ENGM 720 - Lecture 07

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ENGM 720 - Lecture 07

• X-bar, R, S Control Charts
• ARL OC Curves

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Assignment

• Reading
• Chapter 5
• Start Finish reading
• Chapter 6
• Spring Break means reading statistics on the
  beach, right?
• Assignments
• Obtain the draft Ctrl Chart Factors table from
  Materials Page
• Access Excel Template for X-bar, R, S Control
  Charts
• Download Assignment 5 for practice
• Use the Excel sheet to do the charting, and
  verify your hand calculations

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Process for Statistical Control Of Quality

• Removing special causes of variation
• Hypothesis Tests
• Ishikawas Tools
• Managing the process with control charts
• Process Improvement
• Process Stabilization
• Confidence in When to Act

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Moving from Hypothesis Testing to Control Charts

• A control chart is like a sideways hypothesis
  test
• Detects a shift in the process
• Heads-off costly errors by detecting trends

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Test of Hypothesis

• A statistical hypothesis is a statement about the
  value of a parameter from a probability
  distribution.
• Ex. Test of Hypothesis on the Mean
• Say that a process is in-control if its mean is
  m0.
• In a test of hypothesis, use a sample of data
  from the process to see if it has a mean of m0 .
• Formally stated
• H0 m m0 (Process is in-control)
• HA m ? m0 (Process is out-of-control)

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Test of Hypothesis on Mean (Variance Known)

• State the Hypothesis
• H0 m m0
• H1 m ? m0
• Take random sample from process and compute
  appropriate test statistic
• Pick a Type I Error level (a) and find the
  critical value za/2
• Reject H0 if z0 gt za/2

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UCL and LCL are Equivalent to the Test of
Hypothesis

• Reject H0 if
• Case 1
• Case 2
• For 3-sigma limits za/2 3

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Two Types of Errors May Occur When Testing a
Hypothesis

• Type I Error - a
• Reject H0 when we shouldn't
• Analogous to false alarm on control chart, i.e.,
• point lays outside control limits but process is
  truly in-control
• Type II Error - b
• Fail to reject H0 when we should
• Analogous to insensitivity of control chart to
  problems, i.e.,
• point does not lay outside control limits but
  process is never-the-less out-of-control

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Choice of Control LimitsTrade-off Between Wide
or Narrow Control Limits

• Moving limits further from the center line
• Decreases risk of false alarm, BUT increases risk
  of insensitivity
• Moving limits closer to the center line
• Decreases risk of insensitivity, BUT increases
  risk of false alarm

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Consequences of Incorrect Control Limits

• Bad Thing 1
• A control chart that never finds anything wrong
  with the process, but the process produces bad
  product
• Bad Thing 2
• Too many false alarms destroy the operating
  personnels confidence in the control chart, and
  they stop using it

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Differences in Viewpoint Between Test of
Hypothesis Control Charts
Hypothesis Test Control Chart
Checks for the validity of assumptions. (ex. is the actual process mean what we think it is?) Detect departures from assumed state of statistical control
Tests for sustained shift (ex. have we actually reduced the variation like we think we have?) Detects shifts that are short-lived Detects steady drifts Detects trends
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Example Part Dimension

• When a process is in-control, a dimension is
  normally distributed with mean 30 and std dev 1.
  Sample size is 5. Find the control limits for an
  x-bar chart with a false alarm rate of 0.0027.
• r.v. x - dimension of part
• r.v. x - sample mean dimension of part

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Distribution of x vs. Distribution of x
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Ex. Part DimensionCont'd

• Find UCL
• The control limits are

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Ex. Modified Part Limits

• Consider an in-control process. A process
  measurement has mean 30 and std dev 1 and n 5.
• Design a control chart with prob. of false alarm
  0.005
• If the control limits are not 3-Sigma, they are
  called
• "probability limits".

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General ModelShewhart Control Chart

• Suppose x is some quality characteristic, and w
  is a sample statistic of x.
• Suppose mean of w is µw and std dev of w is sw,
  then
• UCL µ w Lsw
• CL µ w
• LCL µ w Lsw
• where L is the distance of the control limits
  from the center line, and expressed in multiples
  (units) of the standard deviation of the
  statistic, i.e. sw.
• This type of chart is called a Shewhart Control
  Chart

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Rational Subgroups

• Subgroups / Samples should be selected so that if
  assignable causes are present
• Chance for differences between samples is
  maximized
• Chance for differences within a sample is
  minimized
• Use consecutive units of production
• Keep sample size small so that
• New events wont occur during sampling
• Inspection is not too expensive
• But size is large enough that x is normally
  distributed

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Why Monitor Both Process Mean and Process
Variability?
Process Over Time
Control Charts
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Teminology

• Causes of Variation
• Assignable / Special Causes
• Keep the process from operating predictably
• Things that we can do something about
• Common / Chance Causes
• Random, inherent variation in the process

• Meaning of Control
• In Specification
• Meets customer constraints on product
• In Statistical Control
• No Assignable Causes of variation present in the
  process

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Statistical Basis of x Chart

• Suppose a quality characteristic is x N(m, s)
• and we know m and s
• If x1, x2, , xn is a random sample of size n
  thenand
• Recall that the probability is ? that either
  or

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Statistical Basis of x ChartCont'd

• which is equivalent to
• Where LCL and UCL are the lower and upper
  control limits, respectively
• In practice, one must estimate m and s from data
  coming from an in-control process

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Statistics of the Range

• R the range is a sample statistic
• If x1, x2, , xn is a random sample of size n
  from a normal distribution then one can estimate
  ? using the range
• where d2 is a function of n and can be found in
  Appendix VI
• Can get a better estimate for ? if using more
  than one sample
• Compute Ri for each of m samples where i 1, ,
  m
• Then use the sample average of Ri

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Computing Trial Control Limits for x Chart

• Assume a quality characteristic x N(m, s)
• Take m ? 20 samples of size n 4, 5, or 6
• For each sample i, compute x and Ri for i 1, ,
  m
• Compute x and R

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Computing Trial Control Limits for x Chart

• General model for x chart
• Substituting estimates for µx and s x and using
  3-sigma limits
• Where A2 comes from Appendix VI and depends on n

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Computing Trial Control Limits for R - Chart

• x and R charts come as a pair
• General model for R chart
• Substituting estimates for mR and sR and using
  3-sigma limits

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Computing Trial Control Limits for R - Chart
(continued)

• where and
• D3 and D4 are tabulated in Appendix VI and depend
  on n
• NOTE R chart is quite sensitive to departures
  from normality

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Control Chart Factors Table (Appendix VI see
Materials Page for Engineering Notebook Copy)

• For a constant sample size (n) and 3s limits
• Table factors derived from Montgomery,D.C.,
  (2005) Statistical Quality Control, 5th Ed.

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Trial Control Chart LimitsGuidelines for
Sampling

• Sample should be of size 3 to 8
• (sizes 4 6 are more common)
• Sample must be homogeneous
• same time (consecutive units)
• same raw materials
• same operator
• same machine
• Time may pass between samples but not within
  samples

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Steps for Trial Control Limits

• Start with 20 to 25 samples
• Use all data to calculate initial control limits
• Plot each sample in time-order on chart.
• Check for out of control sample points
• If one (or more) found, then
• Investigate the process
• Remove the special cause and
• Remove the special cause point and recalculate
  control limits.
• If cant find special cause - drop point
  recalculate anyway

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Control Chart Sensitizing Rules

• Western Electric Rules
• One point plots outside the three-sigma limits
• Eight consecutive points plot on one side of the
  center line
• Two out of three consecutive points plot beyond
  two-sigma warning limits on the same side of the
  center line or
• Four out of five consecutive points plot beyond
  one-sigma warning limits on the same side of the
  center line.
• If chart shows lack of control, investigate for
  special cause

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Control Chart Examples
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Control Chart Sensitizing Rules

• Additional Sensitizing Rules
• One or more points very near a control limit.
• Six points in a row steadily increasing or
  decreasing.
• Eight points in a row on both sides of the center
  line, but none in-between the one-sigma warning
  limits on both sides of the center line.
• Fourteen points in a row alternating above and
  below the center line.
• Fifteen points in a row anywhere between the
  one-sigma warning limits (including either side
  of the center line).
• Any unusual or non-random pattern to the plotted
  points.

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Charts Based on Standard Values, x Chart

• If values for m and s are known (i.e., do not
  need to estimate from data)
• Quantity A is tabulated in Appendix VI

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R - Chart Based on Standard Values

• If values for R and s are known
• We define a random variable W R / s called
  the relative range
• The parameters of the distribution of W are a
  function of sample size (n)
• From the relative range we can compute the mean
  of R

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The Standard Deviation of R

• The standard deviation of R is given as (Text
  does not derive this)
• therefore
• where
• D1 and D2 are constants tabulated in Appendix VI
• Caution Be careful when using standard values
• make sure these values are representative of the
  actual process

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X-Bar R-Charts

• The X-Bar Chart checks variability in location
  between samples

• The R-Chart checks for changes in sample variation

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X-Bar Sigma-Charts

• Used when sample size is greater than 10

• Sigma-Chart Control Limits
• Approximate, asymmetric 3? limits from S table

• X-Bar Control Limits
• Approximate 3? limits are found from S table

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X-Bar Sigma-Charts

• Limits can also be generated from historical data

• Sigma-Chart Control Limits
• Approximate, asymmetric 3? limits from ?0 table

• X-Bar Control Limits
• Approximate 3? limits are found from known ?0
  table

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Operating Characteristic (OC) Curve

• Ability of the x and R charts to detect shifts
  (sensitivity) is described by OC curves
• For x chart say we know s
• Mean shifts fromm0 (in-control
  value) to m1 m0 ks (out-of-control value)
• The probability of NOT detecting the shift on the
  first sample after shift is

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OC Curve for x Chart

• Plot of b vs. shift size (in std dev units) for
  various sample sizes n
• x chart not effective for small shift sizes,
  i.e., k ? 1.5s
• Performance gets better for larger n and larger
  shifts (k)

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OC curve for R Chart

• Uses distribution of relative range r.v., i.e.,
• Suppose
• s0 - in-control std dev
• s1 - out-of-control std dev
• OC curve for R chart plots b vs. ratio of
  in-control to out-of-control standard deviation
  for various sample sizes
• That is, plot ß vs. l s1/s0
• R chart not very effective for detecting shifts
  for small sample sizes (see Fig. 5-14 in fifth
  edition of the text)

(we wont plot that here, but )
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Probability of Detecting Shift for Subsequent
Samples

• After the shift has occurred
• P(NOT detecting shift ON 1st sample)
• P(DETECTING shift ON 1st sample)
• P(DETECTING shift ON 2nd sample)
• P(DETECTING shift ON rth sample)
• P(DETECTING shift BY 2nd sample)
• P(DETECTING shift BY rth sample)

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Average Run Length (ARL)

• Expected number of samples taken before shift is
  detected is called the Average Run Length (ARL)

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Performance of Any Shewhart Control Chart

• In-Control ARL
• Average number of points plotted on control chart
  before a false alarm occurs(ideally, should be
  large)
• Out-of-Control ARL
• Average number of points, after the process goes
  out-of-control, before the control chart detects
  it(ideally, should be small)

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ARL Curve for x Chart

• Plot of ARL1 vs. shift size (in sd units) for
  various sample sizes n
• Average Time to Signal, (ATS)
• Number of time periods that occur until signal is
  generated on control chart
• h - time interval between samples

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Questions Issues