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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.

Actualizado Ejemplos verificados: 9

Probar
Resultado
Resultado: 0.968961
Máximo de decimales: 6; Al más cercano; empates alejándose de cero
Complement 1 − P
0.031039
Mean
0
Variance
1.25

96.8961% of the Student t (ν = 10) distribution lies at or below 2.1.

P(X ≤ 2.1)

00.10.20.3-4-2024xDensityx = 2.1
Cómo se calcula S

  1. Distribution

    X∼Student t (ν = 10)X \sim \text{Student t ($\nu$ = 10)}

  2. CDF

    F(t)=1−12Iν/(ν+t2) ⁣(ν2,12) (t≥0)F(t) = 1 - \tfrac12 I_{\nu/(\nu+t^2)}\!\left(\tfrac{\nu}{2}, \tfrac12\right)\ (t \ge 0)

  3. Area to the left

    P(X≤2.1)=0.968961377899P(X \le 2.1) = 0.968961377899

Acerca de t, chi-square & F distribution calculator

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.

Ejemplos resueltos

t with 10 df, P(T ≤ 2.1)

Distribution
Student t
Degrees of freedom
10
Find
P(X ≤ x)
Value x
2.1
Resultado
0.968961
Complement 1 − P
0.031039

Fuente de comprobación: 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
Resultado
2.228139

Fuente de comprobación: 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
Resultado
18.307038

Fuente de comprobación: 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
Resultado
2.71089

Fuente de comprobación: 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

Preguntas

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.

¿Qué precisión tiene «t, chi-square & F distribution calculator»?

La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 9. Por ejemplo, «t with 10 df, P(T ≤ 2.1)» se comprueba con Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int).

¿De dónde procede el método?

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).

Acerca de esta calculadora

P(X≤x)=∫−∞xf(u) du,xp=F−1(p)P(X \le x) = \int_{-\infty}^{x} f(u)\,du,\qquad x_p = F^{-1}(p)

Fuentes

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

Verificado con las referencias

Esta calculadora incluye 9 ejemplos resueltos con respuestas de fuentes independientes. Forman parte del conjunto de pruebas y también puedes ejecutarlos aquí.

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