Standard Deviation.

Standard Deviation. Standard Deviation, a number representing how closely bunched a set of numbers is around its mean, or average value. The standard deviation is an important concept in statistics because it is a precise indicator of the degree of variability within a set of numbers. A set with a smaller standard deviation consists of more closely bunched numbers than a set with a larger standard deviation. If the test scores of a class were 76, 80, 82, 85, and 91, for example, the standard deviation would be about 5.04, reflecting the fact that the scores are fairly close together. A class with the wildly varying test scores 0, 53, 77, 91, and 100, however, would have a standard deviation of about 35.8. The standard deviation of a set of numbers is calculated using the deviation of individual numbers from the set's mean. Each deviation is found by subtracting the number from the mean. If there are n members in a set and the deviations of the members from the mean are symbolized by x1 through xn, the standard deviation (? ) is given by the formula: To compute the standard deviation of the set {2, 3, 5, 6, 9}, for example, first find the mean. The mean is the sum of all the members of a set divided by the number of members of the set: (2 + 3 + 5 + 6 + 9)/5 = 25/5 = 5. The individual deviations from the mean are 5 - 2 = 3, 5 - 3 = 2, 5 - 5 = 0, 5 - 6 = -1, and 5 - 9 = -4. Square the individual deviations and add them together: 32 + 22 + 02 + (-1)2 + (-4)2 = 9 + 4 + 0 + 1 + 16 = 30. Divide the result by the number of members of the set: 30/5 = 6. Take the square root and round to two decimal places: ?6 = 2.45. The standard deviation of {2, 3, 5, 6, 9}, therefore, is 2.45. This equation works for sets in which all members are specified. A more complex equation is necessary to determine the standard deviation of a set using only a sample of that set's elements. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.