Control Charts for Variables

1
Chapter 5

• Control Charts for Variables

2
5-1. 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.

3
Variables data

• Examples
• Length, width, height
• Weight
• Temperature
• Volume

4
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)

5
Both mean and variability must be controlled

• See Figure 5-1b
• Process mean has shifted
• See Figure 5-1c
• Variability has increased

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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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5-2. 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 the
  average of this sample is
• (X1 X2 Xn)/n
• is normally distributed with mean, ?, and
  standard deviation,

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5-2. Control Charts for and R

• Statistical Basis of the Charts
• If n samples are taken and their average is
  computed, on the average, a ( or fraction) of
  those averages 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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Control Charts for

• When the parameters are not known
• Compute
• Compute Rbar
• Now
• UCL 3s/SQRT(n)
• LCL - 3s/SQRT(n)

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Control Charts for

• But, sest Rbar/d2
• So, 3s/SQRT(n) can be written as
  3(Rbar/d2)/SQRT(n)
• Let A2 3/d2SQRT(n)
• Then, 3(Rbar/d2)/SQRT(n) A2Rbar

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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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Control chart for R

• We need an estimate of sR
• We will use W R/s, R Ws
• Let d3 be the standard deviation of W
• StdDev R StdDev Ws
• sR d3s
• Use R to estimate sR

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Control chart for R

• sest,R d3(Rbar/d2)
• UCL Rbar 3 sest,R Rbar 3d3(Rbar/d2)
• CL Rbar
• LCL Rbar - 3 sest,R Rbar - 3d3(Rbar/d2)
• Let D3 1 3(d3/d2)
• And, D4 1 3(d3/d2)

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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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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.
• sest Rbar/d2
• This is an unbiased estimator of ?

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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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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 either
  1) 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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Example 5-1

• Go over pages 213-215
• Note R chart first
• Process variability has to be in control
• Note Can do all of this with Minitab

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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 lt LSL) P(x gt USL)

20
Example 5-1 Estimating process capability

• Continue the piston ring example
• Determine sest Rbar/d2 .023/2.326
• Where d2 values are given in App. Table VI
• Specification limits are 74.000 .05 mm.
• (73.950, 74.050)
• Assuming N(74.001, .00992)
• Compute p P(x lt 73.950) P(x gt 74.050)

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Example 5-1 Estimating process capability

• P F(73.95074.001/.0099)
  1 F(74.050-74.001/.0099)
  F(-5.15) 1 F(4.04) .00002
• So, about 20 ppm will be out of specification

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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 gt 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 lt 1, then a large number of nonconforming
  items are being produced.

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Example 5-1 Estimating process capability ratio

• Cest,p (74.05 73.95)/6(.0099) 1.68
• Note Assumes that we can adjust the process
  mean so that it is centered at (USL LSL)/2.

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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.

25
Example 5-1 Estimating bandwidth used

• Pest (1/1.68)100 59.5
• The process uses about 60 of the specification
  band

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Process fallout

• See Figure 5-5
• a cp gt 1
• Little fallout
• b cp 1
• Fallout will average 2700 ppm (.27)
• c cp lt 1
• Fallout will be large

27
Example 5-1 Continued

• Pgs. 218 - 221
• 15 additional samples were collected
• Shown in Table 5-2
• Continuations of the Xbar and R charts are shown
  in Figure 5-6
• At the 37th sample and after, Xbar chart shows
  out-of-control condition
• Must have been a shift in the process mean around
  sample 34 or 35

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Example 5-1 Continued

• Pgs. 218 - 221
• Look at Fig. 5-7
• Shows the old mean and the new apparent mean
  (74.015 mm.)
• At a process mean of 74.015 mm.
• 210 ppm will be defective
• Compare this to the original 20 ppm defective
• This calls for a search for the cause of the shift

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5-2. Control Charts for and R

• Control Limits, Specification Limits, and Natural
  Tolerance Limits (See Fig 5-8)
• 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)

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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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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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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
  impracticalalternative types of control charts
  are available for this situation (Chapter 8)

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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 shiftsbut
  this can result in increased false alarms.

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5-2. Control Charts for and R

• Guidelines for the Design of the Control Chart
• R chart is insensitive to shifts in process
  standard deviation when n is small
• When n 5, P(detection on first sample) is about
  .40 for shifts from s to 2s
• The range method becomes less effective as the
  sample size increases
• May want to use S or S2 chart for larger values
  of n

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5-2. Control Charts for and R

• Guidelines for the Design of the Control Chart
• The OC curve is helpful in determining an
  appropriate sample size
• Discussed later in this chapter

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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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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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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

39
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 0.00461on the R-chart. (Recall
  that for , the probability is only 0.0027).

40
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
  b-risk.

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5-2.6 The Operating Characteristic
Function

• The probability of not detecting a shift in the
  process mean on the first sample is b
• PLCL lt Xbar lt UCL m m1 m0 ks
• L multiple of standard error in the control
  limits
• k shift in process mean ( of standard
  deviations).

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Example

• We are using a 3s limit Xbar chart with sample
  size equal to 5
• L 3
• n 5
• Determine the probability of detecting a shift to
  m1 m0 2s on the first sample following the
  shift
• k 2

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Example, continued

• So, bF32 SQRT(5)-F-32 SQRT(5)
• F(-1.47) F(-7.37) .0708
• P(not detecting on first sample) .0708
• P(detecting on first sample) 1- .0708.9292

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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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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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Use of Figure 5-15

• Let k 1.5
• When n 5, b .35
• When n 10, b .12
• When n 15, b .03

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Use of Figure 5-15

• 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)
• ARL1 1/(1-b)
• 1/(1-.35) 1/.65 1.54
• (average out-of-control run length)

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OC curve for the R chart

• Use l s1/s0 in Fig. 5-16
• R chart is insensitive when n is small
• Let l 2
• When n 4, b .79
• When n 5, b .76
• When n 6, b .70
• But, when n is large, W R/s loses efficiency,
  and S chart is better

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ARL (again)

• In control
• ARL0 1/a
• Out-of-control
• ARL1 1/(1-b)
• Fig. 5-17 can be used to determine ARL1
• To detect a shift of 1.5s when n 3, ARL1 3

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Expected number of individual units sampled, I

• See Fig. 5-18
• If the shift is 1.5s, and n 3, then an average
  of 8 units will pass to detect the shift

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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

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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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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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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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Example 5-3

• Pgs. 243-244

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5-3.1 Construction and Operation of
and S Charts

• Estimating Process Standard Deviation
• The process standard deviation, ? can be
  estimated by
• sest Sbar/c4

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Example 5-3

• Sbar .0094 and c4 .94
• Then, sest .0094/.9400 .01

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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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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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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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Example 5-4

• Pages 245-248
• Piston-ring data, but samples are of size n
  3, 4, 5
• For each sample size, compute its own control
  limits
• Use A3, B3 and B4 for the appropriate sample size

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Example 5-4, cont.

• Estimation of s
• Compute average S for samples where n 5 only
• 5 is most frequently occurring value
• Sbar .1605/17 .0094
• Then sest Sbar/c4 .01
• Where c4 is the value for samples of size 5, the
  most commonly occurring value)

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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 gt 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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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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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

66
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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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
• 1 52 6 52
• 2 51 7 50
• 3 54 8 51
• 4 55 9 58
• 5 50 10 51

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5-4. The Shewhart Control Chart for
Individual Measurements

• Example

69
5-4. The Shewhart Control Chart for
Individual Measurements

• Interpretation of the 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.

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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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Assignment

• Work the odd exercises until you learn the
  material that we covered in this chapter

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End