T-Distribution Explorer

Drag the degrees of freedom and watch the t-distribution's tails shrink toward the normal curve as it grows.

Distribution

Degrees of freedom = 5 Overlay the standard normal curve

Find the probability that t is…

Above

t distribution, df = 5 (dashed: standard normal)

t probability

0.051

Normal probability

0.0228

At df = 5, the t distribution gives 0.051 versus 0.0228 for the normal curve — a difference of 0.0282. With few degrees of freedom, the heavier tails of the t distribution matter: it assigns more probability to extreme values than the normal curve does. The 97.5th percentile critical value here is t = 2.571, compared with z = 1.960 for the normal.

How to use this tool

Drag the degrees-of-freedom slider from 1 up to 100.
Turn on the normal overlay to compare the t-curve with the standard normal curve directly.
Choose between, below, or above, and enter the value or values, to shade an area and read its probability.
Compare the t probability with the normal probability for the same region to see how much the heavier tails matter at low degrees of freedom.

Formula Reference

t probability density

proportional to (1 + t^2/df)^-((df+1)/2)

As df grows

the t distribution converges to the standard normal distribution

Critical value example

the 97.5th percentile t-value shrinks toward z = 1.960 as df increases

Articlet-Distribution vs. Normal Articlet-Tests: One-Sample, Two-Sample and Paired TableStudent t-Table Calct-Test Calculator GlossaryGlossary: Degrees of Freedom

Frequently Asked Questions

Why does the t distribution have heavier tails than the normal?+

It arises from using an estimated standard deviation instead of a known one, and that extra uncertainty is largest with few degrees of freedom, spreading more probability into the tails.

At what point does the t distribution basically become the normal distribution?+

There is no sharp cutoff, but by around 30 degrees of freedom the two curves are very close, and many older textbooks simply switch to the normal table beyond that point.

Why does this matter for a confidence interval or a t-test?+

With a small sample, using a normal critical value instead of a t critical value understates the true uncertainty, giving an interval that is too narrow or a p-value that is too optimistic.

What does degrees of freedom mean here?+

For a one-sample or paired t procedure it is the sample size minus one; other t procedures have their own degrees-of-freedom formula, but the shape of the curve always depends on this single number.