alt="images"/> (read as meu). These terms are defined as follows:
(2.5.1)
In (2.5.1),
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
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
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.
Median
We denote the median of a data set by
1 Step 1. Arrange the observations in the data set in an ascending order and rank them from 1 to .
2 Step 2. Find the rank of the median that is given by(2.5.3) We can check manually that the conditions of Definition 2.4.2 are satisfied.
3 Step 3. Find the value of the observation corresponding to the rank of the median found in (2.5.3). If denotes the th largest value in the sample, and if(i) odd, say , then the median is (ii) even, say , then the median is taken as
Note that in the second case, we take median as the average of
We now give examples of each case,
Example 2.5.3 (Alignment pins for the case of n odd,
