# t, chi-square & F distribution calculator

> t, chi-square and F distribution calculator: tail areas, p-values, densities and critical values, plus exponential and uniform distributions.

Interactive version: https://www.calcopenly.com/statistics/t-chi-square-f-distribution-calculator
Subject: Statistics and probability calculators

Each result is read from a continuous distribution's density curve: the cumulative probability F(x) = P(X ≤ x), the upper tail, the area between two values, the density itself, or, in inverse mode, the value x with a given area to its left. For Student t, χ² and F the areas come from the regularized incomplete beta and gamma functions evaluated to 50 digits; exponential and uniform areas have closed forms.

These are the reference distributions of the common tests: t for means, χ² for counts and variances, F for ANOVA and regression. The default, t with 10 degrees of freedom, puts 96.90% of the area below 2.1, so a t statistic of 2.1 has a one-sided p-value of 0.031.

Printed tables round to three or four figures and list only selected degrees of freedom. Here any positive value works, including the fractional degrees of freedom of Welch's t-test.

## Inputs

- **Distribution** (options: Student t, Chi-square χ², F, Exponential, Uniform)
- **Degrees of freedom**
- **Numerator degrees of freedom d₁**
- **Denominator degrees of freedom d₂**
- **Rate λ**: Events per unit time; the mean waiting time is 1/λ.
- **Minimum a**
- **Maximum b**
- **Find** (options: P(X ≤ x), P(X ≥ x), P(a ≤ X ≤ b), Density f(x), x for a left-tail area (critical value))
- **Value x**
- **Lower value**
- **Upper value**
- **Left-tail area p**: For a two-sided test at α = 0.05 use p = 0.975.

## Results

- Result — main result
- Complement 1 − P
- Mean
- Variance

## Formula

$$
P(X \le x) = \int_{-\infty}^{x} f(u)\,du,\qquad x_p = F^{-1}(p)
$$

## Worked examples

### t with 10 df, P(T ≤ 2.1)

- Distribution: Student t
- Degrees of freedom: 10
- Find: P(X ≤ x)
- Value x: 2.1
- **Result: 0.968961**
- **Complement 1 − P: 0.031039**
- Checked against: Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int)

### t critical value, 10 df, 0.975

- Distribution: Student t
- Degrees of freedom: 10
- Find: x for a left-tail area (critical value)
- Left-tail area p: 0.975
- **Result: 2.228139**
- Checked against: t table (NIST e-Handbook §1.3.6.7.2): 2.228; bisection on the A&S closed form gives 2.2281388520

### χ² critical value, 10 df, 0.95

- Distribution: Chi-square χ²
- Degrees of freedom: 10
- Find: x for a left-tail area (critical value)
- Left-tail area p: 0.95
- **Result: 18.307038**
- Checked against: χ² table (NIST e-Handbook §1.3.6.7.4): 18.307; bisection on the A&S 26.4.5 closed form gives 18.3070380533

### F critical value (5, 20), 0.95

- Distribution: F
- Numerator degrees of freedom d₁: 5
- Denominator degrees of freedom d₂: 20
- Find: x for a left-tail area (critical value)
- Left-tail area p: 0.95
- **Result: 2.71089**
- Checked against: F table (NIST e-Handbook §1.3.6.7.3): 2.71; bisection on the A&S 26.6.5 closed form gives 2.7108898372

### χ² upper tail, 1 df, x = 3.84

- Distribution: Chi-square χ²
- Degrees of freedom: 1
- Find: P(X ≥ x)
- Value x: 3.84
- **Result: 0.050044**
- Checked against: A&S 26.4.4 with k = 1: 2·Q(√3.84) = erfc(√1.92), Python 0.0500435212

### Exponential λ = 0.5, P(X ≤ 2)

- Distribution: Exponential
- Rate λ: 0.5
- Find: P(X ≤ x)
- Value x: 2
- **Result: 0.632121**
- **Mean: 2**
- **Variance: 4**
- Checked against: 1 − e^(−1) = 0.6321205588 (Python math.exp)

## Questions

### How do you find a t critical value?

Choose Student t, enter the degrees of freedom, pick 'x for a left-tail area' and enter 1 − α/2 for a two-sided test or 1 − α for a one-sided one. With 10 df and a two-sided α of 0.05, p = 0.975 gives 2.228, the value in the NIST/SEMATECH e-Handbook t table (§1.3.6.7.2). As df grows the value falls towards the normal 1.960; at 30 df it is 2.042.

### How do you get a p-value from a t statistic?

Take the tail area beyond the statistic: P(T ≥ t) for a right-tailed test, P(T ≤ t) for a left-tailed one, and twice the tail beyond |t| for a two-sided test. A t of 2.1 with 10 df gives P(T ≥ 2.1) = 0.0310, so the two-sided p-value is 0.0621, above 0.05. It is the chance of a statistic at least that extreme if the null hypothesis were true, not the chance that the null hypothesis is true.

### What is the chi-square critical value for 1 degree of freedom?

3.841 at α = 0.05, 6.635 at α = 0.01 and 2.706 at α = 0.10, all upper-tail values. With 10 degrees of freedom the 5% value is 18.307, matching the NIST/SEMATECH χ² table (§1.3.6.7.4). To reproduce any of them, choose Chi-square, pick 'x for a left-tail area' and enter 1 − α, such as 0.95.

### Why does the F distribution have two degrees of freedom?

An F statistic is the ratio of two variance estimates, and each has its own degrees of freedom: d₁ for the numerator and d₂ for the denominator. In a one-way ANOVA with k groups and N observations, d₁ = k − 1 and d₂ = N − k. Order matters: the 5% critical value for (5, 20) is 2.711, but for (20, 5) it is 4.558.

### How is the t distribution different from the normal distribution?

It has heavier tails, because it allows for the standard deviation being estimated from the sample. With 5 degrees of freedom, 10.2% of the area lies beyond ±2, against 4.6% for the standard normal. The gap closes as the degrees of freedom grow: the two-sided 5% critical value is 2.228 at 10 df, 2.042 at 30 df and 1.960 for the normal.

### How accurate is the t, chi-square & F distribution calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 9 worked examples whose answers come from independent sources; for example, “t with 10 df, P(T ≤ 2.1)” is checked against Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int).

### Where does the method come from?

NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6 Gallery of distributions and §1.3.6.7 critical value tables; Abramowitz & Stegun, Handbook of Mathematical Functions, chapter 26 (t, χ², F probability integrals).

## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6 Gallery of distributions and §1.3.6.7 critical value tables](https://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm)
- Abramowitz & Stegun, Handbook of Mathematical Functions, chapter 26 (t, χ², F probability integrals)