📐 临界值计算器（z、t、卡方、F）Critical Value Calculator (z, t, chi-square, F)

使用说明与常见问题How-to & FAQ

「临界值计算器（z、t、卡方、F）」是什么？What is the Critical Value Calculator (z, t, chi-square, F) tool?

求标准正态 z、t（自由度 df）、卡方（df）和 F（df₁, df₂）分布在任意显著性水平 α 下的左尾、右尾或双侧临界值；用不完全 Beta / Gamma 函数精确求逆，结果与统计软件一致到小数点后 6 位。本地处理。Critical values of the standard normal z, t (df), chi-square (df) and F (df1, df2) distributions for any significance level α, left-tailed, right-tailed or two-tailed; computed by exactly inverting the incomplete beta / gamma functions, matching statistical software to 6 decimals. Processed locally.

如何使用「临界值计算器（z、t、卡方、F）」？How do I use the Critical Value Calculator (z, t, chi-square, F) tool?

在「临界值计算器（z、t、卡方、F）」中，填写「分布」、「检验方向」、「显著性水平 α」、「自由度 df」、「分子自由度 df₁」等参数，结果会随输入实时计算并显示在下方。In the Critical Value Calculator (z, t, chi-square, F): Fill in "Distribution", "Tail", "Significance level α", "Degrees of freedom df", "Numerator df1"; results are recalculated instantly as you type.

临界值是什么？What is a critical value?

检验统计量的分界点：统计量落在临界值之外（拒绝域）时，在显著性水平 α 下拒绝原假设。例如双侧 z 检验 α = 0.05 时临界值为 ±1.959964。The cut-off for a test statistic: if the statistic falls beyond it (in the rejection region), reject the null hypothesis at significance level α. For a two-tailed z test with α = 0.05 the critical values are ±1.959964.

怎么计算的？How is it computed?

z 用正态分布，t 和 F 用正则化不完全 Beta 函数，卡方用正则化不完全 Gamma 函数，再用二分法精确求出累积概率等于 α（或 1 − α）的点，精度约 10⁻¹⁰，与 R、SciPy 的结果一致。z uses the normal distribution, t and F the regularized incomplete beta function and chi-square the regularized incomplete gamma function; bisection then finds the point whose tail probability equals α (or α/2) to about 10⁻¹⁰, matching R and SciPy.

选哪个分布？Which distribution should I use?

总体标准差已知或大样本均值检验用 z；小样本均值检验（标准差未知）用 t，df = n − 1；方差检验、拟合优度和列联表用卡方；方差比和方差分析用 F。Use z for means with a known population SD or large samples; t for means with an unknown SD (df = n − 1); chi-square for variance tests, goodness of fit and contingency tables; F for variance ratios and ANOVA.

自由度可以是小数吗？Can the degrees of freedom be decimals?

可以，例如 Welch t 检验的自由度常为小数；只要大于 0 即可。Yes — Welch's t test, for example, often gives a fractional df; any value above 0 works.

能举个计算实例吗？Can you show a worked example?

例如在「临界值计算器（z、t、卡方、F）」中填 分布 = 标准正态 z，检验方向 = 双侧，显著性水平 α = 0.05，结果区显示：临界值（双侧）：±1.959964；拒绝域：|z| > 1.959964；分布：z，α = 0.05；ℹ 双侧检验把 α 平均分给两尾（每尾 α/2）；卡方和 F 分布不对称，所以双侧有上下两个不同的临界值。For example, entering Distribution = Standard normal z, Tail = Two-tailed, Significance level α = 0.05 in the Critical Value Calculator (z, t, chi-square, F) gives: Critical values (two-tailed): ±1.959964; Rejection region: |z| > 1.959964; Distribution: z, α = 0.05; ℹ A two-tailed test splits α equally between the tails (α/2 each); chi-square and F are not symmetric, so their two-tailed test has different lower and upper…

哪些输入会被检查？Which inputs are validated?

「临界值计算器（z、t、卡方、F）」会检查这些输入——「显著性水平 α」、「自由度 df」、「分子自由度 df₁」、「分母自由度 df₂」：必须填写；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Critical Value Calculator (z, t, chi-square, F) checks its inputs — "Significance level α", "Degrees of freedom df", "Numerator df1", "Denominator df2": required; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「临界值计算器（z、t、卡方、F）」的设置：「分布」可选 标准正态 z、t 分布、卡方分布、F 分布（默认 标准正态 z）；「检验方向」可选 双侧、右尾、左尾（默认 双侧）；「显著性水平 α」默认 0.05；「自由度 df」默认 10；「分子自由度 df₁」默认 5；「分母自由度 df₂」默认 10。每次生成结果都按当前设置处理。Settings in the Critical Value Calculator (z, t, chi-square, F) — "Distribution": Standard normal z, Student t, Chi-square, F (default Standard normal z); "Tail": Two-tailed, Right-tailed, Left-tailed (default Two-tailed); "Significance level α" default 0.05; "Degrees of freedom df" default 10; "Numerator df1" default 5; "Denominator df2" default 10. The result always uses the current settings.

计算精度如何？How precise are the results?

「临界值计算器（z、t、卡方、F）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Critical Value Calculator (z, t, chi-square, F) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「临界值计算器（z、t、卡方、F）」免费吗？需要注册吗？Is the Critical Value Calculator (z, t, chi-square, F) free? Do I need an account?

「临界值计算器（z、t、卡方、F）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Critical Value Calculator (z, t, chi-square, F) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.