t distribution calculator
The t distribution is the bell curve with a penalty for small samples: heavier tails. Slide the degrees of freedom and watch it become normal.
Solid: t with 10 df. Dashed: standard normal. Watch the tails as you lower the degrees of freedom.
Two-tailed critical t-values
| df | 90% | 95% | 98% | 99% |
|---|---|---|---|---|
| 1 | 6.314 | 12.706 | 31.821 | 63.657 |
| 2 | 2.920 | 4.303 | 6.965 | 9.925 |
| 3 | 2.353 | 3.182 | 4.541 | 5.841 |
| 4 | 2.132 | 2.776 | 3.747 | 4.604 |
| 5 | 2.015 | 2.571 | 3.365 | 4.032 |
| 6 | 1.943 | 2.447 | 3.143 | 3.707 |
| 7 | 1.895 | 2.365 | 2.998 | 3.499 |
| 8 | 1.860 | 2.306 | 2.896 | 3.355 |
| 9 | 1.833 | 2.262 | 2.821 | 3.250 |
| 10 | 1.812 | 2.228 | 2.764 | 3.169 |
| 12 | 1.782 | 2.179 | 2.681 | 3.055 |
| 15 | 1.753 | 2.131 | 2.602 | 2.947 |
| 20 | 1.725 | 2.086 | 2.528 | 2.845 |
| 25 | 1.708 | 2.060 | 2.485 | 2.787 |
| 30 | 1.697 | 2.042 | 2.457 | 2.750 |
| 40 | 1.684 | 2.021 | 2.423 | 2.704 |
| 60 | 1.671 | 2.000 | 2.390 | 2.660 |
| 120 | 1.658 | 1.980 | 2.358 | 2.617 |
| ∞ (z) | 1.645 | 1.960 | 2.326 | 2.576 |
Why the t distribution exists
When you standardise a sample mean using the sample standard deviation s instead of the true σ, you add a second source of randomness: s itself varies from sample to sample. With small samples, s is often too small, which makes the statistic t = (x̄ − μ) / (s/√n) land in the tails more often than z would. The t distribution accounts for this with heavier tails.
William Sealy Gosset worked this out in 1908 while testing small batches of barley at the Guinness brewery; the company made him publish under the pseudonym "Student".
Degrees of freedom
For a one-sample t test or interval, df = n − 1. As df grows the t curve approaches the normal: at df = 30 the 95% critical value is 2.042 against z's 1.960, and by df = 120 it is 1.980. Drag the slider to watch the solid curve close in on the dashed one.
Use the result to get a p-value or build a confidence interval.
Still not clicking? Ask the tutor
An AI tutor that sees the numbers you entered above and explains them. It can make mistakes, so check anything important against the worked steps.
Questions students ask
When do you use the t distribution instead of the normal?
When you estimate a mean and the population standard deviation is unknown, so you use the sample SD. This is almost always the case in practice.
What are degrees of freedom?
For a one-sample t procedure, df = n − 1. Fewer degrees of freedom mean more uncertainty in the SD estimate and heavier tails.
What is the critical t-value for a 95% confidence interval?
It depends on df: 2.228 for df = 10, 2.042 for df = 30 and 1.984 for df = 100, approaching the normal 1.960 as df grows.
Who invented the t distribution?
William Sealy Gosset, a chemist at the Guinness brewery in Dublin, published it in 1908 under the pen name "Student", which is why it is called Student's t.