Glenn Knapp

Standard Deviation

Homework Assignment re: Standard Deviation

Standard Deviation, n. Statistics, a measure of the degree of scattering of a frequency distribution about its arithmetic mean, equal to the square root of the mean of the squared deviations from the distribution mean.

Mean Square Deviation, n. Statistics. Variance; standard deviation

Mean Deviation, n. Statistics, the arithmetic average of the absolute values of the individual deviations in a distribution from a central value.

Average deviation, n. Statistics, the average arithmetic mean of the deviations, taken without regard to sign, from some fixed value, usually the arithmetic mean of the data.

If the observations are measured in units, such as inches, then the variance is given in terms of square units. When you take the nonnegative square root of the variance, you then have a measure of variation that is in the same units as those of the given data. This measure is called the standard deviation.

The range of a collection of data is the difference between the greatest and least values in the list. It is the simplest measure of the variation in the data. The range tells us nothing, however about the data that are scattered or clustered together, in particular, around the mean. For example, consider these two lists of data, A and B.

A: 2, 5, 6, 7, 10

B: 2, 3, 6, 10, 10

Example: Find the mean for each list A & B.

A: 2+5+6+7+10=30 30 divided 5 (by amount numbers)=6 (6 is the mean)

B: 2+3+6+10+10=30 30 divided 5 (by amount numbers)=6 (6 is the mean)

They have the same range, 8, and the same mean, 6. But in list A the number are clustered much more closely about the mean than in list B. You can compare the degree of scattering in two data samples by finding the variance of each one. The variance is the average of the squares of the deviations (differences) of all the measurements from their mean.

Example: Find the variance for each list A & B.

The deviations from the mean are:

A:

-4, -1, 0, +1, +4

((-4)^2+(-1)^2+0^2+(+1)^2+(+4)^2)/5=34/5=6.8

B:

-4, -4, 0, +4, +4

((-4)^2+(-4)^2+0^2+(+4)^2+(+4)^2)/5=64/5=12.8

Since the variance in A is 6.8 whereas in B it is 12.8, the data in B are considerably more scattered from their mean than are the data in A.

Example: Find the standard deviation s (standard deviation is usually denoted by s) for lists A and B.

For A, the variance is 6.8, so: Square Root of 6.8=s =Ö 6.8 = 2.61 (answer)

B, the variance is 12.8, so: Square Root of 12.8=s = Ö12.8 = 3.58 (answer)

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