S
Q
tot
=
∑
i
=
1
n
(
y
i
−
y
¯
)
2
,
{\displaystyle SQ_{\text{tot}}=\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2},}
n
{\displaystyle n}
Partindo de que
y
i
{\displaystyle y_{i}}
y
¯
{\displaystyle {\bar {y}}}
S
Q
res
=
∑
i
=
1
n
(
y
i
−
y
i
^
)
2
,
{\displaystyle SQ_{\text{res}}=\sum _{i=1}^{n}(y_{i}-{\hat {y_{i}}})^{2},}
y
i
^
{\displaystyle {\hat {y_{i}}}}
y
i
{\displaystyle y_{i}}
Esta equação é a soma dos quadrados dos resíduos, que calcula a parte que não é explicada pelo modelo.
S
Q
exp
=
∑
i
=
1
n
(
y
i
^
−
y
¯
)
2
,
{\displaystyle SQ_{\text{exp}}=\sum _{i=1}^{n}({\hat {y_{i}}}-{\bar {y}})^{2},}
onde
y
i
^
{\displaystyle {\hat {y_{i}}}}
y
i
{\displaystyle y_{i}}
Esta equação, a soma dos quadrados explicada, indica-nos a diferença entre a média das observações e o valor estimado para cada observação, e soma os respectivos quadrados. Quanto menor for a diferença, maior poder explicativo detém o modelo.
Em alguns casos temos:
S
Q
tot
=
S
Q
exp
+
S
Q
res
.
{\displaystyle SQ_{\text{tot}}=SQ_{\text{exp}}+SQ_{\text{res}}.}
R
2
=
S
Q
exp
S
Q
tot
=
1
−
S
Q
res
S
Q
tot
.
{\displaystyle R^{2}={\frac {SQ_{\text{exp}}}{SQ_{\text{tot}}}}=1-{\frac {SQ_{\text{res}}}{SQ_{\text{tot}}}}.}
Para provarmos que o Coeficiente de Determinação equivale ao quadrado do Coeficiente de Correlação, precisamos provar inicialmente:
S
Q
tot
=
n
.
(
y
2
¯
−
y
¯
2
)
{\displaystyle SQ_{\text{tot}}=n.{\biggl (}{\overline {y^{2}}}-{\overline {y}}^{2}{\biggr )}}
S
Q
tot
=
(
y
1
−
y
¯
)
2
+
(
y
2
−
y
¯
)
2
+
⋯
+
(
y
n
−
y
¯
)
2
{\displaystyle SQ_{\text{tot}}=(y_{1}-{\overline {y}})^{2}+(y_{2}-{\overline {y}})^{2}+\cdots +(y_{n}-{\overline {y}})^{2}}
=
(
y
1
2
−
2.
y
1
.
y
¯
+
y
¯
2
)
+
⋯
+
(
y
n
2
−
2.
y
n
.
y
¯
+
y
¯
2
)
{\displaystyle =(y_{1}^{2}-2.y_{1}.{\overline {y}}+{\overline {y}}^{2})+\cdots +(y_{n}^{2}-2.y_{n}.{\overline {y}}+{\overline {y}}^{2})}
=
(
∑
y
2
)
−
(
2.
y
¯
.
∑
y
)
+
(
n
.
y
¯
2
)
{\displaystyle ={\bigl (}\sum {y^{2}}{\bigr )}-{\bigl (}2.{\overline {y}}.\sum {y}{\bigr )}+{\bigl (}n.{\overline {y}}^{2}{\bigr )}}
=
(
n
.
y
2
¯
)
−
2.
y
¯
.
(
n
.
y
¯
)
+
n
.
y
¯
2
{\displaystyle =(n.{\overline {y^{2}}})-2.{\overline {y}}.(n.{\overline {y}})+n.{\overline {y}}^{2}}
=
n
.
(
y
2
¯
−
2.
y
¯
2
+
y
¯
2
)
{\displaystyle =n.({\overline {y^{2}}}-2.{\overline {y}}^{2}+{\overline {y}}^{2})}
=
n
.
(
y
2
¯
−
y
¯
2
)
c
.
q
.
d
.
{\displaystyle =n.({\overline {y^{2}}}-{\overline {y}}^{2})\ \ \ \ \ c.q.d.}
S
Q
r
e
s
=
n
.
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
¯
2
−
x
2
¯
+
S
Q
t
o
t
{\displaystyle SQ_{res}=n.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}+SQ_{tot}}
Inicialmente, precisamos reescrever a expressão do valor estimado pela Regressão Linear :
y
^
k
=
A
.
x
k
+
B
{\displaystyle {\hat {y}}_{k}=A.x_{k}+B}
=
A
.
x
k
+
(
y
¯
−
A
.
x
¯
)
{\displaystyle =A.x_{k}+({\overline {y}}-A.{\overline {x}})}
=
A
.
(
x
k
−
x
¯
)
+
y
¯
{\displaystyle =A.(x_{k}-{\overline {x}})+{\overline {y}}}
S
Q
r
e
s
=
(
y
^
1
−
y
1
)
2
+
(
y
^
2
−
y
2
)
2
+
⋯
+
(
y
^
n
−
y
n
)
2
{\displaystyle SQ_{res}=({\hat {y}}_{1}-y_{1})^{2}+({\hat {y}}_{2}-y_{2})^{2}+\cdots +({\hat {y}}_{n}-y_{n})^{2}}
=
[
A
.
(
x
1
−
x
¯
)
+
y
¯
−
y
1
]
2
+
⋯
+
[
A
.
(
x
n
−
x
¯
)
+
y
¯
−
y
n
]
2
{\displaystyle =[A.(x_{1}-{\overline {x}})+{\overline {y}}-y_{1}]^{2}+\cdots +[A.(x_{n}-{\overline {x}})+{\overline {y}}-y_{n}]^{2}}
=
[
A
.
(
x
1
−
x
¯
)
+
(
y
¯
−
y
1
)
]
2
+
⋯
+
[
A
.
(
x
n
−
x
¯
)
+
(
y
¯
−
y
n
)
]
2
{\displaystyle =[A.(x_{1}-{\overline {x}})+({\overline {y}}-y_{1})]^{2}+\cdots +[A.(x_{n}-{\overline {x}})+({\overline {y}}-y_{n})]^{2}}
=
A
2
.
(
x
1
−
x
¯
)
2
+
2.
A
.
(
x
1
−
x
¯
)
.
(
y
¯
−
y
1
)
+
(
y
¯
−
y
1
)
2
+
⋯
+
A
2
.
(
x
n
−
x
¯
)
2
+
2.
A
.
(
x
n
−
x
¯
)
.
(
y
¯
−
y
n
)
+
(
y
¯
−
y
n
)
2
{\displaystyle =A^{2}.(x_{1}-{\overline {x}})^{2}+2.A.(x_{1}-{\overline {x}}).({\overline {y}}-y_{1})+({\overline {y}}-y_{1})^{2}+\cdots +A^{2}.(x_{n}-{\overline {x}})^{2}+2.A.(x_{n}-{\overline {x}}).({\overline {y}}-y_{n})+({\overline {y}}-y_{n})^{2}}
=
A
2
.
(
x
1
2
−
2.
x
1
.
x
¯
+
x
¯
2
)
+
2.
A
.
(
x
1
.
y
¯
−
x
1
.
y
1
−
x
¯
.
y
¯
+
x
¯
.
y
1
)
+
(
y
¯
2
−
2.
y
¯
.
y
1
+
y
1
2
)
+
⋯
{\displaystyle =A^{2}.(x_{1}^{2}-2.x_{1}.{\overline {x}}+{\overline {x}}^{2})+2.A.(x_{1}.{\overline {y}}-x_{1}.y_{1}-{\overline {x}}.{\overline {y}}+{\overline {x}}.y_{1})+({\overline {y}}^{2}-2.{\overline {y}}.y_{1}+y_{1}^{2})+\cdots }
=
A
2
.
x
1
2
−
2
A
2
.
x
1
.
x
¯
+
A
2
.
x
¯
2
+
2.
A
.
x
1
.
y
¯
−
2.
A
.
x
1
.
y
1
−
2.
A
.
x
¯
.
y
¯
+
2.
A
.
x
¯
.
y
1
+
y
¯
2
−
2.
y
¯
.
y
1
+
y
1
2
+
⋯
{\displaystyle =A^{2}.x_{1}^{2}-2A^{2}.x_{1}.{\overline {x}}+A^{2}.{\overline {x}}^{2}+2.A.x_{1}.{\overline {y}}-2.A.x_{1}.y_{1}-2.A.{\overline {x}}.{\overline {y}}+2.A.{\overline {x}}.y_{1}+{\overline {y}}^{2}-2.{\overline {y}}.y_{1}+y_{1}^{2}+\cdots }
=
A
2
.
(
∑
x
2
)
−
2
A
2
.
x
¯
.
(
∑
x
)
+
n
.
A
2
.
x
¯
2
+
2.
A
.
y
¯
.
(
∑
x
)
−
2.
A
.
(
∑
x
.
y
)
−
2.
A
.
n
.
x
¯
.
y
¯
+
2.
A
.
x
¯
.
(
∑
y
)
+
n
.
y
¯
2
−
2.
y
¯
.
(
∑
y
)
+
(
∑
y
2
)
{\displaystyle =A^{2}.(\sum {x^{2}})-2A^{2}.{\overline {x}}.(\sum {x})+n.A^{2}.{\overline {x}}^{2}+2.A.{\overline {y}}.(\sum {x})-2.A.(\sum {x.y})-2.A.n.{\overline {x}}.{\overline {y}}+2.A.{\overline {x}}.(\sum {y})+n.{\overline {y}}^{2}-2.{\overline {y}}.(\sum {y})+(\sum {y^{2}})}
=
A
2
.
(
n
.
x
2
¯
)
−
2
A
2
.
x
¯
.
(
n
.
x
¯
)
+
n
.
A
2
.
x
¯
2
+
2.
A
.
y
¯
.
(
n
.
x
¯
)
−
2.
A
.
(
n
.
x
y
¯
)
−
2.
A
.
n
.
x
¯
.
y
¯
+
2.
A
.
x
¯
.
(
n
.
y
¯
)
+
n
.
y
¯
2
−
2.
y
¯
.
(
n
.
y
¯
)
+
(
n
.
y
2
¯
)
{\displaystyle =A^{2}.(n.{\overline {x^{2}}})-2A^{2}.{\overline {x}}.(n.{\overline {x}})+n.A^{2}.{\overline {x}}^{2}+2.A.{\overline {y}}.(n.{\overline {x}})-2.A.(n.{\overline {xy}})-2.A.n.{\overline {x}}.{\overline {y}}+2.A.{\overline {x}}.(n.{\overline {y}})+n.{\overline {y}}^{2}-2.{\overline {y}}.(n.{\overline {y}})+(n.{\overline {y^{2}}})}
=
n
.
(
A
2
.
x
2
¯
−
2.
A
2
.
x
¯
2
+
A
2
.
x
¯
2
+
2.
A
.
x
¯
.
y
¯
−
2.
A
.
x
y
¯
−
2.
A
.
x
¯
.
y
¯
+
2.
A
.
x
¯
.
y
¯
+
y
¯
2
−
2.
y
¯
2
+
y
2
¯
)
{\displaystyle =n.(A^{2}.{\overline {x^{2}}}-2.A^{2}.{\overline {x}}^{2}+A^{2}.{\overline {x}}^{2}+2.A.{\overline {x}}.{\overline {y}}-2.A.{\overline {xy}}-2.A.{\overline {x}}.{\overline {y}}+2.A.{\overline {x}}.{\overline {y}}+{\overline {y}}^{2}-2.{\overline {y}}^{2}+{\overline {y^{2}}})}
=
n
.
(
A
2
.
x
2
¯
−
A
2
.
x
¯
2
+
2.
A
.
x
¯
.
y
¯
−
2.
A
.
x
y
¯
−
y
¯
2
+
y
2
¯
)
{\displaystyle =n.(A^{2}.{\overline {x^{2}}}-A^{2}.{\overline {x}}^{2}+2.A.{\overline {x}}.{\overline {y}}-2.A.{\overline {xy}}-{\overline {y}}^{2}+{\overline {y^{2}}})}
=
n
.
[
A
2
.
(
x
2
¯
−
x
¯
2
)
+
2
A
.
(
x
¯
.
y
¯
−
x
y
¯
)
+
y
2
¯
−
y
¯
2
]
{\displaystyle =n.[A^{2}.({\overline {x^{2}}}-{\overline {x}}^{2})+2A.({\overline {x}}.{\overline {y}}-{\overline {xy}})+{\overline {y^{2}}}-{\overline {y}}^{2}]}
=
n
.
[
(
x
¯
.
y
¯
−
x
y
¯
x
¯
2
−
x
2
¯
)
2
.
(
x
2
¯
−
x
¯
2
)
+
2.
x
¯
.
y
¯
−
x
y
¯
x
¯
2
−
x
2
¯
.
(
x
¯
.
y
¯
−
x
y
¯
)
]
+
n
.
(
y
2
¯
−
y
¯
2
)
{\displaystyle =n.\left[\left({\dfrac {{\overline {x}}.{\overline {y}}-{\overline {xy}}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}\right)^{2}.({\overline {x^{2}}}-{\overline {x}}^{2})+2.{\dfrac {{\overline {x}}.{\overline {y}}-{\overline {xy}}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}.({\overline {x}}.{\overline {y}}-{\overline {xy}})\right]+n.({\overline {y^{2}}}-{\overline {y}}^{2})}
=
n
.
[
(
x
¯
.
y
¯
−
x
y
¯
)
2
.
−
(
x
¯
2
−
x
2
¯
)
(
x
¯
2
−
x
2
¯
)
2
+
2.
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
¯
2
−
x
2
¯
]
+
S
Q
tot
{\displaystyle =n.\left[{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}.-({\overline {x}}^{2}-{\overline {x^{2}}})}{({\overline {x}}^{2}-{\overline {x^{2}}})^{2}}}+2.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}\right]+SQ_{\text{tot}}}
=
n
.
[
−
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
¯
2
−
x
2
¯
+
2.
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
¯
2
−
x
2
¯
]
+
S
Q
tot
{\displaystyle =n.\left[{\dfrac {-({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}+2.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}\right]+SQ_{\text{tot}}}
=
n
.
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
¯
2
−
x
2
¯
+
S
Q
y
¯
c
.
q
.
d
.
{\displaystyle =n.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}+SQ_{\overline {y}}\ \ \ \ \ \ c.q.d.}
Teorema 3:
R
2
=
(
x
¯
.
y
¯
−
x
y
¯
)
2
(
x
2
¯
−
x
¯
2
)
.
(
y
2
¯
−
y
¯
2
)
{\displaystyle R^{2}={\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{({\overline {x^{2}}}-{\overline {x}}^{2}).({\overline {y^{2}}}-{\overline {y}}^{2})}}}
Prova:
R
2
=
1
−
S
Q
r
e
s
S
Q
tot
=
S
Q
tot
S
Q
tot
−
S
Q
r
e
s
S
Q
tot
=
S
Q
tot
−
S
Q
r
e
s
S
Q
tot
=
S
Q
tot
−
[
n
.
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
¯
2
−
x
2
¯
+
S
Q
tot
]
S
Q
tot
{\displaystyle R^{2}=1-{\dfrac {SQ_{res}}{SQ_{\text{tot}}}}={\dfrac {SQ_{\text{tot}}}{SQ_{\text{tot}}}}-{\dfrac {SQ_{res}}{SQ_{\text{tot}}}}={\dfrac {SQ_{\text{tot}}-SQ_{res}}{SQ_{\text{tot}}}}={\dfrac {SQ_{\text{tot}}-\left[n.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}+SQ_{\text{tot}}\right]}{SQ_{\text{tot}}}}}
=
n
.
(
x
¯
.
y
¯
−
x
y
¯
)
2
x
2
¯
−
x
¯
2
.
1
S
Q
tot
=
n
.
(
x
¯
.
y
¯
−
x
y
¯
)
2
(
x
2
¯
−
x
¯
2
)
.
n
.
(
y
2
¯
−
y
¯
2
)
=
(
x
¯
.
y
¯
−
x
y
¯
)
2
(
x
2
¯
−
x
¯
2
)
.
(
y
2
¯
−
y
¯
2
)
c
.
q
.
d
.
{\displaystyle =n.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x^{2}}}-{\overline {x}}^{2}}}.{\dfrac {1}{SQ_{\text{tot}}}}=n.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{({\overline {x^{2}}}-{\overline {x}}^{2}).n.({\overline {y^{2}}}-{\overline {y}}^{2})}}={\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{({\overline {x^{2}}}-{\overline {x}}^{2}).({\overline {y^{2}}}-{\overline {y}}^{2})}}\ \ \ \ \ \ c.q.d.}
Teorema 4: ( Coeficiente de Correlação)² = Coeficiente de Determinação
Prova: Coeficiente de Correlação =
R
=
∑
(
x
−
x
¯
)
.
(
y
−
y
¯
)
∑
(
x
−
x
¯
)
2
.
∑
(
y
−
y
¯
)
2
{\displaystyle R={\dfrac {\sum {(x-{\overline {x}}).(y-{\overline {y}})}}{{\sqrt {\sum {(x-{\overline {x}})^{2}}}}.{\sqrt {\sum {(y-{\overline {y}})^{2}}}}}}}
Para elevá-lo ao quadrado, façamos separadamente numerador e denominador:
Quadrado do numerador:
[
∑
(
x
−
x
¯
)
.
(
y
−
y
¯
)
]
2
{\displaystyle [\ \sum {(x-{\overline {x}}).(y-{\overline {y}})}\ ]^{2}}
=
[
∑
(
x
.
y
−
x
.
y
¯
−
x
¯
.
y
+
x
¯
.
y
¯
)
]
2
{\displaystyle =[\ \sum {(x.y-x.{\overline {y}}-{\overline {x}}.y+{\overline {x}}.{\overline {y}})}\ ]^{2}}
=
[
(
x
1
.
y
1
−
x
1
.
y
¯
−
x
¯
.
y
1
+
x
¯
.
y
¯
)
+
⋯
+
(
x
n
.
y
n
−
x
n
.
y
¯
−
x
¯
.
y
n
+
x
¯
.
y
¯
)
]
2
{\displaystyle =[\ (x_{1}.y_{1}-x_{1}.{\overline {y}}-{\overline {x}}.y_{1}+{\overline {x}}.{\overline {y}})+\cdots +(x_{n}.y_{n}-x_{n}.{\overline {y}}-{\overline {x}}.y_{n}+{\overline {x}}.{\overline {y}})\ ]^{2}}
=
[
(
∑
x
.
y
)
−
y
¯
.
(
∑
x
)
−
x
¯
.
(
∑
y
)
+
n
.
x
¯
.
y
¯
]
2
{\displaystyle =[\ (\sum {x.y})-{\overline {y}}.(\sum {x})-{\overline {x}}.(\sum {y})+n.{\overline {x}}.{\overline {y}}\ ]^{2}}
=
[
(
n
.
x
.
y
¯
)
−
y
¯
.
(
n
.
x
¯
)
−
x
¯
.
(
n
.
y
¯
)
+
n
.
x
¯
.
y
¯
]
2
{\displaystyle =[\ (n.{\overline {x.y}})-{\overline {y}}.(n.{\overline {x}})-{\overline {x}}.(n.{\overline {y}})+n.{\overline {x}}.{\overline {y}}\ ]^{2}}
=
[
n
.
(
x
.
y
¯
−
x
¯
.
y
¯
)
]
2
{\displaystyle =[\ n.({\overline {x.y}}-{\overline {x}}.{\overline {y}})\ ]^{2}}
=
n
2
.
(
x
.
y
¯
−
x
¯
.
y
¯
)
2
{\displaystyle =n^{2}.({\overline {x.y}}-{\overline {x}}.{\overline {y}})^{2}}
[
∑
(
x
−
x
¯
)
2
.
∑
(
y
−
y
¯
)
2
]
2
{\displaystyle [\ {\sqrt {\sum {(x-{\overline {x}})^{2}}}}.{\sqrt {\sum {(y-{\overline {y}})^{2}}}}\ ]^{2}}
=
[
∑
(
x
−
x
¯
)
2
]
.
[
∑
(
y
−
y
¯
)
2
]
{\displaystyle =[\ \sum {(x-{\overline {x}})^{2}}\ ].[\ \sum {(y-{\overline {y}})^{2}}\ ]}
=
[
∑
(
x
2
−
2.
x
.
x
¯
+
x
¯
2
)
]
.
[
∑
(
y
2
−
2.
y
.
y
¯
+
y
¯
2
)
]
{\displaystyle =[\ \sum {(x^{2}-2.x.{\overline {x}}+{\overline {x}}^{2})}\ ].[\ \sum {(y^{2}-2.y.{\overline {y}}+{\overline {y}}^{2})}\ ]}
=
[
(
∑
x
2
)
−
2.
x
¯
.
(
∑
x
)
+
n
.
x
¯
2
]
.
[
(
∑
y
2
)
−
2.
y
¯
.
(
∑
y
)
+
n
.
y
¯
2
]
{\displaystyle =[\ (\sum {x^{2}})-2.{\overline {x}}.(\sum {x})+n.{\overline {x}}^{2}\ ].[\ (\sum {y^{2}})-2.{\overline {y}}.(\sum {y})+n.{\overline {y}}^{2}\ ]}
=
[
(
n
.
x
2
¯
)
−
2.
x
¯
.
(
n
.
x
¯
)
+
n
.
x
¯
2
]
.
[
(
n
.
y
2
¯
)
−
2.
y
¯
.
(
n
.
y
¯
)
+
n
.
y
¯
2
]
{\displaystyle =[\ (n.{\overline {x^{2}}})-2.{\overline {x}}.(n.{\overline {x}})+n.{\overline {x}}^{2}\ ].[\ (n.{\overline {y^{2}}})-2.{\overline {y}}.(n.{\overline {y}})+n.{\overline {y}}^{2}\ ]}
=
(
n
.
x
2
¯
−
n
.
x
¯
2
)
.
(
n
.
y
2
¯
−
n
.
y
¯
2
)
{\displaystyle =(n.{\overline {x^{2}}}-n.{\overline {x}}^{2}).(n.{\overline {y^{2}}}-n.{\overline {y}}^{2})}
=
[
n
.
(
x
2
¯
−
x
¯
2
)
]
.
[
n
.
(
y
2
¯
−
y
¯
2
)
]
{\displaystyle =[\ n.({\overline {x^{2}}}-{\overline {x}}^{2})\ ].[\ n.({\overline {y^{2}}}-{\overline {y}}^{2})\ ]}
=
n
2
.
(
x
2
¯
−
x
¯
2
)
.
(
y
2
¯
−
y
¯
2
)
{\displaystyle =n^{2}.({\overline {x^{2}}}-{\overline {x}}^{2}).({\overline {y^{2}}}-{\overline {y}}^{2})}
(Coeficiente de Correlação)² =
(
R
)
2
=
n
2
.
(
x
.
y
¯
−
x
¯
.
y
¯
)
2
n
2
.
(
x
2
¯
−
x
¯
2
)
.
(
y
2
¯
−
y
¯
2
)
{\displaystyle (R)^{2}={\dfrac {n^{2}.({\overline {x.y}}-{\overline {x}}.{\overline {y}})^{2}}{n^{2}.({\overline {x^{2}}}-{\overline {x}}^{2}).({\overline {y^{2}}}-{\overline {y}}^{2})}}}
=
(
x
¯
.
y
¯
−
x
y
¯
)
2
(
x
2
¯
−
x
¯
2
)
.
(
y
2
¯
−
y
¯
2
)
{\displaystyle ={\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{({\overline {x^{2}}}-{\overline {x}}^{2}).({\overline {y^{2}}}-{\overline {y}}^{2})}}}
![{\displaystyle =[A.(x_{1}-{\overline {x}})+{\overline {y}}-y_{1}]^{2}+\cdots +[A.(x_{n}-{\overline {x}})+{\overline {y}}-y_{n}]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb6af4b73c5855632eb93688d20b828702fac17)
![{\displaystyle =[A.(x_{1}-{\overline {x}})+({\overline {y}}-y_{1})]^{2}+\cdots +[A.(x_{n}-{\overline {x}})+({\overline {y}}-y_{n})]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6055281ac885c4e6e940a3cdd26a3078a83e0c5d)
![{\displaystyle =n.[A^{2}.({\overline {x^{2}}}-{\overline {x}}^{2})+2A.({\overline {x}}.{\overline {y}}-{\overline {xy}})+{\overline {y^{2}}}-{\overline {y}}^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0c9b4352d2d1dc325b05fa3e4b0bb686cf7827)
![{\displaystyle =n.\left[\left({\dfrac {{\overline {x}}.{\overline {y}}-{\overline {xy}}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}\right)^{2}.({\overline {x^{2}}}-{\overline {x}}^{2})+2.{\dfrac {{\overline {x}}.{\overline {y}}-{\overline {xy}}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}.({\overline {x}}.{\overline {y}}-{\overline {xy}})\right]+n.({\overline {y^{2}}}-{\overline {y}}^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f04b7477694c057eea414fc9ab1d0396fa73e38)
![{\displaystyle =n.\left[{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}.-({\overline {x}}^{2}-{\overline {x^{2}}})}{({\overline {x}}^{2}-{\overline {x^{2}}})^{2}}}+2.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}\right]+SQ_{\text{tot}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b848222af5356060caef0d3ff0aa273691c8667)
![{\displaystyle =n.\left[{\dfrac {-({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}+2.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}\right]+SQ_{\text{tot}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff412dc87b8e6b15e297a8881a9f83d10e313503)
![{\displaystyle R^{2}=1-{\dfrac {SQ_{res}}{SQ_{\text{tot}}}}={\dfrac {SQ_{\text{tot}}}{SQ_{\text{tot}}}}-{\dfrac {SQ_{res}}{SQ_{\text{tot}}}}={\dfrac {SQ_{\text{tot}}-SQ_{res}}{SQ_{\text{tot}}}}={\dfrac {SQ_{\text{tot}}-\left[n.{\dfrac {({\overline {x}}.{\overline {y}}-{\overline {xy}})^{2}}{{\overline {x}}^{2}-{\overline {x^{2}}}}}+SQ_{\text{tot}}\right]}{SQ_{\text{tot}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaa1d21e624bdc42d43785acf806dba0ab57c3d)
![{\displaystyle [\ \sum {(x-{\overline {x}}).(y-{\overline {y}})}\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0135790cedd0a168a1643fbbf57dcb184ef0f63b)
![{\displaystyle =[\ \sum {(x.y-x.{\overline {y}}-{\overline {x}}.y+{\overline {x}}.{\overline {y}})}\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1b2f3c6ac183be010249635548ba1c66f21c30)
![{\displaystyle =[\ (x_{1}.y_{1}-x_{1}.{\overline {y}}-{\overline {x}}.y_{1}+{\overline {x}}.{\overline {y}})+\cdots +(x_{n}.y_{n}-x_{n}.{\overline {y}}-{\overline {x}}.y_{n}+{\overline {x}}.{\overline {y}})\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2794ae9970666f59f8f0ef931ae402593decc299)
![{\displaystyle =[\ (\sum {x.y})-{\overline {y}}.(\sum {x})-{\overline {x}}.(\sum {y})+n.{\overline {x}}.{\overline {y}}\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4adafd2183413c0c16937ff880fb7a45ec739c89)
![{\displaystyle =[\ (n.{\overline {x.y}})-{\overline {y}}.(n.{\overline {x}})-{\overline {x}}.(n.{\overline {y}})+n.{\overline {x}}.{\overline {y}}\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ef7ffc3f13a471fa323f0ae605b82946fb68b1)
![{\displaystyle =[\ n.({\overline {x.y}}-{\overline {x}}.{\overline {y}})\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/036eca152cbeb89b79282e34ebe722e6610d3552)
![{\displaystyle [\ {\sqrt {\sum {(x-{\overline {x}})^{2}}}}.{\sqrt {\sum {(y-{\overline {y}})^{2}}}}\ ]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a258a742c6c2754ebb2cdad145eb22c161d11c)
![{\displaystyle =[\ \sum {(x-{\overline {x}})^{2}}\ ].[\ \sum {(y-{\overline {y}})^{2}}\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eea7a313c5939f2a93a8845d3604f540b8b512ee)
![{\displaystyle =[\ \sum {(x^{2}-2.x.{\overline {x}}+{\overline {x}}^{2})}\ ].[\ \sum {(y^{2}-2.y.{\overline {y}}+{\overline {y}}^{2})}\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2416484acb189930781758d1895bf9002c51d9ac)
![{\displaystyle =[\ (\sum {x^{2}})-2.{\overline {x}}.(\sum {x})+n.{\overline {x}}^{2}\ ].[\ (\sum {y^{2}})-2.{\overline {y}}.(\sum {y})+n.{\overline {y}}^{2}\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c618d2709bc199110e896392db2561c1cf1f8cf5)
![{\displaystyle =[\ (n.{\overline {x^{2}}})-2.{\overline {x}}.(n.{\overline {x}})+n.{\overline {x}}^{2}\ ].[\ (n.{\overline {y^{2}}})-2.{\overline {y}}.(n.{\overline {y}})+n.{\overline {y}}^{2}\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8368951821adec3eb8812a9f112549d414679b8)
![{\displaystyle =[\ n.({\overline {x^{2}}}-{\overline {x}}^{2})\ ].[\ n.({\overline {y^{2}}}-{\overline {y}}^{2})\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da7a3bee67709f7033780c7e4c5319d4bb76ca58)
