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190 CHAPTER 6 RANDOM SAMPLING AND DATA DESCRIPTION

6-1 DATA SUMMARY AND DISPLAY

Well-constructed data summaries and displays are essential to good statistical thinking, be-
cause they can focus the engineer on important features of the data or provide insight about
the type of model that should be used in solving the problem. The computer has become an
important tool in the presentation and analysis of data. While many statistical techniques re-
quire only a hand-held calculator, much time and effort may be required by this approach, and
a computer will perform the tasks much more efficiently.
Most statistical analysis is done using a prewritten library of statistical programs. The
user enters the data and then selects the types of analysis and output displays that are of
interest. Statistical software packages are available for both mainframe machines and
personal computers. We will present examples of output from Minitab (one of the most
widely-used PC packages), throughout the book. We will not discuss the hands-on use of
Minitab for entering and editing data or using commands. This information is found in the
software documentation.
We often find it useful to describe data features numerically. For example, we can char-
acterize the location or central tendency in the data by the ordinary arithmetic average or
mean. Because we almost always think of our data as a sample, we will refer to the arithmetic
mean as the sample mean.

Definition
If the n observations in a sample are denoted by x , x , p , x , the sample mean is
n
2
1
n
a x i
x x p x n i 1
2
1
x (6-1)
n n
EXAMPLE 6-1 Let’s consider the eight observations collected from the prototype engine connectors from
Chapter 1. The eight observations are x 1 12.6, x 2 12.9, x 3 13.4, x 4 12.3, x 5 13.6,
x 13.5, x 12.6, and x 13.1. The sample mean is
8
6
7
8
a x i
x p x 12.6 12.9 p
x 1 2 n i 1 13.1
x
n 8 8
104
13.0 pounds
8
A physical interpretation of the sample mean as a measure of location is shown in the dot
diagram of the pull-off force data. See Figure 6-1. Notice that the sample mean x 13.0 can be
thought of as a “balance point.” That is, if each observation represents 1 pound of mass placed
at the point on the x-axis, a fulcrum located at would exactly balance this system of weights.
x

The sample mean is the average value of all the observations in the data set. Usually,
these data are a sample of observations that have been selected from some larger population
of observations. Here the population might consist of all the connectors that will be manufac-
tured and sold to customers. Recall that this type of population is called a conceptual or