Distributions
T Distribution Calculator
Tail probabilities and critical values for Student's t — the distribution that applies when σ is estimated from the data.
Also called student t distribution calculator, Student's t calculator, t-table calculator.
How to read this result
The t distribution is the normal with an allowance for uncertainty about σ. When the standard deviation is estimated from a small sample, that estimate is sometimes too small, which makes extreme t values more common than a normal would predict. Hence the heavier tails.
Degrees of freedom control how much heavier. At 5 df the 97.5th percentile is 2.571 against the normal's 1.960 — a 31% wider interval. At 30 df it is 2.042. Past about 100 the two are interchangeable for practical purposes.
Critical values are confidence-interval multipliers. The same number that decides significance at α also multiplies the standard error to give the margin of error. That is not a coincidence: they are two statements of one comparison.
Where the degrees of freedom come from. n − 1 for a one-sample test, n₁ + n₂ − 2 for a pooled two-sample test, a fractional Welch-Satterthwaite value when the variances differ, and n − 2 for a regression slope.
The formula
- the degrees of freedom
- the gamma function — factorial, extended to non-integers
As ν grows this converges on the standard normal density. Tail probabilities are computed from the incomplete beta function.
Worked example
How extreme is t = 2.5 on 10 degrees of freedom?
A two-sample test on twelve observations produces t = 2.5 with 10 degrees of freedom. What is the p-value, and what would the threshold have been?
- Find the area to the right of 2.5 under the t curve with 10 df.P(T > 2.5) = 0.01572
- Double it for a two-tailed test.p = 0.03145
- The two-tailed 5% critical value on 10 df.t* = 2.228
- 2.5 exceeds 2.228, which is the same conclusion.significant at α = 0.05
- Compare with the normal, which would have used 1.96.the t threshold is 14% higher, the price of estimating σ
With 100 degrees of freedom the same t = 2.5 would give p = 0.0140 rather than 0.0314 — the small sample costs you real evidence, which is exactly what the distribution is for.
Checked against R's pt(2.5, 10) and qt(0.975, 10).
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
The t distribution applies when the underlying data are normal (or the sample is large enough for the mean to be) and σ is estimated from the same data. With a genuinely known σ — rare outside textbooks and quality control with long historical records — the normal is correct and the t is needlessly conservative.
The heavy tails are about uncertainty in σ, not about robustness to outliers: a t-test on outlier-ridden data is not protected by them.
- You have data rather than a t valueT-testComputes the statistic, the degrees of freedom and the p-value from the data.
- σ is known, or the sample is largeNormal distributionUse z; past about 100 df the two agree.
- You want the p-value from another statisticP-value calculatorz, t, chi-square and F in one place.
- You want a confidence intervalConfidence interval calculatorUses these critical values as the multiplier.
Questions people ask
When do I use t instead of z?
Whenever the standard deviation is estimated from the sample — which is almost always. Use z only when σ is genuinely known in advance, or when the sample is large enough (past roughly 100) that the two agree to three decimal places.
What are degrees of freedom?
The number of values that are free to vary once the estimates are fixed. Estimating the mean from n observations uses up one, leaving n − 1 for the variance. More degrees of freedom means a better-estimated σ and a lighter-tailed distribution.
Why does the t distribution have heavier tails?
Because it accounts for two sources of variation rather than one: the sample mean varies, and so does the estimate of σ used to scale it. Underestimating σ inflates t, and that possibility puts extra weight in the tails.
What is the critical t value for 95% confidence?
It depends on the degrees of freedom: 2.776 at 4 df, 2.228 at 10, 2.042 at 30, 1.984 at 100, approaching 1.960 as df grows. The critical-value mode above computes it for any df, along with a table of common confidence levels.