1. Control Charts for Variables
Chapter 5
Control Charts for Variables
Introduction to Statistical Quality Control, 4th Edition

2. Introduction to Statistical Quality Control, 4th Edition
Variable - a single quality characteristic that can be measured on a numerical scale.
When working with variables, we should monitor both the mean value of the characteristic and the variability associated with the characteristic.
Introduction to Statistical Quality Control, 4th Edition

3. 5-2. Control Charts for 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
= average of the observations in the ith sample (where i = 1, 2, ..., m)
= grand average or “average of the averages (this value is used as the center line of the control chart)
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4. 5-2. Control Charts for and R
Notation and values
Ri = range of the values in the ith sample
Ri = xmax - xmin
= average range for all m samples
 is the true process mean
 is the true process standard deviation
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5. 5-2. Control Charts for 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 he average of this sample is
is normally distributed with mean, , and standard deviation,
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6. 5-2. Control Charts for and R
Statistical Basis of the Charts
The probability is 1 -  that any sample mean will fall between
The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.
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7. 5-2. Control Charts for and R
Control Limits for the chart
A2 is found in Appendix VI for various values of n.
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8. 5-2. Control Charts for and R
Control Limits for the R chart
D3 and D4 are found in Appendix VI for various values of n.
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9. 5-2. Control Charts for and R
Estimating the Process Standard Deviation
The process standard deviation can be estimated using a function of the sample average range.
This is an unbiased estimator of 
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10. 5-2. Control Charts for and R
Trial Control Limits
The control limits obtained from equations (5-4) and (5-5) 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.
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11. 5-2. Control Charts for and R
Trial control limits and the out-of-control process
If points plot out of control, then the control limits must be revised.
Before revising, identify out of control points and look for assignable causes.
If assignable causes can be found, then discard the point(s) and recalculate the control limits.
If no assignable causes can be found then 1) either discard the point(s) as if an assignable cause had been found or 2) retain the point(s) considering the trial control limits as appropriate for current control.
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12. 5-2. Control Charts for and R
Estimating Process Capability
The x-bar and R charts give information about the capability of the process relative to its specification limits.
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)
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13. 5-2. Control Charts for 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.
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.
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14. 5-2. Control Charts for and R
Process-Capability Ratios (Cp)
The percentage of the specification band that the process uses up is denoted by
**The Cp statistic assumes that the process mean is centered at the midpoint of the specification band – it measures potential capability.
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15. 5-2. Control Charts for and R
Control Limits, Specification Limits, and Natural Tolerance Limits
Control limits are functions of the natural variability of the process
Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean)
Specification limits are determined by developers/designers.
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16. 5-2. Control Charts for and R
Control Limits, Specification Limits, and Natural Tolerance Limits
There is no mathematical relationship between control limits and specification limits.
Do not plot specification limits on the charts
Causes confusion between control and capability
If individual observations are plotted, then specification limits may be plotted on the chart.
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17. 5-2. Control Charts for and R
Rational Subgroups
X bar chart monitors the between sample variability
R chart monitors the within sample variability.
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18. 5-2. Control Charts for 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)…which is often impractical…alternative types of control charts are available for this situation…see Chapter 8
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19. 5-2. Control Charts for and R
Guidelines for the Design of the Control Chart
If increasing the sample size is not an option, then sensitizing procedures (such as warning limits) can be used to detect small shifts…but this can result in increased false alarms.
R chart is insensitive to shifts in process standard deviation.(the range method becomes less effective as the sample size increases) may want to use S or S2 chart.
The OC curve can be helpful in determining an appropriate sample size.
Introduction to Statistical Quality Control, 4th Edition

20. 5-2. Control Charts for 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.
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21. 5-2. Control Charts for and R
Changing Sample Size on the and R Charts
In some situations, it may be of interest to know the effect of changing the sample size on the x-bar and R charts. Needed information:
= average range for the old sample size
= average range for the new sample size
nold = old sample size
nnew = new sample size
d2(old) = factor d2 for the old sample size
d2(new) = factor d2 for the new sample size
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22. 5-2. Control Charts for and R
Changing Sample Size on the and R Charts
Control Limits
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23. 5-2.3 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:
Constants are tabulated in Appendix VI
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24. 5-2.4 Interpretation of 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
Mixture
Shift in process level
Trend
Stratification
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25. 5-2.5 The Effects of Nonnormality and R
In general, the chart is insensitive (robust) to small departures from normality.
The R chart is more sensitive to nonnormality than the chart
For 3-sigma limits, the probability of committing a type I error is on the R-chart. (Recall that for , the probability is only ).
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26. 5-2.6 The Operating Characteristic Function
How well the 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 -risk.
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27. 5-2.6 The Operating Characteristic Function
The probability of not detecting a shift in the process mean on the first sample is
L= multiple of standard error in the control limits
k = shift in process mean (#of standard deviations).
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28. 5-2.6 The Operating Characteristic Function
The operating characteristic curves are plots of the value  against k for various sample sizes.
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29. 5-2.6 The Operating Characteristic Function
If  is the probability of not detecting the shift on the next sample, then 1 -  is the probability of correctly detecting the shift on the next sample.
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30. 5-3.1 Construction and Operation of and S Charts
First, S2 is an “unbiased” estimator of 2
Second, S is NOT an unbiased estimator of 
S is an unbiased estimator of c4 
where c4 is a constant
The standard deviation of S is
Introduction to Statistical Quality Control, 4th Edition

31. 5-3.1 Construction and Operation of and S Charts
If a standard  is given the control limits for the S chart are:
B5, B6, and c4 are found in the Appendix for various values of n.
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32. 5-3.1 Construction and Operation of and S Charts
No Standard Given
If  is unknown, we can use an average
sample standard deviation,
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33. 5-3.1 Construction and Operation of and S Charts
Chart when Using S
The upper and lower control limits for the chart are given as
where A3 is found in the Appendix
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34. 5-3.1 Construction and Operation of and S Charts
Estimating Process Standard Deviation
The process standard deviation,  can be estimated by
Introduction to Statistical Quality Control, 4th Edition

35. 5-3.2 The and S Control Charts with Variable Sample Size
The 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
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36. 5-3.2 The and S Control Charts with Variable Sample Size
The grand average can be estimated as:
The average sample standard deviation is:
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37. 5-3.2 The and S Control Charts with Variable Sample Size
Control Limits
If the sample sizes are not equivalent for each sample, then
there can be control limits for each point (control limits may differ for each point plotted)
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38. Introduction to Statistical Quality Control, 4th Edition
5-3.3 The S2 Control Chart
There may be situations where the process variance itself is monitored. An S2 chart is
where and are points found from the chi-square distribution.
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39. 5-4. 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
Repeat measurements on the process differ only because of laboratory or analysis error, as in many chemical processes.
The X and MR charts are useful for samples of sizes
n = 1.
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40. 5-4. The Shewhart Control Chart for Individual Measurements
Moving Range 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.
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41. 5-4. 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
where
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42. 5-4. The Shewhart Control Chart for Individual Measurements
X and Moving Range Charts
The control limits on the moving range chart are:
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43. 5-4. 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
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44. 5-4. The Shewhart Control Chart for Individual Measurements
Example
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45. 5-4. The Shewhart Control Chart for Individual Measurements
Interpretation of the Charts
X Charts can be interpreted similar to charts. MR charts cannot be interpreted the same as or R 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.
More emphasis should be placed on interpretation of the X chart.
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46. 5-4. 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.
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