公式 · 分布 · 判据 速查表

检验统计量的构造逻辑 \[ \text{检验统计量} = \frac{\text{样本估计值} - \text{在 } H_0 \text{ 下的参数值}}{\text{估计量的标准误}} \]

集中趋势与离散度 \[ \bar{x} = \frac{\sum x_i}{n}, \qquad \mu = \frac{\sum x_i}{N}, \qquad s^2 = \frac{\sum (x_i-\bar{x})^2}{n-1}, \qquad s = \sqrt{s^2} \]

变异系数 · z 分数 · 四分位距 \[ CV = 100\times\frac{s}{\bar{x}}, \qquad z_i = \frac{x_i-\bar{x}}{s}, \qquad IQR = Q_3-Q_1, \qquad R = x_{\max}-x_{\min} \]

协方差与相关系数 \[ r = \frac{s_{XY}}{s_X s_Y} = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2}\sqrt{\sum (y_i-\bar{y})^2}}, \qquad -1 \le r \le 1 \]

组宽（频数分布） \[ \text{Bin width} = \frac{x_{\max}-x_{\min}}{k} \quad \text{（向上取整到"好看"的整数）} \]

三条法则与独立性 \[ P(A') = 1-P(A),\quad P(A\cup B)=P(A)+P(B)-P(A\cap B),\quad P(A\mid B)=\frac{P(A\cap B)}{P(B)} \] \[ \text{独立} \iff P(A\mid B)=P(A) \iff P(A\cap B)=P(A)P(B) \]

期望与方差 \[ E[X]=\sum_i x_i P(x_i), \qquad \Var(X)=\sum_i (x_i-\mu)^2 P(x_i), \qquad \sigma=\sqrt{\Var(X)} \]

均匀离散 · Bernoulli · 二项 \[ \text{Uniform } U(a,b):\ P=\frac{1}{b-a+1},\ \mu=\frac{a+b}{2},\ \sigma^2=\frac{(b-a+1)^2-1}{12} \] \[ \text{Bernoulli}:\ E=\pi,\ \Var=\pi(1-\pi) \] \[ \text{Binomial}:\ P(X=x)=\binom{n}{x}\pi^x(1-\pi)^{n-x},\ E=n\pi,\ \Var=n\pi(1-\pi) \]

变换与组合 \[ E[aX+b]=aE[X]+b, \qquad \Var(aX+b)=a^2\Var(X), \qquad \sigma_{aX+b}=|a|\sigma_X \] \[ \Var(X+Y)=\Var(X)+\Var(Y)\ (\text{独立}), \qquad \Var(aX+bY)=a^2\Var(X)+b^2\Var(Y)+2ab\,\mathrm{Cov}(X,Y) \]

均匀连续与正态 \[ \text{Uniform } U(a,b):\ f(x)=\frac{1}{b-a},\ P(c<X<d)=\frac{d-c}{b-a},\ \sigma^2=\frac{(b-a)^2}{12} \] \[ \text{Normal}:\ f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}, \qquad z=\frac{x-\mu}{\sigma},\qquad x=\mu+z\sigma \]

标准误与 CLT \[ \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}, \qquad \sigma_p=\sqrt{\frac{\pi(1-\pi)}{n}}, \qquad \bar{X}\sim N\!\left(\mu,\frac{\sigma}{\sqrt{n}}\right)\ \text{(CLT, }n\ge 30\text{)} \]

四类置信区间 \[ \text{均值 }\sigma\text{ 已知}:\ \bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}} \qquad \text{均值 }\sigma\text{ 未知}:\ \bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}\ (\df=n-1) \] \[ \text{比例}:\ p\pm z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}} \qquad \text{均值差}:\ (\bar{x}_1-\bar{x}_2)\pm t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \] \[ \text{配对}:\ \bar{d}\pm t_{\alpha/2}\frac{s_d}{\sqrt{n}} \qquad \text{两比例差}:\ (p_1-p_2)\pm z_{\alpha/2}\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}} \]

两类错误与功效 \[ \alpha = P(\text{拒绝 } H_0 \mid H_0 \text{ 真}), \qquad \beta = P(\text{不拒绝 } H_0 \mid H_0 \text{ 假}), \qquad \text{Power}=1-\beta \] \[ p<\alpha \Rightarrow \text{拒绝 } H_0 \qquad \text{双尾检验} \iff (1-\alpha)\text{ 置信区间（不含 }\mu_0\text{ 则拒绝）} \]

Welch 自由度（Case 3）与合并比例 \[ \df = \frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}} \quad (\text{向下取整}), \qquad p_c = \frac{x_1+x_2}{n_1+n_2} \]

离差平方和与系数 \[ SS_{xx}=\sum(x_i-\bar{x})^2,\quad SS_{yy}=\sum(y_i-\bar{y})^2,\quad SS_{xy}=\sum(x_i-\bar{x})(y_i-\bar{y}) \] \[ b_1=\frac{SS_{xy}}{SS_{xx}}, \qquad b_0=\bar{y}-b_1\bar{x}, \qquad \hat{y}_i=b_0+b_1x_i, \qquad e_i=y_i-\hat{y}_i \]

拟合优度 \[ \mathrm{SST}=\sum(y_i-\bar{y})^2,\quad \mathrm{SSR}=\sum(\hat{y}_i-\bar{y})^2,\quad \mathrm{SSE}=\sum e_i^2,\quad \mathrm{SST}=\mathrm{SSR}+\mathrm{SSE} \] \[ R^2=\frac{\mathrm{SSR}}{\mathrm{SST}}=1-\frac{\mathrm{SSE}}{\mathrm{SST}}=r^2 \]

回归推断 \[ s_e^2=\frac{\mathrm{SSE}}{n-2}, \qquad s_e=\sqrt{s_e^2}, \qquad s_{b_1}=\frac{s_e}{\sqrt{SS_{xx}}}, \qquad s_{b_0}=s_e\sqrt{\frac{1}{n}+\frac{\bar{x}^2}{SS_{xx}}} \] \[ b_1\pm t_{\alpha/2}s_{b_1}, \qquad t_{calc}=\frac{b_1-\beta_1^0}{s_{b_1}}\ (\df=n-2) \]

条件均值 CI vs 单个值预测区间 \[ \text{条件均值 CI}:\ \hat{y}\pm t_{\alpha/2}\,s_e\sqrt{\frac{1}{n}+\frac{(x-\bar{x})^2}{SS_{xx}}} \] \[ \text{单个值 PI}:\ \hat{y}\pm t_{\alpha/2}\,s_e\sqrt{1+\frac{1}{n}+\frac{(x-\bar{x})^2}{SS_{xx}}} \]