Characteristics of Shewhart Charts
Figure 1 illustrates a typical Shewhart chart.
Figure 1: A Shewhart Control Chart
All Shewhart charts have the following characteristics:
- Each point represents a summary statistic
computed from a
sample of measurements of a quality characteristic. For example, the
summary statistic might be the average value of a critical dimension of
five items selected at random, or it might be the proportion of
nonconforming items in a sample of 100 items.
- The vertical axis of a Shewhart chart is scaled in the same
units as the summary statistic.
The samples from which the summary statistics are computed are referred
to as rational subgroups or subgroup samples. The
organization of the data into subgroups is critical to the
interpretation of a Shewhart chart. Shewhart (1931) advocated selecting
rational subgroups so that variation within subgroups is minimized and
variation among subgroups is maximized; this makes the chart more
sensitive to shifts in the process level. Various approaches to
subgrouping are discussed by Grant and Leavenworth (1980), Montgomery
(1996), and Kume (1985).
The horizontal axis of a Shewhart chart identifies the subgroup
samples. Frequently, the samples are indexed according to the order in
which they were taken or the time at which they were taken. Subgroup
samples can also be assigned labels that indicate some other type of
classification (for example, lot number).
The central line on a Shewhart chart indicates the average
(expected value) of the summary statistic when the process is in
The upper and lower control limits, labeled UCL and LCL,
respectively, indicate the range of variation to be expected in the
summary statistic when the process is in statistical control. The
control limits are commonly computed as 3 limits* representing three standard
variation in the summary statistic above and below the central line.
However, the limits can also be determined using a multiple of the
standard error other than three, or from a specified
probability () that a single summary statistic will exceed the
limits when the process is in statistical control. Limits determined by
the latter method are referred to as probability limits.
The control limits are also determined by the subgroup sample size
because the standard error of the summary statistic is a function of
sample size. If the sample size is constant across subgroups, the
control limits are typically horizontal lines, as in Figure 1.
However, if the sample size varies from subgroup to subgroup,
the limits are usually adjusted to compensate for the effect of sample
size, resulting in step-like boundaries.
Control limits can be estimated from the data being analyzed, or they
can be standard, previously determined values. Estimated limits are
often used when statistical control is being established, and standard
limits are often used when statistical control is being maintained.
A point outside the control limits signals the presence of a
special cause of variation. Additionally, tests
for special causes (also referred to as Western Electric rules
and runs tests) can
signal an out-of-control condition if a statistically unusual
pattern of points is observed in the control chart. For example, one
pattern used to diagnose the existence of a trend is seven
consecutive steadily increasing points.
When the process is in statistical control, a point may fall outside the
control limits purely by chance, resulting in a false out-of-control
signal. However, when the Shewhart chart correctly signals the presence
of a special cause, additional action is needed to determine the nature
of the problem and eliminate it.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.