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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.

更新于 已验证的示例：9

试一试
结果
结果: 0.968961
最大小数位数：6；取最近值，等距时远离零
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
计算方法 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

关于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.

计算示例

t with 10 df, P(T ≤ 2.1)

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

核验来源：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
结果
2.228139

核验来源：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
结果
18.307038

核验来源：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
结果
2.71089

核验来源：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

常见问题

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.

“t, chi-square & F distribution calculator”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：9。 例如，“t with 10 df, P(T ≤ 2.1)”根据Abramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int)进行核验。

这种方法出自哪里？

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).

关于此计算器

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)

来源

  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)

已对照来源验证

此计算器包含 9 个已解示例，答案来自独立来源。这些示例会在测试套件中运行，你也可以在此运行验证。

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