The variance and the standard deviation

A key step in developing a measure of variability that includes all the data items involves the computations of the differences between the data values and the mean for the data set. The difference between and the mean

( for a sample, for a population) is called a deviation about the mean. Since we are seeking a descriptive statistical measure that summarizes the variability or dispersion in the entire data set, we want to consider the deviation of each data value about the mean. Thus for a sample size and data values , we will need to compute the deviations

, , …….. .

We might think of summarizing the dispersion in a data set by computing the average deviation about the mean. The only trouble with such an attempted definition is that it would not give us much information about the variation present in the data; the mean would be zero for every sample, because the sum equals zero for every sample. The positive and negative deviations cancel each other out. Hence if we are to use the deviations from the mean as a measure of dispersion we must find another approach. As we already know, one way is computing the average absolute deviation as a measure of variability. While this measure is sometimes used, the one most often used is based on squaring the deviations to eliminate the negative values. The average of the squared deviations for a data set representing a population or sample is given a special name in statistics. It is called the variance.

The population variance is denoted by the Greek symbol

(pronounced “sigma squared”). The formula for population variance is

(1)

where

.

It is frequently desirable to have a measure of dispersion whose units are the same as those of the observations. Since the variance is given in squared units, the square root of the variance would be given in units that we need.

Thus, if we take the square root of the variance, we have the measure of dispersion that is known as the population standard deviation and denoted by . By definition we have

In many statistical applications, the data set we are working with is a sample. When we compute a measure of variability for the sample, we often are interested in using the sample statistic obtained as an estimate of the population parameter, . At this point it might seem that the average of the squared deviations in the sample would provide a good estimate of the population variance. However, statisticians have found that the average squared deviation for the sample has the undesirable feature that it tends to underestimate the population variance . Because of this tendency toward underestimation we say it provides a biased estimate.

Fortunately, it can be shown that if the sum of the squared deviations in the sample is divided by , and not , then the resulting sample statistic will provide an unbiased estimate of the population variance. For this reason the sample variance is not defined to be the average squared deviation in the sample. Sample variance is denoted by and is defined as follows:

(2)

To find the sample standard deviation (denoted by ), one must take the square root of the sample variance:

Example:

Find the variance and the standard deviation for the sample data

21, 17, 13, 25, 9, 19, 6, and 10

Solution:

When we compute by applying formula (2), the computations can most conveniently be shown in a table. The table will be composed of three columns: a column for the observations , a column for the

deviations of the observations from the sample mean ,

and a column for the squared deviations .(Table1.2)

(Table 1.2)

	21-15=6 17-15=2 13-15=-2 25-15=10 9-15=-6 19-15=4 6-15=-9 10-15=-5

;

;

and .

From the computational point of view, it is easier and more efficient to use short-cut formulas to calculate the variance. By using the short-cut formula, we reduce the computation time and round off errors.

The short-cut formulas for calculating variance are as follows:

and

Example:

Find the variance and the standard deviation for the sample of

16, 19, 15, 15, and 14

Solution:

Let us apply

Step1: Find the sum of values,

Step2: Square each value and find the sum

Step3: Substitute in the formula and calculate

Hence the sample variance is 3.7 and sample standard deviation is 1.9.

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