Introduction to Research Design and Statistics

Measures of Variability

Measures of Variability

Measures that give information regarding individual differences, that is, how persons or events vary in their measured scores.

Range

Range = Highest - Lowest

Interquartile Range

IQR = Q3 - Q1 = P(75) - P(25)

Semi-Interquartile Range

Average Deviation

Note: The Average Deviation always equals zero. If you work with a simple example it is easy to see that this is so.

Mean Deviation

Example: 
 X   M  |X-M|
 5 - 3 =  2
 3 - 3 =  0
 2 - 3 =  1
 4 - 3 =  1
 1 - 3 =  2
        -----
          6
 
MD = 6/5 = 1.2

The mean deviation is infrequently used. It is not readily amenable to algebraic manipulation. Further, it does not have many of the additive properties that are possessed by the variance.

Variance/Standard Deviation

(Also see A Tale of Three Variances)

Example:

 X   M  (X-M)  (X-M)2
 5 - 3 =  2      4
 3 - 3 =  0      0
 2 - 3 = -1      1
 4 - 3 =  1      1
 1 - 3 = -2      4
        ----   -----
          0     10
 
s2 = 10/(5-1) = 10/4 = 2.5  

s = 1.58

Coefficient of Variation

A relative measure of variability, defined as:

Examples: 
Sample 1: mean =  90; s =  13; CV = 14.44
Sample 2: mean = 980; s = 130; CV = 13.27

Which sample has the greater variability?

Three Classroom Example

Consider these data from three different classrooms:


Variable |     Obs      Mean   Std. Dev.     Min      Max
---------+-----------------------------------------------
  classa |      30      49.9   4.710468       41       59  
  classb |      30      49.8   9.418983       32       68  
  classc |      30      55.8   4.536442       47       64

Stata Examples

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

                         math score
-------------------------------------------------------------
      Percentiles      Smallest
 1%           36             33
 5%           39             35
10%           40             37       Obs                 200
25%           45             38       Sum of Wgt.         200

50%           52                      Mean             52.645
                        Largest       Std. Dev.      9.368448
75%           59             72
90%         65.5             73       Variance       87.76781
95%         70.5             75       Skewness       .2844115
99%           74             75       Kurtosis       2.337319
 
return list

scalars:
       r(N)        =  200
       r(sum_w)    =  200
       r(mean)     =  52.645
       r(Var)      =  87.76781407035173
       r(sd)       =  9.368447794077294
       r(skewness) =  .2844114876115678
       r(kurtosis) =  2.337319273907357
       r(sum)      =  10529
       r(min)      =  33
       r(max)      =  75
       r(p1)       =  36
       r(p5)       =  39
       r(p10)      =  40
       r(p25)      =  45
       r(p50)      =  52
       r(p75)      =  59
       r(p90)      =  65.5
       r(p95)      =  70.5
       r(p99)      =  74
 
display 100*r(sd)/r(mean)   /* coefficient of variation */
17.795513
 
display r(max)-r(min)       /* range */
42
 
display r(p75)-r(p25)      /* interquartile range */
14
 
display (r(p75)-r(p25))/2  /* semi-interquartile range)
7
 
generate total = read + write + math  

summarize read write math science socst total

Variable |     Obs        Mean   Std. Dev.       Min        Max
---------+-----------------------------------------------------
    read |     200       52.23   10.25294         28         76  
   write |     200      52.775   9.478586         31         67  
    math |     200      52.645   9.368448         33         75  
 science |     200       51.85   9.900891         26         74  
   socst |     200      52.405   10.73579         26         71  
   total |     200      157.65   25.21142        108        211 
 
tabstat read write math science socst total, stat(n mean sd var range iqr) col(stat)

    variable |         N      mean        sd  variance     range       iqr
-------------+------------------------------------------------------------
        read |       200     52.23  10.25294  105.1227        48        16
       write |       200    52.775  9.478586  89.84359        36      14.5
        math |       200    52.645  9.368448  87.76781        42        14
     science |       200     51.85  9.900891  98.02764        48        14
       socst |       200    52.405  10.73579  115.2573        45        15
       total |       200    157.65  25.21142  635.6156       103        39
--------------------------------------------------------------------------

Intro Home Page

Phil Ender, 30Jun98