Standard Deviation

WHAT IS STANDARD DEVIATION? The standard deviation is the most frequently calculated measure of variability. The standard deviation value represents the average distance of a set of scores from the mean.

Standard deviation and the normal curve

Knowing the standard deviation helps create a more accurate picture of the distribution along the normal curve. A smaller standard deviation represents a data set where scores are very close in value to the mean; a smaller range. A data set with a larger standard deviation has scores with more variance; a larger range. For example, if the average score on a test was 80 and the standard deviation was 2, the scores would be more clustered around the mean than if the standard deviation was 10.

normal curve with standard deviation
Figure 1. The normal curve. Standard deviation is a constant interval from the mean. Roll the mouse over the curve to discover the percentage each portion represents.

Calculating the standard deviation

The figure below displays the formula for calculating the standard deviation. (It is much easier than it looks!)

standard deviation formula

S = standard deviation
Σ = sum of
X = individual score
M = mean of all scores
n = sample size (number of scores)

The best method to calculate the standard deviation by hand is to create a organized chart to perform necessary equations. It is necessary first to compute the mean.

X
M
(X-M)
(X-M)2
1
3
-2
4
2
3
-1
1
3
3
0
0
4
3
1
1
5
3
2
4
Total (Σ)
0
10

Take notice of several key points regarding the calculation of the standard deviation. First, the score minus the mean grand total (third column) should ALWAYS equal zero. This is a good cross-check to ensure that the mean has been correctly calculated. Second, the purpose of squaring the deviations is to eliminate the negative values so that their grand total does not equal zero. Finally, the reason the denominator is n-1 is because the standard deviation is being calculated for a sample. Should the standard deviation be calculated for a population, the denominator would simply be n.

     
formula calculating  

Completing the calculation.
Divide total squared deviations by n-1.
That leaves 10/4.
Take the square root of 2.5.
The standard deviation equals 1.58.

(Refresh browser if calculation remains static.)

Kathleen Barlo
SDSU Educational Technology