


Sherril M. Stone, Ph.D. 

Department of Family Medicine 

OSUCollege of Osteopathic Medicine 





Range 

highest score minus the lowest score 

Outliers will skew your data 

Interquartile Range Ð 25th, 50th, 75th, 90th,
etc 25th quartile Ð 25% of all scores fall below this score 

75th quartile Ð 75% of all scores fall below
this score 

Semiinterquartile Ð Q3ÐQ1 (middle 50%) Ð not affected by
outliers 

2 

These are CRUDE MEASURES OF VARIABILITY  does
not tell the whole story about the raw data. A mean of 5.3 does not tell if
most of the scores were similar. So, we have measures of variability, which
tell us about the spread of the scores on the scale of measurement. 

Deviation 

the distance from the mean 

+ is above mean,  is below the mean 

Deviation scores  MUST sum (S) to zero 





Always report a measure of central tendency
(Mean & SD) and a measure of variability to describe a set of scores 

Variation (SS) = Sum of Squares 

Variance (s^{2}
or S^{2}) = SS/N (deviation scores squared/N) 

Standard deviation (s or SD) = square root of
variance 

Measures of variability provide a quantitative measure
of the degree to which scores in a distribution are spread out or clustered
together on the scale. 

If variance = 0 then all scores are the same, NO
variability exists 

If variance is large  then scores were very far
apart 

If variance is small Ð then scores were very
close together 





Variation, variance, and standard deviation 

measures of variability 

report these measures with the mean 

Population
Sample 

Variation: SS = S(X Ðm )^{2} SS_{x} SS_{x} 

Variance: SS/N = mean of SS
s^{2}
s^{2} 

Standard Deviation: square root of s^{2} s s 





Definitional 

1. Find
N, SX,
and mean 

2. Subtract mean (X) from each X for deviation
score 

3. Square each deviation score (XÐ X)^{2} 

4. Sum the deviation scores S(X Ð X )^{2} =
SS (aka Variation) 

5. Variance (s^{2})
= SS/N 

6. S (SD) = SS/N 

Computational 

1. N, SX, and mean 

2. Square each data score 

3. Compute SS = S X^{2}
 (SX)^{2} 

N 

4. Variance (s^{2}) = SS/N 

5. S (SD) = SS/N 




X X  m (X  m)^{2} 

1 1 1 N = 4 

0 2 4 m = 2 

6 +4 16 

1 1 1 

SX =
8 SX Ðm = 0 S(X Ðm)^{2} = 22 






SX^{2}
Ð (SX)^{2} 

s2 =
n 

N 



X X^{2} 

1 1 N = 4 

0 0 X = 2 

6 36 

1 1 

SX = 8 SX^{2} = 38 

SS =
38 Ð 82/4 =
38 Ð 16 = 22 




Samples tend to underestimate the population
data 

To correct sampling error we subtract 2 from n 

All scores are free to vary except 2 (e.g. when
you know all but 1 data score then it is determined) 

s^{2} = Sample variance = Sx^{2} = SS = 35.00 = 11.67 

N1 N1 4 1 



s = Sample standard deviation = s^{2} = 11.67 = 3.42 

