About Student t Distribution Calculator
This Student t distribution calculator gives tail areas and critical values for any degrees of freedom. It opens on 10 degrees of freedom and finds the t value that leaves 2.5% in the upper tail, the two-sided 95% critical value. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The t distribution looks like a normal curve with heavier tails. The fewer the degrees of freedom, the heavier the tails, and as the degrees of freedom grow it approaches the standard normal.
For a two-sided p-value from a t statistic, find the upper tail P(T > |t|) and double it. For a confidence interval, use the inverse at 1 - α ÷ 2.
How to use Student t Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
t density
The density depends only on the degrees of freedom ν.
f(t) = Γ((ν+1)/2) / (sqrt(νπ) × Γ(ν/2)) × (1 + t²/ν)^(−(ν+1)/2)
mean = 0 for ν > 1
variance = ν / (ν − 2) for ν > 2Worked examples
df = 10: t value with 0.975 below it gives 2.2281
The critical value for a two-sided 95% interval with 10 degrees of freedom.
df = 20: P(T > 2.086) gives 0.024998
About 0.025, which is half of a two-sided 5% test.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I find a two-tailed p-value?
Calculate the upper tail P(T > |t|) with the correct degrees of freedom, then double it.
How do I find the critical value for a confidence interval?
Choose Inverse and enter 1 minus half of α. For 95% that is 0.975, for 99% it is 0.995.
Why does the mean show as undefined for 1 degree of freedom?
With 1 degree of freedom the t distribution is the Cauchy distribution, whose tails are so heavy that it has no mean.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Normal Distribution CalculatorEnter a mean and standard deviation to get tail probabilities, intervals and percentiles on a bell curve.
- Binomial Distribution CalculatorEnter the number of trials and the chance of success to get P(X = k), P(X ≤ k), P(X ≥ k) and ranges.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- Geometric Distribution CalculatorEnter the success probability to get the chance that the first success happens on or by trial k.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Chi-Square Distribution CalculatorEnter the degrees of freedom to get chi-square p-values, critical values and tail areas.
- F Distribution CalculatorEnter both degrees of freedom to get F p-values, critical values and tail areas.