一、数据
数据：
| 数据1 | 数据2 | 数据3 | 子组ID |
|---|---|---|---|
| 601.4 | 598 | 601.6 | 1 |
| 601.6 | 599.8 | 600.4 | 1 |
| 598 | 600 | 598.4 | 1 |
| 601.4 | 599.8 | 600 | 1 |
| 599.4 | 600 | 596.8 | 1 |
| 600 | 600 | 602.8 | 2 |
| 600.2 | 598.8 | 600.8 | 2 |
| 601.2 | 598.2 | 603.6 | 2 |
| 598.4 | 599.4 | 604.2 | 2 |
| 599 | 599.6 | 602.4 | 2 |
| 601.2 | 599.4 | 598.4 | 3 |
| 601 | 599.4 | 599.6 | 3 |
| 600.8 | 600 | 603.4 | 3 |
| 597.6 | 598.8 | 600.6 | 3 |
| 601.6 | 599.2 | 598.4 | 3 |
| 599.4 | 599.4 | 598.2 | 4 |
| 601.2 | 599.6 | 602 | 4 |
| 598.4 | 599 | 599.4 | 4 |
| 599.2 | 599.2 | 599.4 | 4 |
| 598.8 | 600.6 | 600.8 | 4 |
| 601.4 | 598.8 | 600.8 | 5 |
| 599 | 598.8 | 598.6 | 5 |
| 601 | 599.8 | 600 | 5 |
| 601.6 | 599.2 | 600.4 | 5 |
| 601.4 | 599.4 | 600.8 | 5 |
| 601.4 | 600 | 600.8 | 6 |
| 598.8 | 600.2 | 597.2 | 6 |
| 601.4 | 600.2 | 600.4 | 6 |
| 598.4 | 599.6 | 599.8 | 6 |
| 601.6 | 599 | 596.4 | 6 |
| 598.8 | 599 | 600.4 | 7 |
| 601.2 | 599.8 | 598.2 | 7 |
| 599.6 | 600.8 | 598.6 | 7 |
| 601.2 | 598.8 | 599.6 | 7 |
| 598.2 | 598.2 | 599 | 7 |
| 598.8 | 600 | 598.2 | 8 |
| 597.8 | 599.2 | 599.4 | 8 |
| 598.2 | 599.8 | 599.4 | 8 |
| 598.2 | 601.2 | 600.2 | 8 |
| 598.2 | 600.4 | 599 | 8 |
| 601.2 | 600.2 | 599.4 | 9 |
| 600 | 599.6 | 598 | 9 |
| 598.8 | 599.6 | 597.6 | 9 |
| 599.4 | 599.6 | 598 | 9 |
| 597.2 | 600.2 | 597.6 | 9 |
| 600.8 | 599.2 | 601.2 | 10 |
| 600.6 | 599 | 599 | 10 |
| 599.6 | 599.6 | 600.4 | 10 |
| 599.4 | 600.4 | 600.6 | 10 |
| 598 | 600 | 599 | 10 |
| 600.8 | 599 | 602.2 | 11 |
| 597.8 | 599.6 | 599.8 | 11 |
| 599.2 | 599.4 | 599.8 | 11 |
| 599.2 | 599.2 | 601 | 11 |
| 600.6 | 597.8 | 601.6 | 11 |
| 598 | 600.4 | 601.6 | 12 |
| 598 | 599.6 | 600.2 | 12 |
| 598.8 | 600 | 601.8 | 12 |
| 601 | 600.8 | 601.2 | 12 |
| 600.8 | 600.4 | 597.6 | 12 |
| 598.8 | 599.4 | 599.8 | 13 |
| 599.4 | 599 | 602.8 | 13 |
| 601 | 598.4 | 600 | 13 |
| 598.8 | 599 | 599.6 | 13 |
| 599.6 | 599.6 | 602.2 | 13 |
| 599 | 598.8 | 603.8 | 14 |
| 600.4 | 599.2 | 603.6 | 14 |
| 598.4 | 599.6 | 601.8 | 14 |
| 602.2 | 598.6 | 602 | 14 |
| 601 | 599.8 | 603.6 | 14 |
| 601.4 | 599.6 | 600.8 | 15 |
| 601 | 599.2 | 600.2 | 15 |
| 601.2 | 599.6 | 600.4 | 15 |
| 601.4 | 600.2 | 600.2 | 15 |
| 601.8 | 599.8 | 602.2 | 15 |
| 601.6 | 599.6 | 598 | 16 |
| 601 | 600 | 598.4 | 16 |
| 600.2 | 599.6 | 600.8 | 16 |
| 599 | 599.2 | 602.8 | 16 |
| 601.2 | 598.6 | 597.6 | 16 |
| 601.2 | 599.6 | 601.6 | 17 |
| 601.2 | 601.2 | 603.4 | 17 |
| 601.2 | 599.6 | 597 | 17 |
| 601 | 600.2 | 599.8 | 17 |
| 601 | 600 | 597.8 | 17 |
| 601.4 | 600 | 602.4 | 18 |
| 601.4 | 599.4 | 602.2 | 18 |
| 598.8 | 599.8 | 600.6 | 18 |
| 598.8 | 599.2 | 596.2 | 18 |
| 598.8 | 599.6 | 602.4 | 18 |
| 598.2 | 599.4 | 601.4 | 19 |
| 601.8 | 600 | 599.2 | 19 |
| 601 | 600 | 601.6 | 19 |
| 601.4 | 599.2 | 600.4 | 19 |
| 601.4 | 599.4 | 598 | 19 |
| 599 | 599.6 | 601.2 | 20 |
| 601.4 | 599.8 | 604.2 | 20 |
| 601.8 | 599 | 600.2 | 20 |
| 601.6 | 599.6 | 600 | 20 |
| 601.2 | 599.4 | 596.8 | 20 |
数据4：
| 子组ID | PH值 |
|---|---|
| 1 | 6.05 |
| 2 | 5.99 |
| 3 | 6.11 |
| 4 | 6.13 |
| 5 | 5.87 |
| 6 | 6.05 |
| 7 | 6.23 |
| 8 | 6.49 |
| 9 | 6.15 |
| 10 | 5.89 |
| 11 | 5.87 |
| 12 | 5.99 |
| 13 | 6.07 |
| 14 | 6.17 |
| 15 | 5.86 |
| 16 | 6.07 |
| 17 | 6.01 |
| 18 | 5.87 |
| 19 | 5.66 |
| 20 | 5.58 |
| 21 | 5.62 |
| 22 | 5.89 |
| 23 | 6.02 |
| 24 | 5.93 |
| 25 | 6.05 |
数据5：
| 子组ID | \(x_1\) | \(x_2\) | \(x_3\) |
|---|---|---|---|
| 1 | 1.504 | 4.075 | 1.971 |
| 2 | 1.685 | 4.599 | 2.26 |
| 3 | 1.529 | 4.1 | 1.994 |
| 4 | 1.554 | 4.19 | 2.024 |
| 5 | 1.604 | 4.275 | 2.063 |
| 6 | 1.664 | 4.341 | 2.12 |
| 7 | 1.789 | 4.981 | 2.287 |
| 8 | 1.723 | 4.416 | 2.166 |
| 9 | 1.831 | 5.196 | 2.285 |
| 10 | 1.622 | 4.353 | 2.135 |
| 11 | 1.683 | 4.396 | 2.154 |
| 12 | 1.598 | 4.329 | 2.18 |
| 13 | 1.847 | 5.168 | 2.331 |
| 14 | 1.793 | 4.547 | 2.184 |
| 15 | 1.886 | 5.259 | 2.389 |
| 16 | 1.631 | 4.338 | 2.073 |
| 17 | 1.543 | 4.204 | 2.151 |
| 18 | 1.665 | 4.48 | 2.282 |
| 19 | 1.578 | 4.349 | 2.128 |
| 20 | 1.533 | 4.28 | 2.039 |
| 21 | 1.674 | 4.504 | 2.192 |
| 22 | 1.749 | 4.371 | 2.155 |
| 23 | 1.83 | 5.094 | 2.436 |
| 24 | 1.813 | 4.989 | 2.428 |
| 25 | 1.73 | 4.396 | 2.16 |
二、控制限
-
子组变量控制图的 n＞1
-
w 的取值范围 [2, 100]
1. Xbar-R 控制图的样本均值图
\(UCL\ =\ \mu\ +\ k\frac{\sigma}{\sqrt{n_i}}\)
\(CL\ =\ \mu\ =\ \bar{X}\)
\(LCL\ =\ \mu\ -\ k\frac{\sigma}{\sqrt{n_i}}\)
\(\sigma\ =\ \frac{Rbar}{d_2(n_i)}\)
\(μ = \bar{X}\)：过程均值
\(k\)：检验 1 的参数，默认为 3
\(n_{i}\)：子组 i 的观测值个数
\(Rbar\)：子组极差的均值
\(d_2\left(n_i\right)\)：与括号中指定的值相对应的无偏常量 \(d_2\)值
\(d_{2}(n_i)=3.4873+0.0250141 \times n_i-0.00009823 \times n_i^{2}\ \ \ \ n_i\in[51,100]\)
实例：数据1
2. Xbar-S 控制图的样本均值图
\(UCL\ =\ \mu\ +\ k\frac{\sigma}{\sqrt{n_i}}\)
\(CL\ =\ \mu\ =\ \bar{X}\)
\(LCL\ =\ \mu\ -\ k\frac{\sigma}{\sqrt{n_i}}\)
\(\sigma\ =\ \frac{Sbar}{C_4(n_i)}\)
\(μ = \bar{X}\)：过程均值
\(k\)：检验 1 的参数，默认为 3
\(n_{i}\)：子组 i 的观测值个数
\(Sbar\)：子组标准差的均值
\(C_4(·)\)：与括号指定的值相对应的无偏常量 \(C_4\) 值。计算公式为：\(C_4(N)\ =\ \sqrt{\frac{2}{N-1}}\frac{\gamma(\frac{N}{2})}{\gamma(\frac{N-1}{2})}\)
实例：数据1
3. Xbar 控制图
\(UCL\ =\ \mu\ +\ k\frac{\sigma}{\sqrt{n_i}}\)
\(CL\ =\ \mu\ =\ \bar{X}\)
\(LCL\ =\ \mu\ -\ k\frac{\sigma}{\sqrt{n_i}}\)
\(\sigma\ =\ \frac{\sqrt{\mu_v}}{C_4(d+1)}\)
\(μ = \bar{X}\)：过程均值
\(k\)：检验 1 的参数，默认为 3
\(n_{i}\)：子组 i 的观测值个数
\(\mu_v\)：子组方差的均值
\(C_4(·)\)：与括号指定的值相对应的无偏常量 \(C_4\)值。计算公式为：\(C_4(N)\ =\ \sqrt{\frac{2}{N-1}}\frac{\gamma(\frac{N}{2})}{\gamma(\frac{N-1}{2})}\)
\(d\)：自由度。计算公式为：\(\sum{(n_i\ -\ 1})\)
实例：数据1
4. R 控制图
\(UCL\ =\ \mu_R\ +\ k\sigma\)
\(CL\ =\ \mu_R\ =\ Rbar\)
\(LCL\ =\ \mu_R\ -\ k\sigma\)
\(\sigma\ =\ \frac{d_3\left(n_i\right)}{d_2\left(n_i\right)}·Rbar\)
\(μ_R = Rbar\)：子组极差的均值
\(k\)：检验 1 的参数，默认为 3
\(n_{i}\)：子组 i 的观测值个数
\(d_2\left(n_i\right)\)：与括号中指定的值相对应的无偏常量 \(d_2\) 值
\(d_3\left(n_i\right)\)：与括号中指定的值相对应的无偏常量 \(d_3\)值
\(d_{2}(n_i)=3.4873+0.0250141 \times n_i-0.00009823 \times n_i^{2}\ \ \ \ n_i\in[51,100]\)
\(d_{3}(n_i)=0.80818-0.0051871 \times n_i-0.00049243 \times n_i^{3}\ \ \ \ n_i\in[51,100]\)
实例：数据1
5. S 控制图
\(UCL\ =\ \mu_S\ +\ k\sigma\)
\(CL\ =\ \mu_S\ =\ Sbar\)
\(LCL\ =\ \mu_S\ -\ k\sigma\)
\(\sigma\ =\ \frac{C_5\left(n_i\right)}{C_4\left(n_i\right)}·Sbar\)
\(μ_S = Sbar\)：子组标准差的均值
\(k\)：检验 1 的参数，默认为 3
\(n_{i}\)：子组 i 的观测值个数
\(C_4(·)\)：与括号指定的值相对应的无偏常量 \(C_4\) 值。计算公式为：\(C_4(N)\ =\ \sqrt{\frac{2}{N-1}}\frac{\gamma(\frac{N}{2})}{\gamma(\frac{N-1}{2})}\)
\(C_5(·)\)：与括号指定的值相对应的无偏常量 \(C_5\) 值。计算公式为：\(C_5(N)\ =\ \sqrt{1-{C_4(N)}^2}\)
实例：数据1
6. 单值控制图
\(UCL\ =\ \mu\ +\ \frac{k\sigma}{\sqrt{n_i}}\ =\ \bar{X}\ +\ k\frac{\bar{R}}{d_2(w)}\)
\(CL\ =\ \mu\ =\ \bar{X}\)
\(LCL\ =\ \mu\ -\ \frac{k\sigma}{\sqrt{n_i}}\ =\ \bar{X}\ -\ k\frac{\bar{R}}{d_2(w)}\)
\(\sigma\ =\ \frac{\bar{R}}{d_2(w)}\)
\(μ = \bar{X}\)：过程均值
\(k\)：检验 1 的参数，默认为 3
\(n_i\)：子组 i 的观测值个数，取值为 1
\(w\)：移动极差长度，默认为 2
\(\bar{R}\)：移动极差的均值
\(d_2\left(n_i\right)\)：与括号中指定的值相对应的无偏常量 \(d_2\) 值
实例：数据4
7. 移动极差控制图
\(UCL\ =\ \bar{R}\ +\ k\sigma_R\ =\ \bar{R}+\ k\frac{d_3(w)}{d_2(w)}\bar{R}\)
\(CL\ =\ \bar{R}\)
\(LCL\ =\ \bar{R}\ -\ k\sigma_R\ =\ \bar{R}-\ k\frac{d_3(w)}{d_2(w)}\bar{R}\)
\(\sigma\ =\ \frac{\bar{R}}{d_2(w)}\)
\(\sigma_R\ =\ d_3(w)\sigma\ =\ \frac{d_3(w)}{d_2(w)}\bar{R}\)
\(k\)：检验 1 的参数，默认为 3
\(w\)：移动极差长度，默认为 2
\(\bar{R}\)：移动极差的均值
\(d_2\left(n_i\right)\)：与括号中指定的值相对应的无偏常量 \(d_2\)值
\(d_3\left(n_i\right)\)：与括号中指定的值相对应的无偏常量 \(d_3\) 值
实例：数据4
8. MA 控制图
当 \(n_i>1\)时：
若 \(i<mv\)：
\(UCL = \mu\ +\ k(\frac{\sigma}{i})\sqrt{\frac{i}{n_i}}\)
\(LCL = \mu\ -\ k(\frac{\sigma}{i})\sqrt{\frac{i}{n_i}}\)
若 \(i≥mv\)：
\(UCL= \mu\ +\ k(\frac{\sigma}{mv})\sqrt{\frac{mv}{n_i}}\)
\(LCL= \mu\ -\ k(\frac{\sigma}{mv})\sqrt{\frac{mv}{n_i}}\)
\(\sigma\ =\ \frac{\sqrt{\mu_v}}{C_4(d+1)}\)
当 \(n_i=1\) 时：
若 \(i<mv\)：
\(UCL = \mu\ +\ k(\frac{\sigma}{i})\sqrt{\frac{i}{n_i}}\)
\(LCL = \mu\ -\ k(\frac{\sigma}{i})\sqrt{\frac{i}{n_i}}\)
若 \(i≥mv\)：
\(UCL= \mu\ +\ k(\frac{\sigma}{mv})\sqrt{\frac{mv}{n_i}}\)
\(LCL= \mu\ -\ k(\frac{\sigma}{mv})\sqrt{\frac{mv}{n_i}}\)
\(\sigma\ =\ \frac{\bar{R}}{d_2(w)}\)
\(i\)：遍历子组个数的取值（不是子组的观测值个数）
\(mv\)：移动均值长度，默认为3
实例：
数据1：\(n_i>1\)
数据4：\(n_i=1\)
9. EWMA 控制图
当 \(n_i>1\) 时：
\(UCL\ =\ \mu\ +\ k\sigma_Z\ = \mu\ +\ k\frac{\sigma}{\sqrt{n_i}}\sqrt{(\frac{r}{2-r})\left[1-\left(1-r\right)^{2i}\right]}\)
\(CL\ =\ \mu\ =\ \bar{X}\)
\(LCL\ =\ \mu\ -\ k\sigma_Z\ = \mu\ -\ k\frac{\sigma}{\sqrt{n_i}}\sqrt{(\frac{r}{2-r})\left[1-\left(1-r\right)^{2i}\right]}\)
\(\sigma\ =\ \frac{\sqrt{\mu_v}}{C_4(d+1)}\)
当 \(n_i=1\) 时：
\(UCL\ =\ \mu\ +\ k\sigma_Z\ = \mu\ +\ k\frac{\sigma}{\sqrt{n_i}}\sqrt{(\frac{r}{2-r})\left[1-\left(1-r\right)^{2i}\right]}\)
\(CL\ =\ \mu\ =\ \bar{X}\)
\(LCL\ =\ \mu\ -\ k\sigma_Z\ = \mu\ -\ k\frac{\sigma}{\sqrt{n_i}}\sqrt{(\frac{r}{2-r})\left[1-\left(1-r\right)^{2i}\right]}\)
\(\sigma\ =\ \frac{\bar{R}}{d_2(w)}\)
\(r\)：EWMA权重，默认为0.2
实例：
数据1：\(n_i>1\)
数据4：\(n_i=1\)
10. T方控制图
子组中的数据：数据1、数据2
\(UCL = \frac{p(m-1)(n-1)}{mn-m-p+1}F(1-p\alpha,\ p,\ mn-m-p+1)\)
\(LCL = 0\)
单个观测值：数据4
\(UCL=\frac{(m-1)^2}{m}\beta(1-\alpha,\ \frac{p}{2},\ \frac{m-p-1}{2})\)
\(LCL=0\)
\(\alpha\)：固定值 \(0.00135\)
\(p\)：特征数
\(m\)：样本数
\(n\)：样本大小
\(F\)：F 分布 f.ppf(1-p*σ, p, m*n-m-p+1)
\(\beta\)：beta 分布 beta.ppf(1-σ, p/2, (m-p-1)/2)
三、描绘点
1. xbar 控制图
数据1：
第一个描绘点：
\(\frac{601.4+601.6+598.0+601.4+599.4}{5} = 600.36\)
第二个描绘点：
\(\frac{600.0+600.2+601.2+598.4+599.0}{5} = 599.76\)
第三个描绘点：
\(\frac{601.2+601.0+600.8+597.6+601.6}{5} = 600.44\)
第四个描绘点：
\(\frac{599.4+601.2+598.4+599.2+598.8}{5} = 599.4\)
2. S 控制图
数据1：
第一个描绘点：
\(\sqrt[2]\frac{[(601.4-600.36)^2+(601.6-600.36)^2+(598.0-600.36)^2+(601.4-600.36)^2+(599.4-600.36)^2]}{4} = 1.596\)
第二个描绘点：
\(\sqrt[2]\frac{[(600-599.76)^2+(600.2-599.76)^2+(601.2-599.76)^2+(598.4-599.76)^2+(599-599.76)^2]}{4} = 1.090\)
第三个描绘点：
\(\sqrt[2]\frac{[(601.2-600.44)^2+(601-600.44)^2+(600.8-600.44)^2+(597.6-600.44)^2+(601.6-600.44)^2]}{4} = 1.615\)
第四个描绘点：
\(\sqrt[2]\frac{[(599.4-599.4)^2+(601.2-599.4)^2+(598.4-599.4)^2+(599.2-599.4)^2+(598.8-599.4)^2]}{4} = 1.077\)
3. R 控制图
数据1：
第一个描绘点：\(601.6-598 = 3.6\)
第二个描绘点：\(601.2-598.4 = 2.8\)
第三个描绘点：\(601.6-597.6 = 4\)
4. 单值控制图
数据1、数据4：就是原始数据
5. 移动极差控制图
数据1：
第二个描绘点：\(601.6-601.4 = 0.2\)
第三个描绘点：\(601.6-598 = 3.6\)
第四个描绘点：\(601.4-598 = 3.4\)
数据4：
第二个描绘点：\(6.05-5.99=0.06\)
第三个描绘点：\(6.11-5.99 = 0.12\)
第四个描绘点：\(6.13-6.11 = 0.02\)
6. MA 控制图
数据1：
第一个描绘点：
\(\frac{601.4+601.6+598.0+601.4+599.4}{5} = 600.36\)
第二个描绘点：
\(\frac{\frac{601.4+601.6+598.0+601.4+599.4}{5}+\frac{600.0+600.2+601.2+598.4+599.0}{5}}{2} = 600.06\)
第三个描绘点：
\(\frac{\frac{601.4+601.6+598.0+601.4+599.4}{5}+\frac{600.0+600.2+601.2+598.4+599.0}{5}+\frac{601.2+601.0+600.8+597.6+601.6}{5}}{mv} = 600.1876\)
第四个描绘点：
\(\frac{\frac{600.0+600.2+601.2+598.4+599.0}{5}+\frac{601.2+601.0+600.8+597.6+601.6}{5}+\frac{599.4+601.2+598.4+599.2+598.8}{5}}{mv} = 600.1876\)
数据4:
第一个描绘点：\(6.05\)
第二个描绘点：\(\frac{6.05+5.99}{2}=6.02\)
第三个描绘点：\(\frac{6.05+5.99+6.11}{mv}=6.05\)
第四个描绘点：\(\frac{5.99+6.11+6.13}{mv}=6.0767\)
7. EWMA 控制图
数据1：
第一个描绘点中的
np.mean(datas)是对所有值求平均。除第一个描绘点外，其他描绘点的计算都与上一个描绘点有关。
第一个描绘点：\(r\times\frac{601.4+601.6+598.0+601.4+599.4}{5}+(1-r)\times(np.mean(datas))=600.130\)
第二个描绘点：\(r\times\frac{600.0+600.2+601.2+598.4+599.0}{5}+(1-r)\times600.13=600.056\)
第三个描绘点：\(r\times\frac{601.2+601.0+600.8+597.6+601.6}{5}+(1-r)\times600.056=600.133\)
第四个描绘点：\(r\times\frac{599.4+601.2+598.4+599.2+598.8}{5}+(1-r)\times600.133=599.986\)
数据4：
第一个描绘点：\(r\times6.05+(1-r)\times(np.mean(datas))=5.9978\)
第二个描绘点：\(r\times5.99+(1-r)\times5.9978=5.9963\)
第三个描绘点：\(r\times6.11+(1-r)\times5.9963=6.0190\)
第四个描绘点：\(r\times6.13+(1-r)\times6.1090=6.0412\)
8. T方控制图
数据1、数据2：
| 子组ID | \(\overline{x}_1\)（数据1） | \(\overline{x}_2\)（数据2） | \(s_{11}\) | \(s_{12}\) | \(s_{22}\) | 统计量 \(T_i^2\) |
|---|---|---|---|---|---|---|
| 1 | 600.36 | 599.52 | 2.548 | -0.634 | 0.732 | 0.281 |
| 2 | 599.76 | 599.2 | 1.188 | -0.500 | 0.500 | 2.283 |
| 3 | 600.44 | 599.36 | 2.608 | 0.422 | 0.188 | 0.919 |
| 4 | 599.4 | 599.56 | 1.160 | 0.020 | 0.388 | 1.505 |
| 5 | 600.88 | 599.2 | 1.152 | 0.180 | 0.180 | 3.734 |
| 6 | 600.32 | 599.8 | 2.492 | -0.150 | 0.260 | 1.238 |
| 7 | 599.8 | 599.32 | 1.880 | 0.440 | 1.012 | 1.104 |
| 8 | 598.24 | 600.12 | 0.128 | 0.074 | 0.552 | 15.115 |
| 9 | 599.32 | 599.84 | 2.192 | -0.036 | 0.108 | 2.961 |
| 10 | 599.68 | 599.64 | 1.252 | -0.474 | 0.328 | 0.605 |
| 11 | 599.52 | 599. | 1.492 | -0.630 | 0.500 | 5.907 |
| 12 | 599.32 | 600.24 | 2.192 | 0.484 | 0.208 | 8.639 |
| 13 | 599.52 | 599.08 | 0.812 | -0.282 | 0.212 | 4.623 |
| 14 | 600.2 | 599.2 | 2.340 | -0.240 | 0.260 | 1.852 |
| 15 | 601.36 | 599.68 | 0.088 | 0.064 | 0.132 | 5.993 |
| 16 | 600.6 | 599.4 | 1.060 | 0.050 | 0.280 | 1.185 |
| 17 | 601.12 | 600.12 | 0.012 | 0.002 | 0.432 | 9.281 |
| 18 | 599.84 | 599.6 | 2.028 | 0.130 | 0.100 | 0.209 |
| 19 | 600.76 | 599.6 | 2.128 | 0.160 | 0.140 | 1.662 |
| 20 | 601 | 599.48 | 1.300 | -0.110 | 0.092 | 2.886 |
| 均值 | \(\overline{\overline{x}}_1\)=600.072 | \(\overline{\overline{x}}_2\)=599.548 | \(\overline{\overline{s}}_{11}\)=1.5026 | \(\overline{\overline{s}}_{12}\)= -0.0515 | \(\overline{\overline{s}}_{22}\)=0.3302 |
① 样本均值的均值：
\(\overline{\overline{x}}=(\overline{\overline{x}}_1,\ \overline{\overline{x}}_2)'=(600.072,\ 599.548)'\)
② 样本协方差：\(m=20, n=5\)
\(\begin{aligned} & x=[601.4,601.6,598.0,601.4,599.4] \\ & y=[598.0,599.8,600.0,599.8,600.0] \\ & \overline{x}=600.36 \\ & \overline{y}=599.52 \\ \end{aligned}\)
\(\begin{aligned} s_{11} & =\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2 \\ & =\frac{1}{n-1}[(601.4-600.36)^2+(601.6-600.36)^2+(598-600.36)^2+(601.4-600.36)^2+(599.4-600.36)^2] \\ & =2.548 \\ \end{aligned}\)
\(\begin{aligned} s_{12} & = \frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y}) \\ & = \frac{1}{n-1}[(601.4-600.36)(598-599.52)+(601.6-600.36)(599.8-599.52)+(598-600.36)(600-599.52)+(601.4-600.36)(599.8-599.52)+(599.4-600.36)(600-599.52)] \\ & = -0.634 \end{aligned}\)
\(s_1= \begin{bmatrix} s_{11} & s_{12}\\ s_{21} & s_{22} \end{bmatrix} = \begin{bmatrix} 2.548 & -0.634\\ -0.634 & 0.732 \end{bmatrix}\)
③ 样本协方差矩阵均值：\(m\) 个矩阵对应位置的均值。
\(\bar{S} = \begin{bmatrix} \bar{s}_{11} & \bar{s}_{12}\\ \bar{s}_{21} & \bar{s}_{22} \end{bmatrix} =\begin{bmatrix} 1.5026 & -0.0515\\ -0.0515 & 0.3302 \end{bmatrix}\)
④ 样本的统计量 \(T_i^2\)
\(\bar{S}^{-1}=\begin{bmatrix} 0.66908978 & 0.10435531\\ 0.10435531 & 3.04474348 \end{bmatrix}\)
\(\begin{aligned} T_i^2 & = n(\bar{x}-\bar{\bar{x}})'\bar{S}^{-1}(\bar{x}^T-\bar{\bar{x}}) \\ & = 5* \begin{bmatrix} 600.36-600.072 & 599.52-599.548 \end{bmatrix}* \bar{S}^{-1}* \begin{bmatrix} 600.36-600.072 & 599.52-599.548 \end{bmatrix} \\ & = 5* \begin{bmatrix} 0.288 & -0.028 \end{bmatrix}* \bar{S}^{-1}* \begin{bmatrix} 0.288 \\ -0.028 \end{bmatrix} \\ & = 0.281 \end{aligned}\)
数据5：
| 子组ID | \(x_1\) | \(x_2\) | \(x_3\) | 统计量 \(T_i^2\) |
|---|---|---|---|---|
| 1 | 1.504 | 4.075 | 1.971 | 3.6011 |
| 2 | 1.685 | 4.599 | 2.26 | 1.3041 |
| 3 | 1.529 | 4.1 | 1.994 | 2.4936 |
| 4 | 1.554 | 4.19 | 2.024 | 1.9272 |
| 5 | 1.604 | 4.275 | 2.063 | 0.9898 |
| 6 | 1.664 | 4.341 | 2.12 | 0.8281 |
| 7 | 1.789 | 4.981 | 2.287 | 2.1348 |
| 8 | 1.723 | 4.416 | 2.166 | 2.2673 |
| 9 | 1.831 | 5.196 | 2.285 | 7.3106 |
| 10 | 1.622 | 4.353 | 2.135 | 0.3211 |
| 11 | 1.683 | 4.396 | 2.154 | 0.7400 |
| 12 | 1.598 | 4.329 | 2.18 | 2.1391 |
| 13 | 1.847 | 5.168 | 2.331 | 4.0995 |
| 14 | 1.793 | 4.547 | 2.184 | 4.9793 |
| 15 | 1.886 | 5.259 | 2.389 | 4.3210 |
| 16 | 1.631 | 4.338 | 2.073 | 1.1237 |
| 17 | 1.543 | 4.204 | 2.151 | 4.0627 |
| 18 | 1.665 | 4.48 | 2.282 | 4.3832 |
| 19 | 1.578 | 4.349 | 2.128 | 1.5162 |
| 20 | 1.533 | 4.28 | 2.039 | 3.6714 |
| 21 | 1.674 | 4.504 | 2.192 | 0.0990 |
| 22 | 1.749 | 4.371 | 2.155 | 5.3129 |
| 23 | 1.83 | 5.094 | 2.436 | 4.4348 |
| 24 | 1.813 | 4.989 | 2.428 | 4.8074 |
| 25 | 1.73 | 4.396 | 2.16 | 3.1322 |
| \(\bar{x}_1\)=1.6823 | \(\bar{x}_2\)=4.5292 | \(\bar{x}_3\)=2.1835 |
① 样本均值：
\(\bar{x}=(\bar{x}_1,\ \bar{x}_2,\ \bar{x}_3)'=(1.6823,\ 4.5292,\ 2.1835)'\)
② 样本协方差矩阵：
\(\begin{aligned} & x_1=[1.504,1.685,...,1.813,1.73] \\ & x_2=[4.075,4.599,...,4.989,4.396] \\ & x_2=[1.971,2.26,...,2.428,2.16] \\ & \overline{x}_1=1.6823 \\ & \overline{x}_2=4.5292 \\ & \overline{x}_3=2.1835 \\ \end{aligned}\)
\(\begin{aligned} s_{11} & =\frac{1}{m-1}\sum_{i=1}^m(x_{1i}-\bar{x}_1)^2 \\ & =\frac{1}{m-1}[(1.504-1.6823)^2+(1.685-1.6823)^2+...+(1.813-1.6823)^2+(1.73-1.6823)^2] \\ & =0.0128 \\ \end{aligned}\)
\(\begin{aligned} s_{12} & =\frac{1}{m-1}\sum_{i=1}^m(x_{1i}-\bar{x}_1)(x_{2i}-\bar{x}_2) \\ & =\frac{1}{m-1}[(1.504-1.6823)(4.075-4.5292)+(1.685-1.6823)(4.599-4.5292)+...+(1.73-1.6823)(4.396-4.5292)] \\ & =0.0366 \\ \end{aligned}\)
\(S= \begin{bmatrix} s_{11} & s_{12} & s_{13}\\ s_{21} & s_{22} & s_{23}\\ s_{31} & s_{32} & s_{33}\\ \end{bmatrix}= \begin{bmatrix} 0.0128 & 0.0366 & 0.0123\\ 0.0366 & 0.1298 & 0.0412\\ 0.0123 & 0.0412 & 0.0163\\ \end{bmatrix}\)
③ 样本的统计量 \(T_i^2\)
\(\begin{aligned} T_1^2 & =(x_1-\bar{x})'S^{-1}(x_1-\bar{x}) \\ & = \begin{bmatrix} 1.504-1.6823 & 4.075-4.5292 & 1.971-2.1835 \end{bmatrix} *S^{-1} * \begin{bmatrix} 1.504-1.6823 \\ 4.075-4.5292 \\ 1.971-2.1835 \end{bmatrix} \\ & = \begin{bmatrix} -0.1783 & -0.4542 & -0.2125 \end{bmatrix} *S^{-1} * \begin{bmatrix} -0.1783 \\ -0.4542 \\ -0.2125 \end{bmatrix} \\ &=3.6011 \end{aligned}\)