Process
Control Charts
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Common Control Charts
Type
Statistic
Distribution
Reveals
X -bar chart
R -chart
s -chart
I -chart
MR -chart
c -chart
p -chart
np -chart
Zone chart
sample mean
sample range
sample st. dev.
individual X values
moving range
sample mean
sample proportion
sample total
any statistic
normal
unnamed
chi
normal
unnamed
Poisson
binomial
binomial
varies
centrality, dispersion
variation
variation
sample item
variation
number of defects
success rate
number of successes
deviation from target
Comment Variable charts are shown in blue, attribute
charts in yellow, and zone charts in purple.
Control Limits Illustrated
Upper Control Limit Sample
means rarely are above UCL
Centerline The intended target
specification or historical mean
Lower Control Limit Sample
means rarely are below LCL
Control Limits for
X-Bar Chart
UCLX = X + 3
R
= upper control limit of sample mean
d2 n
LCLX = X - 3
R
= lower control limit of sample mean
d2 n
where
X is the empirical sample mean based on repeated sampling (centerlin e of X chart)
R is the empirical average range based on repeated sampling (centerlin e of R chart)
n is the size of each sample taken
d2 is the control chart factor for samples of size n (from a table)
Control Limits for
R-Chart
UCLR = D4 R = upper control limit of sample range
LCLR = D3 R = lower control limit of sample range
where
R is the empirical average range based on repeated sampling
D3 is the control chart factor for samples of size n (from a table)
D4 is the control chart factor for samples of size n (from a table)
Example: Numerical Data
Example Manufactured tubing is to have a mean
diameter of 15.00 mm. The historical standard
deviation of the manufacturing process is 0.002
mm. Samples from 4 extrusion machines taken
every 10 minutes and measured with precise
instruments. The mean and range of the four
sample items are plotted on control charts.
Comment X-Bar and R-charts are usually paired. The former
reveals whether the process is tracking its centerline, while the
latter reveals whether excessive variation exists.
X-Bar/R Chart
Xbar/R Chart for Diameter
Sample Mean
15.0035
UCL=15.00
15.0025
15.0015
15.0005
Mean=15
14.9995
14.9985
14.9975
LCL=15.00
14.9965
Subgroup
Sample Range
0.010
0
10
20
30
40
50
UCL=0.009394
0.005
R=0.004118
0.000
LCL=0
Comment The X-bar chart reveals whether the process is tracking its centerline, while the
R-chart reveals whether excessive variation exists. This process is in control, although the
mean of sample 33 is very near its LCL.
Example: Attribute Data
Example In a large urban hospital, a daily sample of 250
emergency arrivals is taken, showing the number of
patients who were treated within 30 minutes. The
historical proportion is 0.90.
Comment This requires either an np-chart (if we just count the
number treated within 30 minutes) or a p-chart (if the count is
converted to a proportion). The daily sample size must be
constant in order to set the control limits.
Example: np-Chart
NP Chart for Number Treated Within 30 Minutes
Sample Count
240
UCL=239.2
230
NP=225
220
LCL=210.8
210
0
50
100
Sample Number
Comment The centerline is 225 (that is, 90% of 250). This chart indicates that, despite
some fluctuation, the proportion treated within 30 minutes remains within its control limits.
Example: p-Chart
P Chart for Proportion of Patients Seen Within 30 Minutes
UCL=0.9569
Proportion
0.95
0.90
P=0.9
0.85
LCL=0.8431
0
50
100
Sample Number
Comment The centerline is 0.90 (the target proportion seen within 30 minutes). Except for
the scaling (dividing the total by 250) this p-chart is identical to the np-chart. Despite some
fluctuation, the sample proportion remains within its control limits.
Control Limits for
np-Chart
UCL = np  3 np(1 - p) = upper control limit of sample total
LCL = np  3 np(1 - p) = lower control limit of sample total
where
p is the target proportion (centerline)
Control Limits for
p-Chart
UCL = p  3
p(1 - p)
= upper control limit of sample proportion l
n
LCL = p  3
p(1 - p)
= lower control limit of sample proportion
n
where
p is the target proportion (centerline)
I/MR Chart
Some manufacturing processes can be monitored
continuously, obviating sampling. Each individual
measurement is plotted on an I-chart. Since the sample
size is n = 1, there is no range (R=Xmax-Xmin). Instead, we
use a moving range and MR-chart. The control limits are
explained in advanced texts.
Comment I-charts and MR-charts are usually paired. The
former reveals whether the process is tracking its centerline,
while the latter reveals whether excessive variation exists.
I/MR Chart: Example
Example Manufactured tubing is to have a mean
diameter of 15.00 mm. The historical standard
deviation of the manufacturing process is 0.002
mm. A laser monitoring system eliminates the
need for sampling, and measurements are taken
automatically every minute. The observed
diameter measurement and moving range are
plotted on control charts.
Example: Tubing Diameter
Individual Value
I and MR Chart for Diameter
UCL=15.01
15.005
15.000
Mean=15
14.995
Subgroup
LCL=14.99
0
100
200
Moving Range
0.008
UCL=0.007371
0.006
0.004
0.002
R=0.002256
0.000
LCL=0
Comment The process is tracking its centerline (I-chart)
and variation does not appear excessive (MR-chart).
X-Bar Chart
Tests for Pattern
Rule 1 Single point beyond 3 sigma
Rule 2 2 of 3 successive points beyond 2 sigma on
same side of centerline
Rule 3 4 of 5 successive points beyond 1 sigma on
same side of centerline
Rule 4 9 successive points on same side of centerline
Rule 1
Rule 1 Single point beyond 3 sigma
Rule 2
Rule 2 2 of 3 successive points beyond 2 sigma
on same side of centerline
Rule 3
Rule 3 4 of 5 successive points beyond 1 sigma on
same side of centerline
Rule 4
Rule 4 9 successive points on same side of centerline
Other Pattern Tests
Note Other rules of thumb are possible. Here are
MINITAB’s tests for pattern for X-bar/R charts.
Violations
Cycle Observations follow a cyclic pattern (several above centerline
followed by several below the centerline). Detected visually or by runs test.
Oscillation Observations tend to alternate above and below the centerline.
Detected visually or using a runs test.
Instability Observations vary more than expected so the points lie too far
from the centerline. Detected by rules 1-3 or from the R-chart.
Level shift Observations shift abruptly to a level above (or below) the
centerline and then stay there. Detected by rules 1-4. Can resemble trend.
Trend Observations drift slowly either upward or downward. Detected by
rules 1-4 or by visual inspection. Can resemble level shift.
Mixture Observations come from two or more populations which are
different. Similar to instability. May be hard to detect.
Instability
Individual Value
I and MR Chart for Unstable
9
8
7
6
5
4
3
2
1
Subgroup
5
6
Mean=5.05
6
5
LCL=1.45
0
50
100
7
Moving Range
UCL=8.65
1
6
5
4
1
1
1
UCL=4.423
3
2
1
0
Test Results for I Chart
TEST 5. 2 out of 3 points more than 2 sigmas from center line
Test Failed at points: 66 98
TEST 6. 4 out of 5 points more than 1 sigma from center line
Test Failed at points: 87 98 99
R=1.354
LCL=0
Test Results for MR Chart
TEST 1. One point more than 3.00
sigmas from center line.
Test Failed at points: 65 67 77 78
Trend
Individual Value
I and MR Chart for Trend
9
8
7
6
5
4
3
2
1
Subgroup
UCL=8.65
2 2
2 2
2 2 2
2 2
Mean=5.05
2
LCL=1.45
0
50
100
Note Easily
mistaken for
level shift.
Moving Range
5
UCL=4.423
4
3
2
1
0
2
2 2
22
Test Results for I Chart
TEST 2. 9 points in a row on same side of center line.
Test Failed at points: 12 86 87 88 89 90 91 92 93 94
R=1.354
LCL=0
Test Results for MR Chart
TEST 2. 9 points in a row on same side of center line.
Test Failed at points: 10 11 12 13 14
Level Shift
Individual Value
I and MR Chart for Level
9
8
7
6
5
4
3
2
1
Subgroup
UCL=8.65
2
2
2
2
2
Mean=5.05
LCL=1.45
0
50
100
Note Easily
mistaken for
trend.
Moving Range
5
UCL=4.423
4
3
2
1
R=1.354
2
0
Test Results for I Chart
TEST 2. 9 points in a row on same side of center line.
Test Failed at points: 49 87 97 98 99
LCL=0
Test Results for MR Chart
TEST 2. 9 points in a row on same side of center line.
Test Failed at points: 30
Cycle
Visual inspection suggests series of runs above centerline followed by
series of runs below centerline. For a chart showing m samples, the
expected number of centerline crossings would be m/2 +1. For these 50
samples, we see only 19 crossings but would expect 26.
Oscillation
Visual inspection suggests alternating means above and below centerline.
For a chart showing m samples, the expected number of centerline
crossings would be m/2 +1. For these 50 samples, we see 40 crossings
but would expect only 26 (it’s hard to count some of the crossings but
we’ll call it 40).
Runs Test
Patterns of cycle and oscillation may not violate any of the usual
rules. A more sensitive test for cycle or oscillation is the runs test.
For m samples, the expected number of centerline crossings is
m/2+1. If m is reasonably large, we count the actual number of
centerline crossings R and compute a test statistic:
Z
R  (m / 2  1)
m(m  2)
4(m  1)
If m is reasonably large, an approximate result is:
Z
R  (m / 2  1)
m
4
Runs Test: Example
For 50 samples, we observe 40 centerline crossings. The runs test
statistic is:
Z
R  (m / 2  1)
m(m  2)
4(m  1)

40  (50 / 2  1)
50 (50  2)
4(50  1)

40  26
 4.00
50 (48)
4(49 )
Since m is reasonably large, the approximate normal test statistic is:
Z
40  (50 / 2  1)
50
4

40  (50 / 2  1)
50
4

40  26
 3.96
3.5355
Either result shows that the observed number of runs is almost four
standard deviations above its expectation, suggesting that oscillation
exists (positive sign). A negative sign would suggest cycle.
Other Control Charts
Variable charts for subgroups
Time-weighted variable charts
Variable charts for individuals
Attribute charts
Note There are many types of control charts. Here are
some offered by MINITAB. If you want to learn about
them, you’ll need to take a specialized class.
Service Sector?
In services such as health care, the usual control charts may not
apply because
processes may not be repetitive
sample size may vary from sample to sample
sample intervals may be irregular
t here is no a priori benchmark to meet.
Alternatives include (1) a time plot showing a statistic surrounded by
its confidence interval (e.g., mean or proportion), (2) a box plot over
time (for a numerical variable such as waiting time), or (3) a plot of
percentiles of the raw data over time (e.g., 10, 25, 50, 75, 90).
Simple Time Plot
Time Plot of 95% CI for Mean ER Wait Time
30.00
25.00
20.00
Upper 95%
15.00
Mean
10.00
Low er 95%
5.00
Observation Date
12
/1
3/
00
12
/6
/0
0
11
/2
9/
00
11
/2
2/
00
11
/1
5/
00
11
/8
/0
0
0.00
11
/1
/0
0
Waiting TIme (minutes)
35.00
Hi-Lo Plot
Confidence Intervals for Mean ER Wait Time
(Stock Hi-Lo Chart Style)
30
25
20
Upper 95%
15
Mean
Low er 95%
10
5
Sam ple Date
12
/6
/0
0
12
/1
3/
00
0
11
/1
/0
0
11
/8
/0
0
11
/1
5/
00
11
/2
2/
00
11
/2
9/
00
Waiting Time (minutes)
35
Box Plots Over Time
Box Plots of ER Waiting Time
Wait Time (minutes)
50
40
30
20
10
Nov 2
Nov 9
Nov 16
Date
Nov 23
Nov 30
Summary
Control charts help you track change
Control charts were developed for manufacturing
Control charts can also be used in services
Creative modifications may be needed