R入門06

標準化

ｚ得点=(データの値-平均)/標準偏差

> a<-c(13,14,7,12,10,6,8,15,4,14,9,6,10,12,5,12,8,8,12,15)
> a
[1] 13 14  7 12 10  6  8 15  4 14  9  6 10 12  5 12  8  8 12 15
> mean(a)
[1] 10
> sqrt(mean((a-mean(a))^2))
[1] 3.331666

ここまでは標準偏差の式。

> (a-mean(a))/sqrt(mean((a-mean(a))^2))
[1]  0.9004503  1.2006005 -0.9004503  0.6003002  0.0000000 -1.2006005
[7] -0.6003002  1.5007506 -1.8009007  1.2006005 -0.3001501 -1.2006005
[13]  0.0000000  0.6003002 -1.5007506  0.6003002 -0.6003002 -0.6003002
[19]  0.6003002  1.5007506

 

> sqrt(mean(((a-mean(a))/sqrt(mean((a-mean(a))^2))-mean((a-mean(a))/sqrt(mean((a-mean(a))^2))))^2))
[1] 1

 

偏差値＝z得点*10+50
> 10*(a-mean(a))/sqrt(mean((a-mean(a))^2))+50
[1] 59.00450 62.00600 40.99550 56.00300 50.00000 37.99400 43.99700 65.00751 31.99099 62.00600 46.99850 37.99400 50.00000 56.00300
[15] 34.99249 56.00300 43.99700 43.99700 56.00300 65.00751

 

偏差値の平均は50、標準偏差が10になる事を確認
> mean(10*(a-mean(a))/sqrt(mean((a-mean(a))^2))+50)
[1] 50
> sqrt(mean(((10*(a-mean(a))/sqrt(mean((a-mean(a))^2))+50)-mean(10*(a-mean(a))/sqrt(mean((a-mean(a))^2))+50))^2))
[1] 10