Lecture Data ans statistics Applications in Business and Economics

Range
The simplest measure of variability is the range.

Let us refer to the data on starting salaries for business school graduates in Table 3.1. The
largest starting salary is 3925 and the smallest is 3310. The range is 3925 _ 3310 _ 615.
Although the range is the easiest of the measures of variability to compute, it is seldom
used as the only measure. The reason is that the range is based on only two of the observations
and thus is highly influenced by extreme values. Suppose one of the graduates
received a starting salary of $10,000 per month. In this case, the range would be
10,000 _ 3310 _ 6690 rather than 615. This large value for the range would not be especially
descriptive of the variability in the data because 11 of the 12 starting salaries are
closely grouped between 3310 and 3730.
Interquartile Range
A measure of variability that overcomes the dependency on extreme values is the interquartile
range (IQR). This measure of variability is the difference between the third quartile,
Q3, and the first quartile, Q1. In other words, the interquartile range is the range for the
middle 50% of the data.

For the data on monthly starting salaries, the quartiles are Q3 _ 3600 and Q1 _ 3465. Thus,
the interquartile range is 3600 _ 3465 _ 135.
Variance
The variance is a measure of variability that utilizes all the data. The variance is based on the
difference between the value of each observation (xi) and the mean. The difference between
each xi and the mean ( for a sample, μ for a population) is called a deviation about the mean.
For a sample, a deviation about the mean is written (xi-x); for a population, it is written
(xi - μ). In the computation of the variance, the deviations about the mean are squared.
If the data are for a population, the average of the squared deviations is called the population
variance. The population variance is denoted by the Greek symbol σ2. For a population
of N observations and with μ denoting the population mean, the definition of the
population variance is as follows.

In most statistical applications, the data being analyzed are for a sample. When we compute
a sample variance, we are often interested in using it to estimate the population variance
σ2. Although a detailed explanation is beyond the scope of this text, it can be shown
that if the sum of the squared deviations about the sample mean is divided by n-1, and
not n, the resulting sample variance provides an unbiased estimate of the population variance.
For this reason, the sample variance, denoted by s2, is defined as follows.

To illustrate the computation of the sample variance, we will use the data on class size

for the sample of five college classes as presented in Section 3.1. A summary of the data,
including the computation of the deviations about the mean and the squared deviations
about the mean, is shown in Table 3.2. The sum of squared deviations about the mean is
²= 256. Hence, with n-1=4, the sample variance is

Before moving on, let us note that the units associated with the sample variance often
cause confusion. Because the values being summed in the variance calculation, (x_i-x)², are
squared, the units associated with the sample variance are also squared. For instance, the
sample variance for the class size data is s2 =64 (students)2. The squared units associated
with variance make it difficult to obtain an intuitive understanding and interpretation of the
numerical value of the variance. We recommend that you think of the variance as a measure
useful in comparing the amount of variability for two or more variables. In a comparison of
the variables, the one with the largest variance shows the most variability. Further interpretation
of the value of the variance may not be necessary.
As another illustration of computing a sample variance, consider the starting salaries
listed in Table 3.1 for the 12 business school graduates. In Section 3.1, we showed that the
sample mean starting salary was 3540. The computation of the sample variance
(s2 =27,440.91) is shown in Table 3.3.
In Tables 3.2 and 3.3 we show both the sum of the deviations about the mean and the
sum of the squared deviations about the mean. For any data set, the sum of the deviations
about the mean will always equal zero. Note that in Tables 3.2 and 3.3, _(xi -x )=0. The
positive deviations and negative deviations cancel each other, causing the sum of the deviations
about the mean to equal zero.