Introduction to Statistical Quality Control, 5th edition

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Learning Objectives
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P Control Chart

• The most versatile and widely used attributes
  control chart
• -- used when the subgroup size is not constant
• Used to evaluate fraction defective
• Control limits are based on Binomial Distribution

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P Control Chart
P CONTROL CHART ALCAS PEFLOW SOLDER MACHINE
90 80 70 60 50 40 30 20 10 0
PERCENT DEFECTIVE
UCL PERCENT
2 4 6 8
10 12 14 16 18 20
22 24 26
SUBGROUP NUMBER
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P Chart

• Compute the control limits
• The UCL and LCL are not straight lines, they rise
  and fall with respect to the subgroup size

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The P Chart

• When to use
• Only when variable data cannot be obtained.
• When charting fraction rejected as nonconforming
  from a varying sample size.
• When screening multiple characteristics for
  potential monitoring on variable control charts.
• When tracking the quality level of a process
  before any rework is performed.

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The P Chart

• How
• By counting the number of defective items from a
  sample and then plotting the percent that are
  defective.
• Conditions
• In order to be of help, there should be some
  rejects in each observed sample.
• The higher the quality level, the larger the
  sample size must be to contain rejects. For
  example, if 20 of a product is rejectable, a
  sample size of 5 will be needed. However, a
  sample of 1,000 will give an average of only one
  reject per sample if 0.1 of the product is
  rejectable.

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The P Chart
Table 1 Formulas for the P Chart
Chart Control limits Centerline Plot point Sample size
p Varying
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The P Chart

• To calculate Plot Points
• The p plot point is the fraction defective in a
  sample. The centerline is the average fraction
  defective in series of samples. Figure 1 is a
  cross section showing countersunk holes for rivet
  installation.

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The P Chart

• To calculate Plot Points
• In a sheet metal assembly shop a common process
  is bucking rivets. Because of the combined
  variation in the rivets, the drilled holes, and
  the bucking process, there are quality problems.
  After the rivets in an assembly have been bucked
  into place they are checked for nonconformity. A
  P chart is used to track the first-time-through
  fraction defective.

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Figure 1

• Cross section of sheet metal plates with
  countersunk holes for rivet installation

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P Chart
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P Chart
TOTALS
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X Relay
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Fluorocarbon Leak Test
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NP Control Chart

• Number of defectives
• Used when subgroup size is constant (n)
• The actual number of defects is represented by pn
  (or np)

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NP Control Chart

• NP Control Chart plating defects of assembled
  parts

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
UCL
2 4 6 8 10 12 14 16 18 20
22 24 26 28 30
LOT NUMBER
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NP Control Charts

• Computations
• Control limits are based on the Binomial
  Distribution
• 1) Central line
• 2)
• 3)

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NP Charts
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4 Steps in control chat for Fraction Rejected

• I Preparatory Decisions
• II Starting the control chart
• III Continuing the control chart
• IV Reports and Action based on control chart

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I. Preparatory Decisions

• Purpose
• Select Quality Characteristic
• Selection of subgroup
• P or NP chart
• Control limit calculated

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II. Starting the control chart

• Recording data
• , Calculation
• Trial limits
• Plotting points

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III Continuing the control chart

• Selection of P0
• Calculation of control limits
• Plotting the points/limits
• Interpretation of lack of control
• Periodic Review/Revision of Pi

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IV Reports and Action based on control chart

• Action to bring process into control-Pareto, high
  spot/low spot.
• Review of Design and specification
• Information to Management(Quality level)
• Sensitivity of p chart 0.1 requires 1000s

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Example of Attribute control chart
LOT Number inspecting Number of defectives P
1 500 27 0.054
2 50 12 0.240
3 800 12 0.015
4 100 14 0.140
5 150 15 0.100
Total 1600 80
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Example of Attribute Control Chart
Subgroup Subgroup Size n of Defectives Percent Defective P () UCL() LCL
1 115 15 13.0 17.7 1.
2 220 18 8.2 15.4 3.
3 210 23 10.9 15.6 3.
4 220 22 10.0 15.4 3.
5 255 18 7.0 15.0 4.
6 365 15 4.1 4.
7 255 44 15.0 4.
8 300 13 4.3 4.
9 280 33 11.7 14.8
10 330 42 12.7 14..3
Total 2550 243
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• Purpose of maintaining attribute charts is to
  continuously improve the processes for defect
  free production by highlighting the key problems.
• How to work on getting desirable pattern
• Recommend working on 3 top problems for the day
  or week of month as the time permits and solve by
  a systematic problem solving method namely
• Define problems
• Find key causes
• Solution to cure the key causes demonstrated by
  statistics

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Design of the Fraction Nonconforming Chart

• Three parameters must be specified
• The sample size
• The frequency of sampling
• The width of the control limits
• Common to base chart on 100 inspection of all
  process output over time
• Rational subgroups may also play role in
  determining sampling frequency

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Average sample size approach
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Charts for Nonconformities
C cart and U chart

• Often times there is interest in knowing how many
  defects an item has
• -- C charts total number of nonconformities
  in subgroups of fixed size (defects per square
  yard of cloth)
• -- U charts average number of nonconformities
  per unit (defects per TV set)

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Charts for Nonconformities

• Used during inspection of complex assemblies
• Control limits based on Poisson Distribution

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C Control Chart

• UCL and LCL are straight lines because of fixed
  sample sizes
• CL
• UCL
• LCL
• Where is the central line and equal to

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The data in table 2 represent the types of
defects found on the first two boxes. Box 1 has
nine defects and box 2 has twelve. Notice that
the types and quantity of defects are different
for the two boxes nevertheless, the total number
of defects is plotted on the C chart.
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Note A C chart (as well as any other attribute
chart) should only be used when there is
absolutely no way to obtain variable data from
the characteristic in which measurable data is
available. To get started, this data can be
analyzed on an attribute chart to get ideas
(using the Pareto analysis) on the
characteristics.
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U Control Chart

• As the subgroup size varies, UCL and LCL varies
• CL
• UCL
• LCL
• where is the central line and equal to

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The U Chart

• When to use
• Only when variable data cannot be obtained.
• When plotting the average number of defects found
  per unit.
• When screening multiple characteristics for
  potential monitoring on variable control charts.
• How
• Each unit is examined and the average number of
  defects found are plotted.

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The U Chart

• Conditions
• Constant unit size, but any convenient number of
  units per plot point. Unit size is different from
  samples size. For example, one unit could be
  defined as 1 square-foot of material.
• On a particular day, 12.3 square-feet of material
  is inspected. The plot point would represent the
  average number of defects per unit, but the
• sample size would be 12.3.

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The U Chart

• There are potentially several different types of
  defects per unit, but none of which would
  necessarily render the part a defective.
• For example paint blemishes on a skin panel, or
  various electrical faults on a circuit board. A
  unit can be single part, an assembly of several
  parts, an area of material, or any rational
  grouping in which the likelihood of defect(s) is
  constant from
• unit to unit.

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• In order for this type of analysis to be of help,
  there should be some defects in each observed
  unit.

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The U Chart

• To Calculate Plot Points
• The u plot point is the average number of defects
  per unit in a sample of n units. The centerline
  is the average of all the plot points on the
  chart. Figure 6-17 shows a roll of composite
  material/dyed cloth with potential multiple
• defects.

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The U Chart

• Composite materials are generally made up of
  several layers of materials that are bonded to
  form the desired characteristics.
• Prior to lay up, are used each day, the number of
  rolls inspected for defects. Since different
  amounts of the material are used each day, the
  number of rolls inspected also changes
• daily.

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Attribute Control Charts
Chart type Names/remarks Central line Control Limits
Fraction defective
Number defectives
Number of defects per subgroup
Number of defects per inspection unit
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Attribute data
Variable data
Ngt50 N not constant Fraction defective
X-bar chart R chart
P chart
nlt6
Ngt50 N is constant Fraction defective
X-bar chart S chart
ngt6
np chart
ngt25
n?1 n not constant Defects per unit
X-bar chart S2 chart
u chart
n1
n?1 Is constant Defects per unit
Individual chart
c chart
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Low Defect Levels
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Actions taken to improve a process
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Learning Objectives