Statistical Indices of Data Variability
Measures of Dispersion
Range
The range gives you the most basic information about the spread of scores. It is calculated by the difference between the
lowest and highest scores.
Interquartile Range:
The difference between the score representing the 75th percentile and the score
representing the 25th percentile is the interquartile range. This value gives you the range of the middle 50% of the values in the data set.
Variance and Standard Deviation:
The standard deviation is the square root of the average squared deviation from the
mean. The average squared deviation from the mean is also known as the variance.
Understanding and Calculating the Standard Deviation
Computers are used extensively for calculating the standard deviation and other
statistics.
However, calculating the standard deviation by hand once or twice can be helpful in developing an understanding of its meaning.
Calculating the variance and
standard deviation
Consider the observations 8,25,7,5,8,3,10,12,9.
 First, determine n, which is the number of data values.
 Second, calculate the arithmetic mean, which is the sum of scores divided by n. For this example, the mean = (8+25+7+5+8+3+10+12+9) / 9 or 9.67
 Then, subtract the mean from each individual score to find the individual deviations.
 Then, square the individual deviations.
 Then, find the sum of the squares of the deviations...can you see why we squared them before adding the values?
 Divide the sum of the squares of the deviations by n1. This is the Variance!
 Take the square root of the variance to obtain the standard deviation, which has the same units as the original data.
Score 
Mean 
Deviation^{*} 
Squared Deviation 
8 
9.67 
1.67 
2.79 
25 
9.67 
+15.33 
235.01 
7 
9.67 
2.67 
7.13 
5 
9.67 
4.67 
21.81 
8 
9.67 
1.67 
2.79 
3 
9.67 
6.67 
44.49 
10 
9.67 
+.33 
.11 
12 
9.67 
+2.33 
5.43 
9 
9.67 
.67 
.45 

Sum of squared dev = 320.01 

^{*}Deviation = Score  Mean 
Standard Deviation = Square root(sum of squared deviations / (N1)

= Square root(320.01/(91)) 

= Square root(40) 

= 6.32 
Raw score method for calculating
standard deviation
Again, consider the observations 8,25,7,5,8,3,10,12,9.
 First, square each of the scores.
 Determine N, which is the number of scores.
 Compute the sum of X and the sum of Xsquared.
 Then, calculate the standard deviation as illustrated below.
Score 
X^{2} 

8 
64 

25 
625 

7 
49 
N=9 
5 
25 

8 
64 
Sum of X=87 
3 
9 

10 
100 
Sum of X^{2}=1161 
12 
144 

9 
81 

 
 

87 
1161 

Standard Deviation = square root[(sum of X^{2})((sum of X)*(sum of X)/N)/(N1)]

= square root[(1161)(87*87)/9)/(91)] 

= square root[(1161(7569/9)/8)] 

= square root[(1161841)/8] 

= square root[320/8] 

= square root[40] 

= 6.32 
Even simple statistics, such as the standard
deviation, are tedious to calculate "by hand".
Copyright © 1997 T. Lee Willoughby
