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.

1. First, determine n, which is the number of data values.
2. 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
3. Then, subtract the mean from each individual score to find the individual deviations.
4. Then, square the individual deviations.
5. Then, find the sum of the squares of the deviations...can you see why we squared them before adding the values?
6. Divide the sum of the squares of the deviations by n-1. This is the Variance!
7. Take the square root of the variance to obtain the standard deviation, which has the same units as the original data.
 Score Mean Deviation* SquaredDeviation 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 / (N-1)
 = Square root(320.01/(9-1)) = 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.

1. First, square each of the scores.
2. Determine N, which is the number of scores.
3. Compute the sum of X and the sum of X-squared.
4. Then, calculate the standard deviation as illustrated below.

 Score X2 8 64 25 625 7 49 N=9 5 25 8 64 Sum of X=87 3 9 10 100 Sum of X2=1161 12 144 9 81 --- --- 87 1161
Standard Deviation = square root[(sum of X2)-((sum of X)*(sum of X)/N)/(N-1)]

 = square root[(1161)-(87*87)/9)/(9-1)] = square root[(1161-(7569/9)/8)] = square root[(1161-841)/8] = square root[320/8] = square root[40] = 6.32

Even simple statistics, such as the standard deviation, are tedious to calculate "by hand".