Raw Scores, Deviation Scores & Standard Scores

Introduction to Research Design and Statistics

Raw Scores, Deviation Scores & Standard Scores

Some Definitions

Raw scores are the observed values of the variables.

Deviation scores are obtained by subtracting the mean from the raw scores, deviation score = x = (X - mean). Deviation scores have a mean = 0 and the same standard deviation as the raw scores.

Standard scores (z-scores) are obtained by dividing deviation scores by the standard deviation, z-score = (X - mean)/sd = x/sd. Standard Scores have a mean = 0 and sd = 1.

Using Stata

use http://www.philender.com/courses/data/hsb2, clear
 
summarize write

Variable |     Obs        Mean   Std. Dev.       Min        Max
---------+-----------------------------------------------------
   write |     200      52.775   9.478586         31         67 
 
return list

scalars:
       r(N)        =  200
       r(sum_w)    =  200
       r(mean)     =  52.775
       r(Var)      =  89.84359296482413
       r(sd)       =  9.47858602138653
       r(min)      =  31
       r(max)      =  67
       r(sum)      =  10555
 
generate dwrite = write - r(mean)
 
generate zwrite1 = dwrite/r(sd)

/* also -- zwrite1 = (write - r(mean))/r(sd) */
 
egen zwrite2 = std(write)
 
list write dwrite zwrite1 zwrite2

         write     dwrite    zwrite1    zwrite2 
  1.        52      -.775  -.0817632  -.0817633  
  2.        59      6.225   .6567435   .6567435  
  3.        33    -19.775  -2.086282  -2.086282  
  4.        44     -8.775  -.9257709   -.925771  
  5.        52      -.775  -.0817632  -.0817633  
  6.        52      -.775  -.0817632  -.0817633  
  7.        59      6.225   .6567435   .6567435  
  8.        46     -6.775  -.7147691  -.7147691  
  9.        57      4.225   .4457416   .4457416  
 10.        55      2.225   .2347396   .2347396  
 ...
 
summarize write dwrite zwrite1 zwrite2

Variable |     Obs        Mean   Std. Dev.       Min        Max
---------+-----------------------------------------------------
   write |     200      52.775   9.478586         31         67  
  dwrite |     200    1.58e-07   9.478586    -21.775     14.225  
 zwrite1 |     200    1.84e-08          1  -2.297284   1.500751  
 zwrite2 |     200    5.53e-09          1  -2.297284   1.500751 
   
correlate read write dwrite zwrite1 zwrite2
(obs=200)

         |     read    write   dwrite  zwrite1  zwrite2
---------+---------------------------------------------
    read |   1.0000
   write |   0.5968   1.0000
  dwrite |   0.5968   1.0000   1.0000
 zwrite1 |   0.5968   1.0000   1.0000   1.0000
 zwrite2 |   0.5968   1.0000   1.0000   1.0000   1.0000

Computing Z-scores

Say that you have raw scores of 31, 37, 24, and 28 from a distribution with a mean = 30 and sd = 5.5. What are the the standard scores (z-scores) for each of the raw scores?

zcalc 31 30 5.5     /* findit zcalc*/

z-score for sample observations

      (X - m)       (31 - 30)
 z = ---------  =  ------------------  =  0.18
         s              5.5
 
zcalc 37 30 5.5

z-score for sample observations

      (X - m)       (37 - 30)
 z = ---------  =  ------------------  =  1.27
         s              5.5
 
zcalc 24 30 5.5

z-score for sample observations

      (X - m)       (24 - 30)
 z = ---------  =  ------------------  =  -1.09
         s              5.5
 
zcalc 28 30 5.5

z-score for sample observations

      (X - m)       (28 - 30)
 z = ---------  =  ------------------  =  -0.36
         s              5.5

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Phil Ender, 15Jan98