차이

문서의 선택한 두 판 사이의 차이를 보여줍니다.

--- 삼원배치법_모수모형_반복없음 [2012/07/25 10:08]
moonrepeat [각 수준의 모평균의 추정 (주효과만이 유의한 경우)]
+++ 삼원배치법_모수모형_반복없음 [2021/03/10 21:42] (현재)
@@ 줄 23: / 줄 23: @@
   * $k$  : [[인자]] $C$ 의 [[수준]] 수 $( k = 1,2, \cdots ,n )$
 ===== 자료의 구조 =====
- ||<|2> [인자] $$B$$ ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ ||
+^  [[인자]]\\ $B$  ^  [[인자]]\\ $C$  ^  [[인자]] $A$  ||||
- || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ ||
+^:::^:::^  $$A_{1}$$  ^  $$A_{2}$$  ^  $$\cdots$$  ^  $$A_{l}$$  |
- |||||||||||| ||
+^  $$B_{1}$$  ^  $$C_{1}$$  |  $$y_{111}$$  |  $$y_{211}$$  |  $$\cdots$$  |  $$y_{l11}$$  |
- ||<|4> $$B_{1}$$ || $$C_{1}$$ || $$y_{111}$$ || $$y_{211}$$ || $$\cdots$$ || $$y_{l11}$$ ||
+^:::^  $$C_{2}$$  |  $$y_{112}$$  |  $$y_{212}$$  |  $$\cdots$$  |  $$y_{l12}$$  |
- || $$C_{2}$$ || $$y_{112}$$ || $$y_{212}$$ || $$\cdots$$ || $$y_{l12}$$ ||
+^:::^  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |
- || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ ||
+^:::^  $$C_{n}$$  |  $$y_{11n}$$  |  $$y_{21n}$$  |  $$\cdots$$  |  $$y_{l1n}$$  |
- || $$C_{n}$$ || $$y_{11n}$$ || $$y_{21n}$$ || $$\cdots$$ || $$y_{l1n}$$ ||
+^  $$B_{2}$$  ^  $$C_{1}$$  |  $$y_{121}$$  |  $$y_{221}$$  |  $$\cdots$$  |  $$y_{l21}$$  |
- ||<|4> $$B_{2}$$ || $$C_{1}$$ || $$y_{121}$$ || $$y_{221}$$ || $$\cdots$$ || $$y_{l21}$$ ||
+^:::^  $$C_{2}$$  |  $$y_{122}$$  |  $$y_{222}$$  |  $$\cdots$$  |  $$y_{l22}$$  |
- || $$C_{2}$$ || $$y_{122}$$ || $$y_{222}$$ || $$\cdots$$ || $$y_{l22}$$ ||
+^:::^  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |
- || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ ||
+^:::^  $$C_{n}$$  |  $$y_{12n}$$  |  $$y_{22n}$$  |  $$\cdots$$  |  $$y_{l2n}$$  |
- || $$C_{n}$$ || $$y_{12n}$$ || $$y_{22n}$$ || $$\cdots$$ || $$y_{l2n}$$ ||
+^  $$\vdots$$  ||  $$\vdots$$  ||||
- |||| $$\vdots$$ |||||||| $$\vdots$$ ||
+^  $$B_{m}$$  ^  $$C_{1}$$  |  $$y_{1m1}$$  |  $$y_{2m1}$$  |  $$\cdots$$  |  $$y_{lm1}$$  |
- ||<|4> $$B_{m}$$ || $$C_{1}$$ || $$y_{1m1}$$ || $$y_{2m1}$$ || $$\cdots$$ || $$y_{lm1}$$ ||
+^:::^  $$C_{2}$$  |  $$y_{1m2}$$  |  $$y_{2m2}$$  |  $$\cdots$$  |  $$y_{lm2}$$  |
- || $$C_{2}$$ || $$y_{1m2}$$ || $$y_{2m2}$$ || $$\cdots$$ || $$y_{lm2}$$ ||
+^:::^  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |
- || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ ||
+^:::^  $$C_{n}$$  |  $$y_{1mn}$$  |  $$y_{2mn}$$  |  $$\cdots$$  |  $$y_{lmn}$$  |
- || $$C_{n}$$ || $$y_{1mn}$$ || $$y_{2mn}$$ || $$\cdots$$ || $$y_{lmn}$$ ||
-  $$AB$$ 2원표
+ $AB$ 2원표
-  ||<|2> [인자] $$B$$ |||||||| [인자] $$A$$ ||<|2> 합계 ||
+^  [[인자]]\\ $B$  ^  [[인자]] $A$  ^^^^  합계  |
-  || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ ||
+^:::^  $$A_{1}$$  ^  $$A_{2}$$  ^  $$\cdots$$  ^  $$A_{l}$$  ^:::|
-  |||||||||||| ||
+|  $$B_{1}$$  |  $$T_{11.}$$  |  $$T_{21.}$$  |  $$\cdots$$  |  $$T_{l1.}$$  |  $$T_{.1.}$$  |
-  || $$B_{1}$$ || $$T_{11.}$$ || $$T_{21.}$$ || $$\cdots$$ || $$T_{l1.}$$ || $$T_{.1.}$$ ||
+|  $$B_{2}$$  |  $$T_{12.}$$  |  $$T_{22.}$$  |  $$\cdots$$  |  $$T_{l2.}$$  |  $$T_{.2.}$$  |
-  || $$B_{2}$$ || $$T_{12.}$$ || $$T_{22.}$$ || $$\cdots$$ || $$T_{l2.}$$ || $$T_{.2.}$$ ||
+|  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |  $$\vdots$$  |
-  || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ ||
+|  $$B_{m}$$  |  $$T_{1m.}$$  |  $$T_{2m.}$$  |  $$\cdots$$  |  $$T_{lm.}$$  |  $$T_{.m.}$$  |
-  || $$B_{m}$$ || $$T_{1m.}$$ || $$T_{2m.}$$ || $$\cdots$$ || $$T_{lm.}$$ || $$T_{.m.}$$ ||
+^  합계  ^  $$T_{1..}$$  ^  $$T_{2..}$$  ^  $$\cdots$$  ^  $$T_{l..}$$  ^  $$T$$  |
-  |||||||||||| ||
-  || 합계 || $$T_{1..}$$ || $$T_{2..}$$ || $$\cdots$$ || $$T_{l..}$$ || $$T$$ ||
-  $$AC$$ 2원표
+ $AC$ 2원표
-  ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ ||<|2> 합계 ||
+^  [[인자]]\\ $C$  ^  [[인자]] $A$  ^^^^  합계  |
-  || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ ||
+^:::^  $$A_{1}$$  ^  $$A_{2}$$  ^  $$\cdots$$  ^  $$A_{l}$$  ^:::|
-  |||||||||||| ||
+|  $$C_{1}$$  |  $$T_{1.1}$$  |  $$T_{2.1}$$  |  $$\cdots$$  |  $$T_{l.1}$$  |  $$T_{..1}$$  |
-  || $$C_{1}$$ || $$T_{1.1}$$ || $$T_{2.1}$$ || $$\cdots$$ || $$T_{l.1}$$ || $$T_{..1}$$ ||
+|  $$C_{2}$$  |  $$T_{1.2}$$  |  $$T_{2.2}$$  |  $$\cdots$$  |  $$T_{l.2}$$  |  $$T_{..2}$$  |
-  || $$C_{2}$$ || $$T_{1.2}$$ || $$T_{2.2}$$ || $$\cdots$$ || $$T_{l.2}$$ || $$T_{..2}$$ ||
+|  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |  $$\vdots$$  |
-  || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ ||
+|  $$C_{n}$$  |  $$T_{1.n}$$  |  $$T_{2.n}$$  |  $$\cdots$$  |  $$T_{l.n}$$  |  $$T_{..n}$$  |
-  || $$C_{n}$$ || $$T_{1.n}$$ || $$T_{2.n}$$ || $$\cdots$$ || $$T_{l.n}$$ || $$T_{..n}$$ ||
+^  합계  ^  $$T_{1..}$$  ^  $$T_{2..}$$  ^  $$\cdots$$  ^  $$T_{l..}$$  ^  $$T$$  |
-  |||||||||||| ||
-  || 합계 || $$T_{1..}$$ || $$T_{2..}$$ || $$\cdots$$ || $$T_{l..}$$ || $$T$$ ||
-  $$BC$$ 2원표
+ $BC$ 2원표
-  ||<|2> [인자] $$C$$ |||||||| [인자] $$B$$ ||<|2> 합계 ||
+^  [[인자]]\\ $C$  ^  [[인자]] $B$  ^^^^  합계  |
-  || $$B_{1}$$ || $$B_{2}$$ || $$\cdots$$ || $$B_{m}$$ ||
+^:::^  $$B_{1}$$  ^  $$B_{2}$$  ^  $$\cdots$$  ^  $$B_{m}$$  ^:::|
-  |||||||||||| ||
+|  $$C_{1}$$  |  $$T_{.11}$$  |  $$T_{.21}$$  |  $$\cdots$$  |  $$T_{.m1}$$  |  $$T_{..1}$$  |
-  || $$C_{1}$$ || $$T_{.11}$$ || $$T_{.21}$$ || $$\cdots$$ || $$T_{.m1}$$ || $$T_{..1}$$ ||
+|  $$C_{2}$$  |  $$T_{.12}$$  |  $$T_{.22}$$  |  $$\cdots$$  |  $$T_{.m2}$$  |  $$T_{..2}$$  |
-  || $$C_{2}$$ || $$T_{.12}$$ || $$T_{.22}$$ || $$\cdots$$ || $$T_{.m2}$$ || $$T_{..2}$$ ||
+|  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |  $$\vdots$$  |
-  || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ ||
+|  $$C_{n}$$  |  $$T_{.1n}$$  |  $$T_{.2n}$$  |  $$\cdots$$  |  $$T_{.mn}$$  |  $$T_{..n}$$  |
-  || $$C_{n}$$ || $$T_{.1n}$$ || $$T_{.2n}$$ || $$\cdots$$ || $$T_{.mn}$$ || $$T_{..n}$$ ||
+^  합계  ^  $$T_{.1.}$$  ^  $$T_{.2.}$$  ^  $$\cdots$$  ^  $$T_{.m.}$$  ^  $$T$$  |
-  |||||||||||| ||
-  || 합계 || $$T_{.1.}$$ || $$T_{.2.}$$ || $$\cdots$$ || $$T_{.m.}$$ || $$T$$ ||
-   || $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{y}_{i..} = \frac{T_{i..}}{mn}$$ ||
+| $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{i..} = \frac{T_{i..}}{mn}$$ |
-   || $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{y}_{.j.} = \frac{T_{.j.}}{ln}$$ ||
+| $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{.j.} = \frac{T_{.j.}}{ln}$$ |
-   || $$T_{..k} = \sum_{i=1}^{l} \sum_{j=1}^{m} y_{ijk}$$ || $$\overline{y}_{..k} = \frac{T_{..k}}{lm}$$ ||
+| $$T_{..k} = \sum_{i=1}^{l} \sum_{j=1}^{m} y_{ijk}$$ | $$\overline{y}_{..k} = \frac{T_{..k}}{lm}$$ |
-   || $$T_{ij.} = \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{y}_{ij.} = \frac{T_{ij.}}{n}$$ ||
+| $$T_{ij.} = \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{ij.} = \frac{T_{ij.}}{n}$$ |
-   || $$T_{i.k} = \sum_{j=1}^{m} y_{ijk}$$ || $$\overline{y}_{i.k} = \frac{T_{i.k}}{m}$$ ||
+| $$T_{i.k} = \sum_{j=1}^{m} y_{ijk}$$ | $$\overline{y}_{i.k} = \frac{T_{i.k}}{m}$$ |
-   || $$T_{.jk} = \sum_{i=1}^{l} y_{ijk}$$ || $$\overline{y}_{.jk} = \frac{T_{.jk}}{l}$$ ||
+| $$T_{.jk} = \sum_{i=1}^{l} y_{ijk}$$ | $$\overline{y}_{.jk} = \frac{T_{.jk}}{l}$$ |
-   || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{\overline{y}} = \frac{T}{lmn} = \frac{T}{N}$$ ||
+| $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{\overline{y}} = \frac{T}{lmn} = \frac{T}{N}$$ |
-   || $$N = lmn$$ || $$CT = \frac{T^{2}}{lmn} = \frac{T^{2}}{N}$$ ||
+| $$N = lmn$$ | $$CT = \frac{T^{2}}{lmn} = \frac{T^{2}}{N}$$ |
 ===== 제곱합 =====
- 개개의 데이터&nbsp&nbsp $$y_{ijk}$$ 와 총편균&nbsp&nbsp $$\overline{\overline{y}}$$ 의 차이는 다음과 같이 7부분으로 나뉘어진다.
+ 개개의 데이터 $y_{ijk}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 7부분으로 나뉘어진다.
-  $$\begin{displaymath}\begin{split} (y_{ijk}-\overline{\overline{y}}) &= (\overline{y}_{i..} - \overline{\overline{y}}) + (\overline{y}_{.j.} - \overline{\overline{y}}) + (\overline{y}_{..k} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}}) + (\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}}) + (\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}}) \\ &+ (y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}}) \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} (y_{ijk}-\overline{\overline{y}}) &= (\overline{y}_{i..} - \overline{\overline{y}}) + (\overline{y}_{.j.} - \overline{\overline{y}}) + (\overline{y}_{..k} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}}) + (\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}}) + (\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}}) \\ &+ (y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}}) \end{split}\end{displaymath}$$
- 양변을 제곱한 후에 모든&nbsp&nbsp $$i, \ j, \ k$$ 에 대하여 합하면 아래의 등식을 얻을 수 있다.
+ 양변을 제곱한 후에 모든 $i, \ j, \ k$에 대하여 합하면 아래의 등식을 얻을 수 있다.
-  $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i..} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.j.} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{..k} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}})^{2} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i..} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.j.} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{..k} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}})^{2} \end{split}\end{displaymath}$$
- 위 식에서 왼쪽 항은 총변동 $$S_{T}$$ 이고, 오른쪽 항은 차례대로&nbsp&nbsp $$A$$ 의 [변동],&nbsp&nbsp $$B$$ 의 [변동],&nbsp&nbsp $$C$$ 의 [변동],&nbsp&nbsp $$A, \ B$$ 의 [교호작용]의 변동,&nbsp&nbsp $$A, \ C$$ 의 [교호작용]의 변동,&nbsp&nbsp $$B, \ C$$ 의 [교호작용]의 변동, [오차변동]인&nbsp&nbsp $$S_{A}$$ , $$S_{B}$$ , $$S_{C}$$ , $$S_{A \times B}$$ , $$S_{A \times C}$$ , $$S_{B \times C}$$ , $$S_{E}$$ 가 된다.
+ 위 식에서 왼쪽 항은 총변동 $S_{T}$이고, 오른쪽 항은 차례대로 $A$의 [[변동]], $B$의 [[변동]], $C$의 [[변동]], $A, \ B$의 [[교호작용]]의 변동, $A, \ C$의 [[교호작용]]의 변동, $B, \ C$의 [[교호작용]]의 변동, [[오차변동]]인 $S_{A}$, $S_{B}$, $S_{C}$, $S_{A \times B}$, $S_{A \times C}$, $S_{B \times C}$, $S_{E}$가 된다.
+ $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mn}-CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mn}-CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{ln}-CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{ln}-CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{..k}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k}^{ \ 2}}{lm}-CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{..k}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k}^{ \ 2}}{lm}-CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{n} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{n} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{y}_{i..}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{y}_{i..}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k}^{ \ 2}}{m} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k}^{ \ 2}}{m} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{y}_{.j.}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{y}_{.j.}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk}^{ \ 2}}{l} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk}^{ \ 2}}{l} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{y}_{ij.}-\overline{y}_{i.k}-\overline{y}_{.jk}+\overline{y}_{i..}+\overline{y}_{.j.}+\overline{y}_{..k}-\overline{\overline{y}})^{2} \\ &= S_{T}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{y}_{ij.}-\overline{y}_{i.k}-\overline{y}_{.jk}+\overline{y}_{i..}+\overline{y}_{.j.}+\overline{y}_{..k}-\overline{\overline{y}})^{2} \\ &= S_{T}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$
 ===== 자유도 =====
  $$\nu_{A}=l-1$$
@@ 줄 220: / 줄 212: @@
  (단, $S_{E}\acute{}=S_{E}+S_{A \times B}+S_{A \times C}+S_{B \times C}, \ \nu_{E}\acute{}=\nu_{E}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C}, \ V_{E}\acute{}=S_{E}\acute{}/\nu_{E}\acute{}$이다.)
- * '''[인자]&nbsp&nbsp $$A$$ 의 [모평균]에 관한 [추정]'''
+ [[인자]] $A$의 [[모평균]]에 관한 [[추정]]
-  $$i$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i})$$ 의 [점추정]값
+ $i$ [[수준]]에서의 [[모평균]] $\mu(A_{i})$의 [[점추정]]값
-   $$\hat{\mu}(A_{i})=\widehat{\mu + a_{i}} = \overline{y}_{i..}$$
+ $$\hat{\mu}(A_{i})=\widehat{\mu + a_{i}} = \overline{y}_{i..}$$
+ $i$ [[수준]]에서의 [[모평균]] $\mu(A_{i})$의 $100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다.
-  $$i$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
+ $\hat{\mu}(A_{i})= \left( \overline{y}_{i..} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \ , \ \overline{y}_{i..} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \right)$
-   $$\hat{\mu}(A_{i})= \left( \overline{y}_{i..} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \ , \ \overline{y}_{i..} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \right)$$
 ----
- * '''[인자]&nbsp&nbsp $$B$$ 의 [모평균]에 관한 [추정]'''
+ [[인자]] $B$의 [[모평균]]에 관한 [[추정]]
-  $$j$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(B_{j})$$ 의 [점추정]값
+ $j$ [[수준]]에서의 [[모평균]] $\mu(B_{j})$의 [[점추정]]값
-   $$\hat{\mu}(B_{j})=\widehat{\mu + b_{j}} = \overline{y}_{.j.}$$
+ $\hat{\mu}(B_{j})=\widehat{\mu + b_{j}} = \overline{y}_{.j.}$
+ $j$ [[수준]]에서의 [[모평균]] $\mu(B_{j})$의 $100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다.
-  $$j$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(B_{j})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
+ $\hat{\mu}(B_{j})= \left( \overline{y}_{.j.} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \ , \ \overline{y}_{.j.} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \right)$
-   $$\hat{\mu}(B_{j})= \left( \overline{y}_{.j.} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \ , \ \overline{y}_{.j.} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \right)$$
 ----
- * '''[인자]&nbsp&nbsp $$C$$ 의 [모평균]에 관한 [추정]'''
+ [[인자]] $C$의 [[모평균]]에 관한 [[추정]]
-  $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(C_{k})$$ 의 [점추정]값
+ $k$ [[수준]]에서의 [[모평균]] $\mu(C_{k})$의 [[점추정]]값
-   $$\hat{\mu}(C_{k})=\widehat{\mu + c_{k}} = \overline{y}_{..k}$$
+ $$\hat{\mu}(C_{k})=\widehat{\mu + c_{k}} = \overline{y}_{..k}$$
+ $k$ [[수준]]에서의 [[모평균]] $\mu(C_{k})$의 $100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다.
-  $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(C_{k})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
+ $$\hat{\mu}(C_{k})= \left( \overline{y}_{..k} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \ , \ \overline{y}_{..k} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \right)$$
-   $$\hat{\mu}(C_{k})= \left( \overline{y}_{..k} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \ , \ \overline{y}_{..k} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \right)$$
 ----
- * '''[인자]&nbsp&nbsp $$A$$ 와&nbsp&nbsp $$B$$ &nbsp&nbsp그리고&nbsp&nbsp $$C$$ 의 [모평균]에 관한 [추정]'''
+ [[인자]] $A$와 $B$ 그리고 $C$의 [[모평균]]에 관한 [[추정]]
-  $$A$$ [인자]의&nbsp&nbsp $$i$$ [수준]과&nbsp&nbsp $$B$$ [인자]의&nbsp&nbsp $$j$$ [수준],&nbsp&nbsp $$C$$ [인자]의&nbsp&nbsp $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i}B_{j}C_{k})$$ 의 [점추정]값
-   $$\hat{\mu}(A_{i}B_{j}C_{k})=\widehat{\mu+a_{i}+b_{j}+c_{k}}=\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2 \overline{\overline{y}}$$
+ $A$ [[인자]]의 $i$ [[수준]]과 $B$ [[인자]]의 $j$ [[수준]], $C$ [[인자]]의 $k$ [[수준]]에서의 [[모평균]] $\mu(A_{i}B_{j}C_{k})$의 [[점추정]]값
+ $\hat{\mu}(A_{i}B_{j}C_{k})=\widehat{\mu+a_{i}+b_{j}+c_{k}}=\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2 \overline{\overline{y}}$
-  $$A$$ [인자]의&nbsp&nbsp $$i$$ [수준]과&nbsp&nbsp $$B$$ [인자]의&nbsp&nbsp $$j$$ [수준],&nbsp&nbsp $$C$$ [인자]의&nbsp&nbsp $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i}B_{j}C_{k})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
+ $A$ [[인자]]의 $i$ [[수준]]과 $B$ [[인자]]의 $j$ [[수준]], $C$ [[인자]]의 $k$ [[수준]]에서의 [[모평균]] $\mu(A_{i}B_{j}C_{k})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다.
-   $$\hat{\mu}(A_{i}B_{j}C_{k})= \left( (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \ , \ (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \right)$$
+ $\hat{\mu}(A_{i}B_{j}C_{k})= \left( (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \ , \ (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \right)$
-   단,&nbsp&nbsp $$n_{e}$$ 는 [유효반복수]이고&nbsp&nbsp $$n_{e} = \frac{lmn}{l+m+n-2}$$ 이다.
+ 단, $n_{e}$는 [[유효반복수]]이고 $n_{e} = \frac{lmn}{l+m+n-2}$이다.
 ----
   * [[실험계획법]]
   * [[삼원배치법]]

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