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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.

更新日 検証済みの例：9

試す
結果
結果: 0.968961
小数点以下の最大桁数：6；最も近い値へ、等距離ならゼロから遠い値へ
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
計算方法 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

t, chi-square & F distribution calculatorについて

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.

計算例

t with 10 df, P(T ≤ 2.1)

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

照合元：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
結果
2.228139

照合元：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
結果
18.307038

照合元：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
結果
2.71089

照合元：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

よくある質問

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」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：9。 例えば、「t with 10 df, P(T ≤ 2.1)」はAbramowitz & Stegun 26.7.4 closed form for even ν, evaluated in Python (pyref.t_cdf_int)と照合しています。

この計算方法の出典は何ですか？

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).

この計算機について

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)

出典

  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)

出典と照合済み

この計算機には、独立した出典の解答を使った計算例が 9 件あります。テストに組み込まれており、ここでも実行できます。

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