especially for small df. The CDF gives P(T ≤ x). The inverse CDF returns the x that has probability p to its left, which is how critical values are found.

CDF(x) = P(T ≤ x)      quantile(p) = x such that CDF(x) = p

Worked example

df = 10

Input
CDF, df = 10, x = 2
Output
0.963306
Walkthrough
Inverse CDF at p = 0.975 gives 2.228139, matching the standard 2.228 table value. The PDF at x = 0 is 0.389108.

Limitations

  • Central t only; no non-central version.
  • df can be fractional (as in Welch) but must be positive.

Questions

What is P(T ≤ 2) for 10 df?
0.9633.
What is the 97.5th percentile for 10 df?
2.2281.
How does t differ from normal?
It has heavier tails; with few df you need a larger cut-off than 1.96.

Features

  • PDF, CDF, quantile
  • Plotted curve
  • Runs in your browser

Guides

Related resources

Guides and videos from around the web—opens on YouTube or the publisher’s site.

  • Hypothesis testing and p-values

    StatQuest with Josh Starmer

    Intuition for t/z/χ² decision rules.

    Watch on YouTube

  • Mean, Variance and Standard Deviation

    StatQuest with Josh Starmer

    Descriptive stats behind table and test tools.

    Watch on YouTube

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