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The mean of a sample or a population is calculated by dividing the sum of the data measurements by the number of measurements in the data. The sample mean is also known as sample average and is denoted by (read as X bar), and the population mean is denoted by the Greek letter (read as meu). These terms are defined as follows:

(2.5.1)

(2.5.2)

In (2.5.1), denotes the value of the variable possessed by the th member of the population, . In (2.5.2), the denotes the th measurement made in a sample of size . Here, and denote the population and sample size, respectively, and . The symbol (read as sigma) denotes the summation over all the measurements. Note that here is a statistic, and is a parameter.

Example 2.5.1 (Workers' hourly wages) The data in this example give the hourly wages (in dollars) of randomly selected workers in a manufacturing company:

8, 6, 9, 10, 8, 7, 11, 9, 8

Find the sample average and thereby estimate the mean hourly wage of these workers.

Solution: Since wages listed in these data are for only some of the workers in the company, the data represent a sample. Thus, we have , and the observed is

Thus, the sample average is observed to be

In this example, the average hourly wage of these employees is $8.44 an hour.

Example 2.5.2 (Ages of employees) The following data give the ages of all the employees in a city hardware store:

22, 25, 26, 36, 26, 29, 26, 26

Find the mean age of the employees in that hardware store.

Solution: Since the data give the ages of all the employees of the hardware store, we are dealing with a population. Thus, we have

so that the population mean is

In this example, the mean age of the employees in the hardware store is 27 years.

Even though the formulas for calculating sample average and population mean are very similar, it is important to make a clear distinction between the sample mean or sample average and the population mean for all application purposes.

Sometimes, a data set may include a few observations that are quite small or very large. For examples, the salaries of a group of engineers in a big corporation may include the salary of its CEO, who also happens to be an engineer and whose salary is much larger than that of other engineers in the group. In such cases, where there are some very small and/or very large observations, these values are referred to as extreme values or outliers. If extreme values are present in the data set, then the mean is not an appropriate measure of centrality. Note that any extreme values, large or small, adversely affect the mean value. In such cases, the median is a better measure of centrality since the median is unaffected by a few extreme values. Next, we discuss the method to calculate the median of a data set.