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202 Part IV: Quality Assurance
EXAMPLE 18.2
The following data give the ages of all the employees in a hardware store.
22, 25, 26, 36, 26, 29, 26, 26
Find the mean age of the employees in the hardware store.
Solution:
Since the data give the ages of all the employees of the hardware store, we have a population.
Thus, we have
So the population mean is
N = 8
x i = 22 + 25 + 26 + 36 + 26 + 29 + 26 + 26 = 216.
xi 216
m = = = 27 years.
N 8
In this example, the mean age of the employees in the hardware store is 27 years.
Part IV.A.1
Note that even though the formulas for calculating sample mean and population
mean are very similar, it is important to make a very clear distinction between
the sample mean x – and the population mean m for all application purposes.
Sometimes a data set may include a few observations or measurements that
are very small or very large. For example, the salaries of a group of engineers in a
big corporation may also include the salary of its CEO, who also happens to be an
engineer and whose salary is much larger than the 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. If extreme values are present in the
data set then their mean is not an appropriate measure of centrality. It is important
to note that any extreme value, large or small, adversely affects 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 of calculating the
median of a data set.
Median. We denote the median of a data set by M d . To determine the median of a
data set we take the following steps:
Step 1. Arrange the measurements in the data set in ascending order and
rank them from 1 to n.
Step 2. Find the rank of the median, which is equal to (n +1)/2.
Step 3. Find the value corresponding to the rank (n + 1)/2 of the median.
This value represents the median of the data set.
It is important to note that the median may or may not be one of the values of the
data set. Whenever the sample size is odd, the median is the center value, and
whenever it is even, the median is always the average of the two middle values
where the data are arranged in ascending order.