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Once the typical underlying pattern (the inherent variability) has been described, the statistical prop-
erties of the deviations of observations from this typical pattern need to be characterized. If the deviations
are random, independent, and have constant variance, we can construct a control chart that will examine
these deviations. The average value of the deviations will be zero, and symmetrical control limits, calculated
in the classical way, can be drawn above and below zero.
The general steps in constructing a control chart are these:

1. Sample the process at specific times (t, t − 1, t − 2,…) to obtain … y t , y t−1 , and y t−2 . These
typically are averages of subgroups of n observations, but they may be single observations.
2. Calculate a quantity V t , which is a function of the observations. The definition of V t depends
on the type of control chart.
3. Plot values V t in a time sequence on the control chart.
4. Using appropriate control limits and rules, plot new observations and decide whether to take
corrective action or to investigate.

Kinds of Control Charts
2
What has been said so far is true for control charts of all kinds. Now we look at the Shewhart chart
(1931), cumulative sum chart (Cusum), and moving average charts. Moving averages were used for
smoothing in Chapter 4.

Shewhart Chart
The Shewhart chart is used to detect a change in the level of a process. It does not indicate a change
in the variability. A Range chart (Chapter 11) is often used in conjunction with a Shewhart or other chart
that monitors process level.
The quantity plotted on the Shewhart chart at each recording interval is an average, y t , of the subgroup
of n observations y t made at time t to calculate:

n
V t = y t = 1 ∑ y t
---
n
i=1
If only one observation is made at time t, plot V t = y t . This is an I-chart (I for individual observation)
instead of an X chart. Making only one observation at each sampling reduces the power of the chart to
detect a shift in performance.
The central line on the control chart measures the general level of the process (i.e., the long-term
average of the process). The upper control limit is drawn at 3s above the central control line; the lower
limit is 3s below the central line. s is the standard error of averages of n observations used to calculate
the average value at time t. This is determined from measurements made over a period of time when
the process is in a state of stable operation.

Cumulative Sum Chart
The cumulative sum, or Cusum, chart is used to detect a change in the level of the process. It does not
indicate a change in the variability. The Cusum chart will detect a change sooner (in fewer sampling
intervals) than a Shewhart chart. It is the best chart for monitoring changes in process level.

2
In Chapter 10, Shewhart charts were also called X (X-bar) charts and X was the notation used to indicate a measurement from
a laboratory quality control setting. In all other parts of the book, we have used y to indicate the variable. Because the term Y-bar
chart is not in common use and we wish to use y instead of x, in this chapter we will call these X-bar charts Shewhart charts.
© 2002 By CRC Press LLC