t, chi-square and F distribution calculator: tail areas, p-values, densities and critical values, plus exponential and uniform distributions.
Mis à jour Exemples vérifiés : 9
96.8961% of the Student t (ν = 10) distribution lies at or below 2.1.
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.
Source de vérification : Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int)
Source de vérification : t table (NIST e-Handbook §1.3.6.7.2): 2.228; bisection on the A&S closed form gives 2.2281388520
Source de vérification : 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
Source de vérification : 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
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.
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.
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.
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.
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.
La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 9. Par exemple, « t with 10 df, P(T ≤ 2.1) » est vérifié à l’aide de Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int).
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).
Ce calculateur comprend 9 exemples résolus dont les réponses proviennent de sources indépendantes. Ils font partie de la suite de tests et peuvent aussi être exécutés ici.
Normal distribution calculator: z-scores, the probability below, above or between values, and the value at any percentile (inverse normal).
t-test calculator for one-sample, two-sample (Welch or pooled) and paired tests, plus z-tests for a mean or proportion: p-value, critical value, decision.
Binomial distribution calculator: exact P(X = k), P(X ≤ k), tails and ranges for binomial, Poisson, geometric, negative binomial and hypergeometric.