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coefficient of linear correlation theorem

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Coefficient of linear Correlation: Theorem: In the case of linear regression we have that R 2 =r 2 Proof: R ! = !∗ ! ! ! = ! ! − ! ! ! ! = ! ! − ( 1 ! ! − 1 ! − 1 ! ! ) ! ! = − = R ! = ! ! − ( 1 ! ! − 1 ( − !" ! ! ) ! − !" ! ! 1 ! ! ) ! ! = = ! ! − ( 1 ! ! − ( − !" ! ! ) − !" ! ! 1 ! ! ) ! ! = = ! ! − ( 1 ! ! − ! + !" ! ! − !" ! ! 1 ! ! ) ! ! = = ! ! − ( ! ! + !" ! ! − !" ! ! 1 ! ! ) ! ! = ! ! − ! ! − !" ! ! + !" ! ! 1 ! ! ! ! = = − !" ! ! + !" ! ! 1 ! ! ! ! = !" ! ! − + 1 ! ! ! ! = !" ! ! !" ! ! = !! ! ! ! ! ! = R ! = =

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Page 1: Coefficient of linear correlation theorem

Coefficient  of  linear  Correlation:    Theorem:  In  the  case  of  linear  regression  we  have  that  R2  =  r2        Proof:  

R! =𝑆!∗!

𝑆!!=𝑆!! − 𝑆!!

𝑆!!=𝑆!! − (

1𝑁 𝑦!! − 𝑎

1𝑁 𝑦! − 𝑏

1𝑁 𝑥!𝑦!)

𝑆!!  

   

𝐴𝑠      𝒂 = 𝒚−𝑺𝒙𝒚𝑺𝒙𝟐

𝒙      𝑎𝑛𝑑      𝒃 =𝑺𝒙𝒚𝑺𝒙𝟐  

   

R! =𝑆!! − (

1𝑁 𝑦!! −

1𝑁 (𝑦 −

𝑆!"𝑆!!

𝑥) 𝑦! −𝑆!"𝑆!!

1𝑁 𝑥!𝑦!)

𝑆!!  =            

 

=𝑆!! − (

1𝑁 𝑦!! − 𝑦(𝑦 −

𝑆!"𝑆!!

𝑥)−𝑆!"𝑆!!

1𝑁 𝑥!𝑦!)

𝑆!!=  

 

=𝑆!! − (

1𝑁 𝑦!! − 𝑦! +

𝑆!"𝑆!!

𝑦𝑥 −𝑆!"𝑆!!

1𝑁 𝑥!𝑦!)

𝑆!!=  

   

=𝑆!! − (𝑆!! +

𝑆!"𝑆!!

𝑦𝑥 −𝑆!"𝑆!!

1𝑁 𝑥!𝑦!)

𝑆!!=𝑆!! − 𝑆!! −

𝑆!"𝑆!!

𝑦𝑥 +𝑆!"𝑆!!

1𝑁 𝑥!𝑦!

𝑆!!  =  

   

=−𝑆!"𝑆!!

𝑦𝑥 +𝑆!"𝑆!!

1𝑁 𝑥!𝑦!

𝑆!!=

𝑆!"𝑆!!

−𝑦𝑥 + 1𝑁 𝑥!𝑦!

𝑆!!  =

𝑆!"𝑆!!

𝑆!"

𝑆!!=

𝑆!!!𝑆!!𝑆!!

=  R!  

   

𝐑𝟐 =𝑺𝒙𝒚𝟐

𝑺𝒚𝟐𝑺𝒙𝟐= 𝒓𝟐  

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