ν 自由度计算器（t 检验、卡方、方差分析、回归）Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression)
按检验类型计算自由度：单样本或配对 t 检验 (n − 1)、两独立样本 t 检验（合并方差 n₁ + n₂ − 2 或 Welch 近似）、卡方拟合优度 (k − 1)、列联表 (r − 1)(c − 1)、单因素方差分析（组间 k − 1、组内 N − k）和线性回归 (n − p − 1)。本地处理。Get the degrees of freedom for your test: one-sample or paired t (n − 1), two independent samples (pooled n₁ + n₂ − 2 or the Welch approximation), chi-square goodness of fit (k − 1), contingency table (r − 1)(c − 1), one-way ANOVA (between k − 1, within N − k) and linear regression (n − p − 1). Processed locally.
使用说明与常见问题How-to & FAQ
「自由度计算器（t 检验、卡方、方差分析、回归）」是什么？What is the Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) tool?
按检验类型计算自由度：单样本或配对 t 检验 (n − 1)、两独立样本 t 检验（合并方差 n₁ + n₂ − 2 或 Welch 近似）、卡方拟合优度 (k − 1)、列联表 (r − 1)(c − 1)、单因素方差分析（组间 k − 1、组内 N − k）和线性回归 (n − p − 1)。本地处理。Get the degrees of freedom for your test: one-sample or paired t (n − 1), two independent samples (pooled n₁ + n₂ − 2 or the Welch approximation), chi-square goodness of fit (k − 1), contingency table (r − 1)(c − 1), one-way ANOVA (between k − 1, within N − k) and linear regression (n − p − 1). Processed locally.
如何使用「自由度计算器（t 检验、卡方、方差分析、回归）」？How do I use the Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) tool?
在「自由度计算器（t 检验、卡方、方差分析、回归）」中，填写「检验」、「样本量 n（或 n₁）」、「样本量 n₂」、「标准差 s₁」、「标准差 s₂」等参数，结果会随输入实时计算并显示在下方。In the Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression): Fill in "Test", "Sample size n (or n₁)", "Sample size n₂", "Std dev s₁", "Std dev s₂"; results are recalculated instantly as you type.
自由度是什么？What are degrees of freedom?
可以自由变化的独立数值个数。估计了多少个参数，就要从样本量里减去多少，例如估计均值后，n 个数据只剩 n − 1 个自由度。The number of values free to vary. Each estimated parameter uses one up, so after estimating the mean, n values leave n − 1 degrees of freedom.
Welch 自由度怎么算？How is the Welch df calculated?
df = (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]。结果通常不是整数，查表时可保守地向下取整。df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]. It is usually not a whole number; round down to be conservative when using tables.
方差分析的 N 是什么？What is N in ANOVA?
N 是所有组的观测值总数，k 是组数。组间自由度 k − 1，组内自由度 N − k，二者相加等于总自由度 N − 1。N is the total number of observations across all groups and k the number of groups. Between-groups df is k − 1, within-groups df is N − k, and together they make N − 1.
能举个计算实例吗？Can you show a worked example?
例如在「自由度计算器（t 检验、卡方、方差分析、回归）」中填 检验 = 单样本 / 配对 t，样本量 n（或 n₁） = 30，结果区显示：自由度：29；公式：n − 1For example, entering Test = One-sample or paired t, Sample size n (or n₁) = 30 in the Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) gives: Degrees of freedom: 29; Formula: n − 1
哪些输入会被检查？Which inputs are validated?
「自由度计算器（t 检验、卡方、方差分析、回归）」会检查这些输入——「样本量 n（或 n₁）」、「样本量 n₂」：所选方式用到时必须填写，须大于 0；「标准差 s₁」、「标准差 s₂」：所选方式用到时必须填写，不能为负数；「类别数或组数 k」、「自变量个数 p」：所选方式用到时必须填写；「行数」、「列数」：所选方式用到时必须填写，须为不小于 2 的整数；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 Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) checks its inputs — "Sample size n (or n₁)", "Sample size n₂": required when the chosen option uses it, must be greater than 0; "Std dev s₁", "Std dev s₂": required when the chosen option uses it, cannot be negative; "Categories or groups k", "Predictors p": required when the chosen option uses it; "Rows", "Columns": required when the chosen option uses it, must be a whole number of at least 2; numbers above 10¹⁵ in absolute value are…
有哪些选项可以调整？What options can I adjust?
「自由度计算器（t 检验、卡方、方差分析、回归）」的设置：「检验」可选 单样本 / 配对 t、两独立样本（合并方差）、两独立样本（Welch）、卡方拟合优度、卡方列联表、单因素方差分析等（默认 单样本 / 配对 t）；「样本量 n（或 n₁）」默认 30；「样本量 n₂」默认 25；「标准差 s₁」默认 4；「标准差 s₂」默认 6；「类别数或组数 k」默认 4；「行数」默认 3；「列数」默认 4；「自变量个数 p」默认 2。每次生成结果都按当前设置处理。Settings in the Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) — "Test": One-sample or paired t, Two samples, pooled, Two samples, Welch, Chi-square goodness of fit, Chi-square contingency table, One-way ANOVA, etc. (default One-sample or paired t); "Sample size n (or n₁)" default 30; "Sample size n₂" default 25; "Std dev s₁" default 4; "Std dev s₂" default 6; "Categories or groups k" default 4; "Rows" default 3; "Columns" default 4; "Predictors p" default 2. The result always uses the…
计算精度如何？How precise are the results?
「自由度计算器（t 检验、卡方、方差分析、回归）」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 Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「自由度计算器（t 检验、卡方、方差分析、回归）」免费吗？需要注册吗？Is the Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) free? Do I need an account?
「自由度计算器（t 检验、卡方、方差分析、回归）」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 Degrees of Freedom Calculator (t-tests, Chi-square, ANOVA, Regression) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.