3. Control Charts for and
x R
1 2
: quantity of interest ( , )
, , :samples of
n
x x N
x x x x
,
x N
n
4. Subgroup Data with Unknown and
1 2
, , : ranges of samples
m
R R R m
: grand average of , best estimate for
x x
6. Phase I Application of and R Charts
• Equations 5-4 and 5-5 are trial control limits.
– Determined from m initial samples.
• Typically 20-25 subgroups of size n between 3 and 5.
– Any out-of-control points should be examined for assignable
causes.
• If assignable causes are found, discard points from calculations and
revise the trial control limits.
• Continue examination until all points plot in control.
• Adopt resulting trial control limits for use.
• If no assignable cause is found, there are two options.
1. Eliminate point as if an assignable cause were found and revise limits.
2. Retain point and consider limits appropriate for control.
– If there are many out-of-control points they should be examined
for patterns that may identify underlying process problems.
x
10. Cp: Process Capability Ration (PCR)
2
Note: 6 spread is the basic definition of process capability. 3 above mean and 3 below.
R
ˆ ˆ
If is unknown, we can use = . in the example is 0.1398.
d
P : % of specification band the process uses up. P can be estimated as:
12. Revision of Control Limits and Center Lines
• Effective use of control charts requires periodic review and revision
of control limits and center lines.
• Sometimes users replace the center line on the chart with a target
value.
• When R chart is out of control, out-of-control points are often
eliminated to re-compute a revised value of which is used to
determine new limits and center line on R chart and new limits on
chart.
x
x
R
13. Phase II Operation of Charts
• Use of control chart for monitoring future production, after a set of
reliable limits are established, is called phase II of control chart
usage (Figure 5-4).
• A run chart showing individuals observations in each sample, called
a tolerance chart or tier diagram (Figure 5-5), may reveal patterns or
unusual observations in the data.
17. Control vs. Specification Limits
• Control limits are derived from
natural process variability, or
the natural tolerance limits of a
process.
• Specification limits are
determined externally, for
example by customers or
designers.
• There is no mathematical or
statistical relationship between
the control limits and the
specification limits.
18. Rational Subgroups
• charts monitor between-sample variability.
• R charts measure within-sample variability.
• Standard deviation estimate of used to construct
control limits is calculated from within-sample variability.
• It is not correct to estimate using
x
19. Guidelines for Control Chart Design
• Control chart design requires specification of sample size, control
limit width, and sampling frequency.
– Exact solution requires detailed information on statistical characteristics
as well as economic factors.
– The problem of choosing sample size and sampling frequency is one of
allocating sampling effort.
• For chart, choose as small a sample size consistent with
magnitude of process shift one is trying to detect. For moderate to
large shifts, relatively small samples are effective. For small shifts,
larger samples are needed.
• For small samples, R chart is relatively insensitive to changes in
process standard deviation. For larger samples (n > 10 or 12), s or
s2
charts are better choices.
• NOTE: Skip Section on Changing Sample Size (pages 209-212)
x
20. Charts Based on Standard Values
D1 = d2 - 3d3
D2 = d2 + 3d3
d2 : mean of distribution of relative range
d3 : standard deviation of distribution of relative range
22. • An assumption in performance properties is that the underlying
distribution of quality characteristic is normal.
– If underlying distribution is not normal, sampling distributions can be
derived and exact probability limits obtained.
• Usual normal theory control limits are very robust to normality
assumption.
• In most cases, samples of size 4 or 5 are sufficient to ensure
reasonable robustness to normality assumption for chart.
• Sampling distribution of R is not symmetric, thus symmetric 3-sigma
limits are an approximation and -risk is not 0.0027. R chart is
more sensitive to departures from normality than chart.
• Assumptions of normality and independence are not a primary
concern in Phase I.
x
x
Effect of Nonnormality on and Charts
x R
23. Operating Characteristic (OC) Function
σ is known. In-control mean: μ0 out of control mean: μ1 = μ0 + kσ
Probability of not detecting shift: β-risk
L: number of σ’s
For L = 3, n = 5, k = 2.
25. Average run length (r): shift is detected in the rth
sample.
In the example.
Expected number of samples for detecting shift = 4.
26. Average Run Length for Chart
x
For Shewhart control chart:
Average time to signal (ATS)
Average number of individual units sampled for detection (I)
28. Use the and charts instead of the and charts when:
x s x R
Control Charts for and
x s
2 2
5 4 4 6 4 4
3 1 and 3 1
B c c B c c
29. th
Assume no standard is given for . Need to estimate.
preliminary samples, each of size .
: standard deviation for sample
i
m n
s i
4
: unbiased estimator for
chart has the following parameters:
S
c
s
2 2
3 4 4 4
4 4
3 3
Note: 1 1 and 1 1 Then:
B c B c
c c
30. 4
When is used to estimate , chart has the following parameters:
S
x
c
3
4
3
Define . Then:
A
c n
38. Shewhart Control Chart for Individual Measurements
What if there is only one observation for each sample.
Use the moving range between two successive samples for range.
44. Average Run Lengths
• ARL0 of combined individuals and moving-range chart with
conventional 3-sigma limits is generally much less than Shewhart
control chart.
• Ability of individuals chart to detect small shifts is very poor.
45. Normality
• In-control ARL is dramatically
affected by nonnormal data.
• One approach for nonnormal
data is to determine control
limits for individuals control
chart based on percentiles of
correct underlying distribution.
