Standard Deviation and Variance

The standard deviation (abbreviated as s or SD) is the average amount of variability that a set of scores contains and is the average distance from the mean. The larger the standard deviation, the more variability is in the set of data and the farther the average point is from the mean of the set of scores.

The formula for computing the standard deviation is as follows:

where

• s is the standard deviation
• Σ is sigma, which tells you to find the sum of what follows
• X is each individual score
• X¯ is the mean of all the scores
• n is the sample size.

The variance is simply the square of the standard deviation.

Given the following data set,

• 56
• 42
• 44
• 67
• 78
• 76
• 56
• 45
• 65
• 61

the steps for computing the standard deviation are as follows. See Table 1 for the corresponding value for each step.

• 1.
  List each score.
• 2.
  Compute the mean of the group (in this example, it is 59).
• 3.
  Subtract the mean from each score.
• 4.
  Square each individual difference. The result is the column marked(X X¯)2.
• 5.
  Sum all of the squared deviations about the mean (in this example, it is 1,482).
• 6.
  Divide the sum by n 1.
• 7.
  Compute the square root. That is, the standard deviation for this set of 10 scores is 12.83. The variance would be this value squared or 164.61.

Given these results, each score in this set of 10 differs from the mean by an average of 12.83 points.

The value of 1 is subtracted from the denominator of the standard deviation formula because s is an estimate of the population standard deviation and is unbiased. By subtracting 1 from the denominator, the standard deviation is forced to be larger than it would be otherwise. This is done to ensure that if any error [Page 935]is made, it will be an overestimate of the population value.