Download Introduction to Statistical Quality Control, 4th Edition

CHAPTER 6
Control Charts for
Variables
6-1. INTRODUCTION
Variable - a single quality characteristic that can be
measured on a numerical scale.
 We monitor both the mean value of the
characteristic and the variability associated with
the characteristic.

VARIABLES DATA

Examples
Length, width, height
 Weight
 Temperature
 Volume

BOTH MEAN AND VARIABILITY
CONTROLLED
MUST BE
6-2. CONTROL CHARTS FOR x AND R
Notation for variables control charts
 n - size of the sample (sometimes called a subgroup)
chosen at a point in time
 m - number of samples selected
 x i = average of the observations in the ith sample
(where i = 1, 2, ..., m)
 x = grand average or “average of the averages (this
value is used as the center line of the control chart)
6-2. CONTROL CHARTS FOR x AND R
Notation and values
 Ri = range of the values in the ith sample
Ri = xmax - xmin
 R = average range for all m samples
  is the true process mean
  is the true process standard deviation
6-2. CONTROL CHARTS FOR x AND R
Statistical Basis of the Charts
 Assume the quality characteristic of interest is
normally distributed with mean , and standard
deviation, .
 If x1, x2, …, xn is a sample of size n, then the
average of this sample is
x
= (X1 + X2 + … + Xn)/n
x is normally distributed with mean, , and
standard deviation,    / n
x
6-2. CONTROL CHARTS FOR x AND R
Statistical Basis of the Charts
 If n samples are taken and their average is computed,
the probability is 1- α that any sample mean will fall

between:
  Z / 2  x    Z / 2
n
and
  Z / 2  x    Z / 2


n
The above can be used as upper and lower control
limits on a control chart for sample means, if the
process parameters are known.
CONTROL CHARTS FOR

When the parameters are not known
Compute x
 Compute R(bar)


x
Now


UCL =
LCL =
3
X
n
3
X
n
CONTROL CHARTS FOR x

But, est = R / d 2
3R
 So, 3 / n can be written as:
d2 n

Let A2 =

Then,
3
d2 n
3R
 A2 R
d2 n
6-2. CONTROL CHARTS FOR x AND R
Control Limits for the
x chart
UCL  x  A2 R
Center Line  x
LCL  x  A2 R

A2 is found in Appendix VI for various values of n.
CONTROL CHART FOR R
We need an estimate of R
We will use relative range W = R/, R = W
Let d3 be the standard deviation of W
StdDev [R] = StdDev [W]
 R  d3 
Use R to estimate R
CONTROL CHART FOR R
est,R = d3(R(bar)/d2)
UCL = R(bar) + 3 est,R = R(bar) + 3d3(R(bar)/d2)
CL = R(bar)
LCL = R(bar) - 3 est,R = R(bar) - 3d3(R(bar)/d2)
Let D3 = 1 – 3(d3/d2)
And, D4 = 1 + 3(d3/d2)
6-2. CONTROL CHARTS FOR x AND R
Control Limits for the R chart
UCL  D4 R
Center Line  R
LCL  D3 R

D3 and D4 are found in Appendix VI for various
values of n.
6-2. CONTROL CHARTS FOR x AND R
Trial Control Limits
 The control limits obtained from equations (6-4) and
(6-5) should be treated as trial control limits.
 If this process is in control for the m samples
collected, then the system was in control in the past.
 If all points plot inside the control limits and no
systematic behavior is identified, then the process was
in control in the past, and the trial control limits are
suitable for controlling current or future production.
6-2. CONTROL CHARTS FOR x AND R
Trial control limits and the out-of-control process

Eqns 6.4 and 6.5 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
If no assignable cause is found, there are two options
1.
2.

Eliminate point as if an assignable cause were found and
revise limits
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
EXAMPLE 6.1 THE HARD BAKE PROCESS
6-2. CONTROL CHARTS FOR x AND R
Estimating Process Capability




The x-bar and R charts give information about the
performance or Process Capability of the process.
Assumes a stable process.
We can estimate the fraction of nonconforming items
for any process where specification limits are
involved.
Assume the process is normally distributed, and x is
normally distributed, the fraction nonconforming can
be found by solving:
P(x < LSL) + P(x > USL)
EXAMPLE 6-1: ESTIMATING PROCESS
CAPABILITY

Determine est = R(bar)/d2 = .32521/2.326=0.1398


Specification limits are 1.5 + .5 microns.


Where d2 values are given in App. Table VI
(1.00, 2.00)
Assuming N(1.5056, 0.13982)

Compute p = P(x < 1.00) + P(x > 2.00)
EXAMPLE 6-1: ESTIMATING PROCESS
CAPABILITY
P = F([1.00–1.5056]/0.1398) +1 – F([2.001.5056]/.1398) = F(-3.6166) + 1 – F(3.53648) =
.00035
 These values from the APPENDIX 2 / 693
 So, about 350 ppm will be out of specification

6-2. CONTROL CHARTS FOR x AND R
Process-Capability Ratios (Cp)
 Used to express process capability.
 For processes with both upper and lower control
limits,
Use an estimate of  if it is unknown.
USL  LSL
Cp 
6

Cest,p = (2.0 – 1.0)/[6(.1398)] = 1.192
6-2. CONTROL CHARTS FOR x AND R
If Cp > 1, then a low # of nonconforming items will
be produced.
 If Cp = 1, (assume norm. dist) then we are
producing about 0.27% nonconforming.
 If Cp < 1, then a large number of nonconforming
items are being produced.

6-2. CONTROL CHARTS FOR x AND R
Process-Capability Ratios (Cp)
 The percentage of the specification band that the
process uses up is denoted by
 1 
P̂   100%
C 
 p
**The Cp statistic assumes that the process mean is
centered at the midpoint of the specification band
– it measures potential capability.
EXAMPLE 6-1: ESTIMATING BANDWIDTH
USED

Pest = (1/1.192)100% = 83.89%

The process uses about 84% of the specification band.
PHASE II OPERATION OF CHARTS
Use of control chart for monitoring future
production, once a set of reliable limits are
established, is called phase II of control chart
usage (Figure 6.4)
 A run chart showing individuals observations in
each sample, called a tolerance chart or tier
diagram (Figure 6.5), may reveal patterns or
unusual observations in the data

6-2. CONTROL CHARTS FOR x AND R
Control Limits, Specification Limits, and
Natural Tolerance Limits (See Fig 6.6)
 Control limits are functions of the natural
variability of the process (LCL, UCL)
 Natural tolerance limits represent the natural
variability of the process (usually set at 3-sigma
from the mean) (LNTL, UNTL)
 Specification limits are determined by
developers/designers. (LSL, USL)
6-2. CONTROL CHARTS FOR x AND R
6-2. CONTROL CHARTS FOR x AND R
Control Limits, Specification Limits, and
Natural Tolerance Limits

There is no mathematical relationship between
control limits and specification limits.
RATIONAL SUBGROUPS
x
charts monitor between-sample variability
(variability in the process over time)
 R charts measure within-sample variability
(the instantaneous process variability at a given
time)
 Standard deviation estimate of  used to construct
control limits is calculated from within-sample
variability
 It is not correct to estimate  using

6-2. CONTROL CHARTS FOR x AND R
Guidelines for the Design of the Control Chart
 Specify sample size, control limit width, and
frequency of sampling
 If the main purpose of the x-bar chart is to detect
moderate to large process shifts, then small sample
sizes are sufficient (n = 4, 5, or 6)
 If the main purpose of the x-bar chart is to detect
small process shifts, larger sample sizes are
needed (as much as 15 to 25).
6-2. CONTROL CHARTS FOR x AND R
Guidelines for the Design of the Control Chart

R chart is insensitive to shifts in process standard
deviation when n is small
The range method becomes less effective as the sample
size increases
 May want to use S or S2 chart for larger values of n > 10

6-2. CONTROL CHARTS FOR x AND R
Guidelines for the Design of the Control Chart
Allocating Sampling Effort
 Choose a larger sample size and sample less
frequently? Or, choose a smaller sample size and
sample more frequently?
 The method to use will depend on the situation. In
general, small frequent samples are more desirable.
CHANGING SAMPLE SIZE OF X-BAR AND R
CHARTS
1- variable sample size: the center line changes
continuously and the chart is difficult to
interpret. (X-bar and s is more appropriate)
 2- Permanent or semi-permanent change:

UCL  X  A2 [
d 2 (new)
]Rold
d 2 (old )
CL  X
LCL  X  A2 [
d 2 (new)
]Rold
d 2 (old )
for  R  chart
d (new)
UCL  D4 [ 2
]Rold
d 2 (old )
CL  Rnew  [
LCL  D3 [
d 2 (new)
]Rold
d 2 (old )
d 2 (new)
]Rold
d 2 (old )
Introduction to Statistical Quality Control, 4th
Edition
for  X  chart
6-2.3 CHARTS BASED ON STANDARD
VALUES
If the process mean and variance are known or can
be specified, then control limits can be developed
using these values:
R  chart
X  chart
UCL  D 2 
UCL    A
CL  d 2 
CL  
LCL  D1
LCL    A
 Constants are tabulated in Appendix VI

6-2.4 INTERPRETATION OF x AND R
CHARTS

Patterns of the plotted points will provide useful
diagnostic information on the process, and this
information can be used to make process
modifications that reduce variability.
Cyclic Patterns ( systematic environmental changes )
Mixture ( tend to fall near or slightly outside the control
limits – over control and adjustment too often)
 Shift in process level ( new workers, raw material,…)
 Trend ( continuous movement in one direction- wear out )
 Stratification ( a tendency for the point to cluster artificially
around the center line- e.g. Error in control limits
computations )


6-2.6 THE OPERATING
CHARACTERISTIC FUNCTION


How well the x and R charts can detect process
shifts is described by operating characteristic
(OC) curves.
Consider a process whose mean has shifted from
an in-control value by k standard deviations. If
the next sample after the shift plots in-control,
then you will not detect the shift in the mean.
The probability of this occurring is called the brisk.
6-2.6 THE OPERATING
CHARACTERISTIC FUNCTION

The probability of not detecting a shift in the
process mean on the first sample is b
= P{LCL < Xbar < UCL  = 1 = 0 + k}
b  F (L  k n )  F ( L  k n )

L= multiple of standard error in the control limits
k = shift in process mean (# of standard
deviations).
EXAMPLE

We are using a 3 limit Xbar chart with sample
size equal to 5
L=3
 n=5


Determine the probability of detecting a shift to
1 = 0 + 2 on the first sample following the shift

k=2
EXAMPLE, CONTINUED
So, b=F[3–2 SQRT(5)]-F[-3–2 SQRT(5)]
= F(-1.47) – F(-7.37) = .0708
P(not detecting on first sample) = .0708
P(detecting on first sample) =1- .0708=.9292
6-2.6 THE OPERATING
CHARACTERISTIC FUNCTION

The operating characteristic curves are plots of the
value b against k for various sample sizes
If the shift is 1.0σ and the
sample size is n = 5, then
β = 0.75.
USE OF FIGURE 6-13

Let k = 1.5

When n = 5, b = .35
P(detection on 2nd sample) = .35(.65)
 P(detection on 3rd sample) = .352(.65)

P(detection on rth sample) = br-1(1-b)
 ARL = 1/(1-b)


1/(1-.35) = 1/.65 = 1.54
OC CURVE FOR THE R CHART
Use l  1/0 ( the ratio of new to old process
standard deviation ) in Fig. 6.14
 R chart is insensitive when n is small
 But, when n is large, W = R/
loses efficiency, and S chart is better

53
6-3.1 CONSTRUCTION AND
OPERATION OF x AND S CHARTS
S is an estimator of c4 
where c4 is a constant depends on sample size
n.
 The standard deviation of S is  1 c 2

4
6-3.1 CONSTRUCTION AND
OPERATION OF x AND S CHARTS

If a standard  is given the control limits for the S
chart are:
UCL  B 
6
CL  c 4 
LCL  B5

B5, B6, and c4 are found in the Appendix VI for
various values of n.
6-3.1 CONSTRUCTION AND
OPERATION OF x AND S CHARTS
No Standard Given
 If  is unknown, we can use an average sample
standard deviation
1 m
S
UCL  B4 S
CL  S
LCL  B3 S
 Si
m i 1
6-3.1 CONSTRUCTION AND
OPERATION OF x AND S CHARTS
x
Chart when Using S
The upper and lower control limits for the x chart are
given as
UCL  x  A 3 S
CL  x
LCL  x  A 3 S
where A3 is found in the Appendix VI
6-3.2 THE x AND S CONTROL CHARTS
WITH VARIABLE SAMPLE SIZE
The x and S charts can be adjusted to account for
samples of various sizes.
 A “weighted” average is used in the calculations of
the statistics.
m = the number of samples selected.
ni = size of the ith sample

6-3.2 THE x AND S CONTROL CHARTS
WITH VARIABLE SAMPLE SIZE

The grand average can be estimated as:
m
x
 nixi
i 1
m
 ni
i 1
 The average sample standard deviation is:
m
 (n i  1)Si
S  i 1m
 ni  m
i 1
2
6-3.2 THE x AND S CONTROL CHARTS
WITH VARIABLE SAMPLE SIZE

Control Limits
UCL  x  A 3 S
UCL  B 4 S
CL  x
CL  S
LCL  x  A 3 S
LCL  B3 S
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS

What if you could not get a sample size
greater than 1 (n =1)? Examples include
Automated inspection and measurement technology is
used, and every unit manufactured is analyzed.
 The production rate is very slow, and it is
inconvenient to allow samples sizes of n > 1 to
accumulate before analysis


The X and MR charts are useful for samples
of sizes n = 1.
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS
Moving Range Control Chart
 The moving range (MR) is defined as the absolute
difference between two successive observations:
MRi = |xi - xi-1|
which will indicate possible shifts or changes in the
process from one observation to the next.
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS
X and Moving Range Charts
 The X chart is the plot of the individual
observations. The control limits are
UCL  x  3
MR
d2
CL  x
where
LCL  x  3
m
 MR i
MR  i 1
m
MR
d2
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS
X and Moving Range Charts
 The control limits on the moving range chart are:
UCL  D 4 MR
CL  MR
LCL  0
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS
Example
Ten successive heats of a steel alloy are tested for
hardness. The resulting data are
Heat Hardness
Heat
Hardness
1
52
6
52
2
51
7
50
3
54
8
51
4
55
9
58
5
50
10
51
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS
Example
Individuals
I and MR Chart for hardness
62
3.0SL=60.97
52
X=52.40
-3.0SL=43.83
42
Observation 0
Moving Range
10
1
2
3
4
5
6
7
8
9
10
3.0SL=10.53
5
R=3.222
0
-3.0SL=0.000
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS
Interpretation of the Charts
 MR charts cannot be interpreted the same as R or x
charts.
 Since the MR chart plots data that are “correlated”
with one another, then looking for patterns on the
chart does not make sense.
 MR chart cannot really supply useful information
about process variability.
6-4. THE SHEWHART CONTROL CHART
FOR INDIVIDUAL MEASUREMENTS

The normality
assumption is often
taken for granted.
When using the
individuals chart, the
normality assumption is
very important to chart
performance.
.999
.99
.95
Probability

Normal Probability Plot
.80
.50
.20
.05
.01
.001
50
51
52
53
54
55
56
57
58
hardness
Average: 52.4
StDev: 2.54733
N: 10
Anderson-Darling Normality Test
A-Squared: 0.648
P-Value: 0.063
END