Variance for data with multiple-observation values

Suppose that a data set contains values occurring with frequencies, respectively.

1. For a population of observations, so that

The variance is

The standard deviation is .

2. For a sample of observations, so that

The variance is

The standard deviation is .

The arithmetic is most conveniently set out in tabular form.

Example:

The score for the sample of 25 students on a 5-point quiz are shown below.

Find a sample variance and standard deviation.

Score	Frequency

Solution:

Remark: The denominator in the formula is obtained by summing the frequencies . It is not number of classes.

To calculate variance we need three columns to display the computation of the quantities a column for the a column for the and a column for the . We also need a column for and a final column for the products . (Table 1.5)

The necessary computations for finding are shown below.

Table 1.5

Score	Frequency
		0-2.7=-2.7 1-2.7=-1.7 2-2.7=-0.7 3-2.7=0.3 4-2.7=1.3 5-2.7=2.3	7.29 2.89 0.49 0.09 1.69 5.29	0· 7.29=0 1· 2.89=2.89 2· 0.49=0.98 3· 0.09=0.27 4· 1.69=6.76 5· 5.29=26.45

Thus we have

.

Example:

The number of television sets sold per month over a two year period is reported below. Find the variance and standard deviation for the data.

Number of sets sold	Frequency (month)

Solution:

Let us apply .

Make a table as shown below

Sets	Frequency

To find standard deviation we take the square root of variance

.

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