variation.html

Standard Deviation

The range only uses the largest and smallest value in a data set. The standard deviation measures the variation by taking each value into account.

Definition . The standard deviation for a sample of size n is given by

The standard deviation measures how much the data varies from the mean.

Related to the standard deviation is a number called the variance. The variance is the square of the standard deviation or putting it another way, the standard deviation is the square root of the variance.

Definition . The variance for a sample of size n is given by

Example 1

Find the standard deviation and variance for the sample 2, 6, 10, 14, 18

Solution .

Using the definition we must first calculate the mean. This is done in the first column of the table that appears below. After the sum (50) of the values is obtained, it is divided by 5 which results in the mean of 10. The mean is then subtracted from each x and the result squared. This is shown in the second column. The total of these squares is 160. This is divided by 4 which is one less than the sample size, resulting in the variance of 40. The square root of 40 is taken to get the standard deviation.

       x    ((x - mean)^2)

--------------------------------

       2.     64.

       6.     16.

       10.    0

       14.    16.

       18.    64.

SUM    50.    160.

COUNT  5.

       10.    40.

       MEAN   VARIANCE

              6.324555320

              STANDARD DEVIATION

Example 2

Find the variance and standard deviation for the sample 5, 7, 10, 13, 15.

Solution .

The results are shown in the table below.

       x    ((x - mean)^2)

--------------------------------

       5.     25.

       7.     9.

       10.    0

       13.    9.

       15.    25.

SUM    50.    68.

COUNT  5.

       10.    17.

       MEAN   VARIANCE

              4.123105626

              STANDARD DEVIATION

When we looked at A={2, 6, 10, 14, 18} and B={5, 7, 10, 13, 15} in the section on range we noted that they had identical means, medians, and midranges. There seems to be more variation in the data in A than there is in B. The standard deviations computed in the last two examples were 6.32 for A and 4.12 for B.

Example 3

Find the standard deviation of the sample 3.4, 3.4, 3.4, 3.4, 3.4, 3.4,

Solution .

Since there is no variation from the mean for this set of numbers, the standard deviation is zero.

This is confirmed in the following table.

       x          ((x - mean)^2)

--------------------------------------

       3.4          0

       3.4          0

       3.4          0

       3.4          0

       3.4          0

       3.4          0

SUM    20.4         0

COUNT  6.

       3.400000000  0

       MEAN         VARIANCE

                    STANDARD DEVIATION

When the standard deviation and variance are computed for a population, there is a slight change in the formula.

For a population of size N, the standard deviation is where is the mean of the population.

The variance for a population is