# 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.

Versione interattiva: https://www.calcopenly.com/it/statistics/t-chi-square-f-distribution-calculator
Argomento: Calcolatori di statistica e probabilità

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.

## Dati

- **Distribution** (opzioni: 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** (opzioni: 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.

## Risultati

- Risultato — risultato principale
- Complement 1 − P
- Mean
- Variance

## Formula

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

## Esempi svolti

### t with 10 df, P(T ≤ 2.1)

- Distribution: Student t
- Degrees of freedom: 10
- Find: P(X ≤ x)
- Value x: 2.1
- **Risultato: 0.968961**
- **Complement 1 − P: 0.031039**
- Fonte di verifica: 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
- **Risultato: 2.228139**
- Fonte di verifica: 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
- **Risultato: 18.307038**
- Fonte di verifica: 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
- **Risultato: 2.71089**
- Fonte di verifica: 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
- **Risultato: 0.050044**
- Fonte di verifica: 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
- **Risultato: 0.632121**
- **Mean: 2**
- **Variance: 4**
- Fonte di verifica: 1 − e^(−1) = 0.6321205588 (Python math.exp)

## Domande

### 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.

### Quanto è preciso «t, chi-square & F distribution calculator»?

La precisione dipende dai dati inseriti e dalle ipotesi del metodo. Il calcolo decimale usa 50 cifre significative, ma stime, metodi numerici e dati di origine possono essere meno precisi; l’arrotondamento visualizzato non elimina questi limiti. Esempi svolti verificati con fonti indipendenti: 9. Per esempio, «t with 10 df, P(T ≤ 2.1)» viene verificato con Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int).

### Da dove proviene il metodo?

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).

## Fonti

- [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)