Lecture Data ans statistics Applications in Business and Economics

Standard Deviation
The standard deviation is defined to be the positive square root of the variance. Following
the notation we adopted for a sample variance and a population variance, we use s to
denote the sample standard deviation and σ to denote the population standard deviation. The
standard deviation is derived from the variance in the following way.

Recall that the sample variance for the sample of class sizes in five college classes is
s2 _ 64. Thus, the sample standard deviation is

For the data on starting

salaries, the sample standard deviation is s= 27,440.91=165.65.
What is gained by converting the variance to its corresponding standard deviation? Recall
that the units associated with the variance are squared. For example, the sample variance
for the starting salary data of business school graduates is s² =27,440.91 (dollars)2.
Because the standard deviation is the square root of the variance, the units of the variance,
dollars squared, are converted to dollars in the standard deviation. Thus, the standard deviation
of the starting salary data is $165.65. In other words, the standard deviation is measured
in the same units as the original data. For this reason, the standard deviation is more
easily compared to the mean and other statistics that are measured in the same units as the
original data.
Coefficient of Variation
In some situations we may be interested in a descriptive statistic that indicates how large
the standard deviation is relative to the mean. This measure is called the coefficient of
v ariation and is usually expressed as a percentage.

For the class size data, we found a sample mean of 44 and a sample standard deviation

of 8. The coefficient of variation is [(8/44) _ 100]% _ 18.2%. In words, the coefficient of
variation tells us that the sample standard deviation is 18.2% of the value of the sample
mean. For the starting salary data with a sample mean of 3540 and a sample standard deviation
of 165.65, the coefficient of variation, [(165.65/3540) _ 100]% _ 4.7%, tells us the
sample standard deviation is only 4.7% of the value of the sample mean. In general, the coefficient
of variation is a useful statistic for comparing the variability of variables that have
different standard deviations and different means.