Control Charts for Variables

1
Control Charts for Variables

• EBB 341 Quality Control

2
Variation

• There is no two natural items in any category are
  the same.
• Variation may be quite large or very small.
• If variation very small, it may appear that items
  are identical, but precision instruments will
  show differences.

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3 Categories of variation

• Within-piece variation
• One portion of surface is rougher than another
  portion.
• Apiece-to-piece variation
• Variation among pieces produced at the same time.
• Time-to-time variation
• Service given early would be different from that
  given later in the day.

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Source of variation

• Equipment
• Tool wear, machine vibration,
• Material
• Raw material quality
• Environment
• Temperature, pressure, humadity
• Operator
• Operator performs- physical emotional

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Control Chart Viewpoint

• Variation due to
• Common or chance causes
• Assignable causes
• Control chart may be used to discover assignable
  causes

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Some Terms

• Run chart - without any upper/lower limits
• Specification/tolerance limits - not statistical
• Control limits - statistical

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Control chart functions

• Control charts are powerful aids to understanding
  the performance of a process over time.

Output
PROCESS
Input
Whats causing variability?
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Control charts identify variation

• Chance causes - common cause
• inherent to the process or random and not
  controllable
• if only common cause present, the process is
  considered stable or in control
• Assignable causes - special cause
• variation due to outside influences
• if present, the process is out of control

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Control charts help us learn more about processes

• Separate common and special causes of variation
• Determine whether a process is in a state of
  statistical control or out-of-control
• Estimate the process parameters (mean, variation)
  and assess the performance of a process or its
  capability

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Control charts to monitor processes

• To monitor output, we use a control chart
• we check things like the mean, range, standard
  deviation
• To monitor a process, we typically use two
  control charts
• mean (or some other central tendency measure)
• variation (typically using range or standard
  deviation)

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Types of Data

• Variable data
• Product characteristic that can be measured
• Length, size, weight, height, time, velocity
• Attribute data
• Product characteristic evaluated with a discrete
  choice
• Good/bad, yes/no

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Control chart for variables

• Variables are the measurable characteristics of a
  product or service.
• Measurement data is taken and arrayed on charts.

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Control charts for variables

• X-bar chart
• In this chart the sample means are plotted in
  order to control the mean value of a variable
  (e.g., size of piston rings, strength of
  materials, etc.).
• R chart
• In this chart, the sample ranges are plotted in
  order to control the variability of a variable.
• S chart
• In this chart, the sample standard deviations are
  plotted in order to control the variability of a
  variable.
• S2 chart
• In this chart, the sample variances are plotted
  in order to control the variability of a
  variable.

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X-bar and R charts

• The X- bar chart is developed from the average of
  each subgroup data.
• used to detect changes in the mean between
  subgroups.
• The R- chart is developed from the ranges of each
  subgroup data
• used to detect changes in variation within
  subgroups

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Control chart components

• Centerline
• shows where the process average is centered or
  the central tendency of the data
• Upper control limit (UCL) and Lower control limit
  (LCL)
• describes the process spread

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The Control Chart Method
X bar Control Chart UCL XDmean A2 x Rmean
LCL XDmean - A2 x Rmean CL
XDmean 

• R Control Chart
• UCL D4 x Rmean
• LCL D3 x Rmean
• CL Rmean 
• Capability Study
• PCR (USL - LSL)/(6s) where s Rmean /d2

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Control Chart Examples
UCL
Nominal
Variations
LCL
Sample number
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How to develop a control chart?
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Define the problem

• Use other quality tools to help determine the
  general problem thats occurring and the process
  thats suspected of causing it.
• Select a quality characteristic to be measured
• Identify a characteristic to study - for example,
  part length or any other variable affecting
  performance.

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Choose a subgroup size to be sampled

• Choose homogeneous subgroups
• Homogeneous subgroups are produced under the same
  conditions, by the same machine, the same
  operator, the same mold, at approximately the
  same time.
• Try to maximize chance to detect differences
  between subgroups, while minimizing chance for
  difference with a group.

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Collect the data

• Generally, collect 20-25 subgroups (100 total
  samples) before calculating the control limits.
• Each time a subgroup of sample size n is taken,
  an average is calculated for the subgroup and
  plotted on the control chart.

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Determine trial centerline

• The centerline should be the population mean, ?
• Since it is unknown, we use X Double bar, or the
  grand average of the subgroup averages.

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Determine trial control limits - Xbar chart

• The normal curve displays the distribution of the
  sample averages.
• A control chart is a time-dependent pictorial
  representation of a normal curve.
• Processes that are considered under control will
  have 99.73 of their graphed averages fall within
  6?.

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UCL LCL calculation
25
Determining an alternative value for the standard
deviation
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Determine trial control limits - R chart

• The range chart shows the spread or dispersion of
  the individual samples within the subgroup.
• If the product shows a wide spread, then the
  individuals within the subgroup are not similar
  to each other.
• Equal averages can be deceiving.
• Calculated similar to x-bar charts
• Use D3 and D4 (appendix 2)

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Example Control Charts for Variable Data

• Slip Ring Diameter (cm)
• Sample 1 2 3 4 5 X R
• 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
• 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
• 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
• 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
• 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
• 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
• 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
• 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
• 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
• 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
• 50.09 1.15

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Calculation

• From Table above
• Sigma X-bar 50.09
• Sigma R 1.15
• m 10
• Thus
• X-Double bar 50.09/10 5.009 cm
• R-bar 1.15/10 0.115 cm

Note The control limits are only preliminary
with 10 samples. It is desirable to have at least
25 samples.
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Trial control limit

• UCLx-bar X-D bar A2 R-bar 5.009
  (0.577)(0.115) 5.075 cm
• LCLx-bar X-D bar - A2 R-bar 5.009 -
  (0.577)(0.115) 4.943 cm
• UCLR D4R-bar (2.114)(0.115) 0.243 cm
• LCLR D3R-bar (0)(0.115) 0 cm
• For A2, D3, D4 see Table B, Appendix

n 5
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3-Sigma Control Chart Factors
Sample size X-chart
R-chart n A2 D3 D4 2 1.88 0 3.27 3 1.02
0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48
0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86
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X-bar Chart
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R Chart
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Run Chart
34

• Another Example of X-bar R chart

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Given Data (Table 5.2)
Subgroup X1 X2 X3 X4 X-bar UCL-X-bar X-Dbar LCL-X-bar R UCL-R R-bar LCL-R
1 6.35 6.4 6.32 6.37 6.36 6.47 6.41 6.35 0.08 0.20 0.0876 0
2 6.46 6.37 6.36 6.41 6.4 6.47 6.41 6.35 0.1 0.20 0.0876 0
3 6.34 6.4 6.34 6.36 6.36 6.47 6.41 6.35 0.06 0.20 0.0876 0
4 6.69 6.64 6.68 6.59 6.65 6.47 6.41 6.35 0.1 0.20 0.0876 0
5 6.38 6.34 6.44 6.4 6.39 6.47 6.41 6.35 0.1 0.20 0.0876 0
6 6.42 6.41 6.43 6.34 6.4 6.47 6.41 6.35 0.09 0.20 0.0876 0
7 6.44 6.41 6.41 6.46 6.43 6.47 6.41 6.35 0.05 0.20 0.0876 0
8 6.33 6.41 6.38 6.36 6.37 6.47 6.41 6.35 0.08 0.20 0.0876 0
9 6.48 6.44 6.47 6.45 6.46 6.47 6.41 6.35 0.04 0.20 0.0876 0
10 6.47 6.43 6.36 6.42 6.42 6.47 6.41 6.35 0.11 0.20 0.0876 0
11 6.38 6.41 6.39 6.38 6.39 6.47 6.41 6.35 0.03 0.20 0.0876 0
12 6.37 6.37 6.41 6.37 6.38 6.47 6.41 6.35 0.04 0.20 0.0876 0
13 6.4 6.38 6.47 6.35 6.4 6.47 6.41 6.35 0.12 0.20 0.0876 0
14 6.38 6.39 6.45 6.42 6.41 6.47 6.41 6.35 0.07 0.20 0.0876 0
15 6.5 6.42 6.43 6.45 6.45 6.47 6.41 6.35 0.08 0.20 0.0876 0
16 6.33 6.35 6.29 6.39 6.34 6.47 6.41 6.35 0.1 0.20 0.0876 0
17 6.41 6.4 6.29 6.34 6.36 6.47 6.41 6.35 0.12 0.20 0.0876 0
18 6.38 6.44 6.28 6.58 6.42 6.47 6.41 6.35 0.3 0.20 0.0876 0
19 6.35 6.41 6.37 6.38 6.38 6.47 6.41 6.35 0.06 0.20 0.0876 0
20 6.56 6.55 6.45 6.48 6.51 6.47 6.41 6.35 0.11 0.20 0.0876 0
21 6.38 6.4 6.45 6.37 6.4 6.47 6.41 6.35 0.08 0.20 0.0876 0
22 6.39 6.42 6.35 6.4 6.39 6.47 6.41 6.35 0.07 0.20 0.0876 0
23 6.42 6.39 6.39 6.36 6.39 6.47 6.41 6.35 0.06 0.20 0.0876 0
24 6.43 6.36 6.35 6.38 6.38 6.47 6.41 6.35 0.08 0.20 0.0876 0
25 6.39 6.38 6.43 6.44 6.41 6.47 6.41 6.35 0.06 0.20 0.0876 0
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Calculation

• From Table 5.2
• Sigma X-bar 160.25
• Sigma R 2.19
• m 25
• Thus
• X-double bar 160.25/29 6.41 mm
• R-bar 2.19/25 0.0876 mm

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Trial control limit

• UCLx-bar X-double bar A2R-bar 6.41
  (0.729)(0.0876) 6.47 mm
• LCLx-bar X-double bar - A2R-bar 6.41
  (0.729)(0.0876) 6.35 mm
• UCLR D4R-bar (2.282)(0.0876) 0.20 mm
• LCLR D3R-bar (0)(0.0876) 0 mm
• For A2, D3, D4 see Table B Appendix, n 4.

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X-bar Chart
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R Chart
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Revised CL Control Limits

• Calculation based on discarding subgroup 4 20
  (X-bar chart) and subgroup 18 for R chart
• (160.25 - 6.65 -
  6.51)/(25-2)
• 6.40 mm
• (2.19 - 0.30)/25 - 1
• 0.079 0.08 mm

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New Control Limits

• New value
• Using standard value, CL 3? control limit
  obtained using formula

42

• From Table B
• A 1.500 for a subgroup size of 4,
• d2 2.059, D1 0, and D2 4.698
• Calculation results

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Trial Control Limits Revised Control Limit
Revised control limits
UCL 6.46
CL 6.40
LCL 6.34
UCL 0.18
CL 0.08
LCL 0
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Revise the charts

• In certain cases, control limits are revised
  because
• out-of-control points were included in the
  calculation of the control limits.
• the process is in-control but the within subgroup
  variation significantly improves.

45
Revising the charts

• Interpret the original charts
• Isolate the causes
• Take corrective action
• Revise the chart
• Only remove points for which you can determine an
  assignable cause

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Process in Control

• When a process is in control, there occurs a
  natural pattern of variation.
• Natural pattern has
• About 34 of the plotted point in an imaginary
  band between 1s on both side CL.
• About 13.5 in an imaginary band between 1s and
  2s on both side CL.
• About 2.5 of the plotted point in an imaginary
  band between 2s and 3s on both side CL.

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The Normal Distribution
? Standard deviation
48

• 34.13 of data lie between ? and 1? above the
  mean (?).
• 34.13 between ? and 1? below the mean.
• Approximately two-thirds (68.28 ) within 1? of
  the mean.
• 13.59 of the data lie between one and two
  standard deviations
• Finally, almost all of the data (99.74) are
  within 3? of the mean.

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Normal Distribution Review

• Define the 3-sigma limits for sample means as
  follows
• What is the probability that the sample means
  will lie outside 3-sigma limits?
• Note that the 3-sigma limits for sample means are
  different from natural tolerances which are at

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Common Causes
51
Process Out of Control

• The term out of control is a change in the
  process due to an assignable cause.
• When a point (subgroup value) falls outside its
  control limits, the process is out of control.

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Assignable Causes
(a) Mean
Average
Grams
53
Assignable Causes
Average
(b) Spread
Grams
54
Assignable Causes
Average
(c) Shape
Grams
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Control Charts
Assignable causes likely
UCL
Nominal
LCL
1 2
3 Samples
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Control Chart Examples
UCL
Nominal
Variations
LCL
Sample number
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Control Limits and Errors
Type I error Probability of searching for a
cause when none exists
(a) Three-sigma limits
UCL
Process average
LCL
58
Control Limits and Errors
Type I error Probability of searching for a
cause when none exists
(b) Two-sigma limits
UCL
Process average
LCL
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Control Limits and Errors
Type II error Probability of concluding that
nothing has changed
(a) Three-sigma limits
UCL
Shift in process average
Process average
LCL
60
Control Limits and Errors
Type II error Probability of concluding that
nothing has changed
(b) Two-sigma limits
UCL
Shift in process average
Process average
LCL
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Achieve the purpose

• Our goal is to decrease the variation inherent in
  a process over time.
• As we improve the process, the spread of the data
  will continue to decrease.
• Quality improves!!

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Improvement
63
Examine the process

• A process is considered to be stable and in a
  state of control, or under control, when the
  performance of the process falls within the
  statistically calculated control limits and
  exhibits only chance, or common causes.

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Consequences of misinterpreting the process

• Blaming people for problems that they cannot
  control
• Spending time and money looking for problems that
  do not exist
• Spending time and money on unnecessary process
  adjustments
• Taking action where no action is warranted
• Asking for worker-related improvements when
  process improvements are needed first

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Process variation

• When a system is subject to only chance causes of
  variation, 99.74 of the measurements will fall
  within 6 standard deviations
• If 1000 subgroups are measured, 997 will fall
  within the six sigma limits.

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Chart zones

• Based on our knowledge of the normal curve, a
  control chart exhibits a state of control when
• Two thirds of all points are near the center
  value.
• The points appear to float back and forth across
  the centerline.
• The points are balanced on both sides of the
  centerline.
• No points beyond the control limits.
• No patterns or trends.