# Statistical Process Control Statistical

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```					Statistical Process
Control
Evolution of Quality Control

Acceptance            Process        Continuous
Sampling             Control       Improvement

Inspection of materials    Corrective        Quality
before and product      action during     built into
after production        production      the process

reactive                    proactive
Cereal Manufacturer Example

   Your company produces boxes of cereal
   Each box of cereal is supposed to be 16.0 oz
   How cereal boxes are filled:
 an empty box is moved forward on a conveyer until it is positioned
under the filling funnel
 a door at the base of the funnel is opened
 after a certain period of time the door is closed
 the conveyer moves the boxes forward again and the process repeats

   Will each box be exactly 16.0 oz, or will there be at least a little
variation in the weight?
   How can we tell if the box filling machine is working properly or not?
Getting Started

 Mark out this section of the notes:
“How does variation relate to quality?
 “unusual”variation
 “too much” variation
What we will cover …

   Statistical process control
   Used during processing by comparing measurements to
standards
   Used to detect problems with the process (by looking at output of
process) so process can be corrected
   Process capability
   Used before processing
   Used to determine if a particular process is able to meet
customer specifications

   Difference between them?
   Purpose & when used
   Basis (what is examined)
Getting started …
   Penny placement – round 1
   Place penny on paper and trace outline of it
   Now pick it up, hold it about 6 inches above the paper and try to
place it in exactly the same place 5 times
   Was it in exactly the same place every time?

   Penny placement – round 2
   Close your eyes and pick up the penny
   Now try to place the penny in exactly the same place 5 times in a
row (open your eyes to check your placement each time but close
them before picking it up again)
   Was it in exactly the same place every time?

   Is there a difference between the 1st and 2nd rounds?
   What accounts for that difference?
Why use samples in SPC?

Individual observations tend to be too erratic
to make trends quickly visible
Only natural variation present

   Output from process forms
distributions that are stable and
predictable over time
 Same mean and standard deviation
 Day 1 – Day 4 samples from the
same population

Day 1         Day 2   Day 3   Day 4
   Process is considered “in control”
(operating as expected)

Distribution of
sample (n=5) means
from day 1
Assignable cause of variation present

   Output from process forms
distributions that are not stable or
predictable over time
 Vary in central tendency (mean),
standard deviation (variation) and
shape
 Day 4 samples not from the same
population as Day 1
Day 1   Day 2   Day 3   Day 4
   Process considered “out of control”
(not performing as expected)
Statistical Process Control Chart
   A graphic representation of process data over time
   Time-ordered sequence of sample statistics

Upper limit
Target

Lower limit

   SPC – monitor a process by comparing sample
statistics to standards (limits) that are based on
natural variation being present
What sample statistic?

   X -chart (mean)            X
   R-chart (range)            R
   P-chart (% defective)      p
   C-chart (# defective)      c
Statistical Process Control Chart

 3σ X           UCL
 2σ X
Natural or inherent
 1σ X         variation

X
 1σ X     Natural or inherent
variation
 2σ X
- 3σ X           LCL

U,LCL = target value + “acceptable” variation

= overall mean + (# of std deviations)(std deviation)
Statistical Process Control Chart

Day 1   Day 2   Day 3   Day 4

If only natural variation present
where should most sample
values fall?

Day 1    Day 2   Day 3   Day 4

If assignable causes of variation
present where should at least
some sample values fall?
Statistical Process Control Chart

1. Establish control limits based on prior samples

2. Use control chart to monitor process by
comparing sample values to established control
limits
Let’s play “Is it a variable or an attribute?”

   Diameter of a steel bar
   # of lawn mowers per 20 that start on 1st pull
   # of complaints at a fast-food joint per week
   Length of a piece of metal
   # of bolts of fabric with too little fabric per
week
Which chart?
How is quality
assessed?
Measured                              Defects are
(variable data)                           counted
(weight, length,                       (attribute data)
Density, etc.)
X and R-charts                         Is the count out of some
known or set number?

Yes                                    No
(# of defects out of a                (# of defects each day or
sample of n units – so              # of defects in a windshield -
you can determine the                  so you do not know the
% defective)                       sample size and can’t
calculate the % defective)
P-chart                                  C-chart
Control Charts for Variables

shift in mean

shift in
variation
Variable Charts

   X-chart – detect shift in central tendency
 If process (population) standard deviation known
 If process (population) standard deviation not known
(not enough data)

   R-chart – detect shift in the variation
What is a standard deviation?
95.45% of values will lie within + 2 standard deviations
99.73% of values will lie within + 3 standard deviations

95.45% of area
under the curve

99.73% of area
under the curve

-3     -2    -1     0     +1    2     +3
In-class example

   Based on 100 samples of 5 cereal boxes each collected
over time, you have estimated the population mean as
16.0 oz and the population standard deviation as 1 oz.
   Determine 3s control limits for an X-chart

 X-chart:
 1 
U,LCL= μ  zσ X  16.0  3   
 5
 16.0  1.3
Hint:
If you have 5 samples,
In-class example                    what is your sample size?

   You have 5 samples that have been collected for the
cereal-box filling line
Sample                 Weights
Time      Box 1   Box 2       Box 3   Box 4
9 am      15.8     16.2       15.9    16.2
10 am      16.1     16.2       15.9    15.8

11 am      15.9     16.0       16.3    16.1

Noon       15.7     15.8       16.1    16.2

1 pm      15.7     16.1       15.9    16.1

   Determine control limits for all necessary charts
In-class example
Sample                      Weight
Time    Box 1   Box 2   Box 3   Box 4   Mean    Range
9 am    15.8    16.2    15.9    16.2    16.02    0.4
10 am   16.1    16.2    15.9    15.8    16.00    0.4
11 am   15.9    16.0    16.3    16.1    16.08    0.4    n4
Noon    15.7    15.8    16.1    16.2    15.95    0.5    X  16.0
1 pm    15.7    16.1    15.9    16.1    15.95    0.4
R  0.42
Factors for 3s Control Limits
Sample Size   Mean Factor   Upper Range   Lower Range
n             A2            D4            D3
2          1.880          3.268             0
3          1.023          2.574             0
4           .729          2.282             0
5           .577          2.115             0
6           .483          2.004             0
7           .419          1.924         0.076
8           .373          1.864         0.136
9           .337          1.816         0.184
10           .308          1.777         0.223
12           .266          1.716         0.284
In-class example
Sample                      Weight
Time                                                    n4
Box 1   Box 2   Box 3   Box 4   Mean    Range
9 am    15.8    16.2    15.9    16.2    16.02    0.4    X  16.0
10 am   16.1    16.2    15.9    15.8    16.00    0.4    R  0.42
11 am   15.9    16.0    16.3    16.1    16.08    0.4    A 2  0.729
Noon    15.7    15.8    16.1    16.2    15.95    0.5    D 4  2.282
1 pm    15.7    16.1    15.9    16.1    15.95    0.4
D3  0

U, LCLX  X  A 2 R  16.0  .729.42
UCL R  D 4 R  2.282(.42) 0.96
LCLR  D3 R  0(.42)  0
In-class Example – Munchkins

   What reflects quality?

   Set up control chart
   Take samples when you think process is working properly
   Calculate appropriate sample statistics
   Determine control limits for all appropriate charts
   Using the control chart
   Draw samples of pre-defined size at pre-determined time
intervals
   Plot appropriate sample statistics on control chart
   Draw conclusion – what type of variation is present?
   If “out of control”, what could be assignable cause of variation?
What can be done to fix it?
   Did corrective action remove assignable cause?
“In control” Process
   both central tendency and variation must be “in
control” for process to be “in control”

Mean = 1.0

Mean = 1.0

Variation =   0.2

Variation = 0.2
Using the Control Chart

√   Set control limits based on data collected in prior
periods
   Use control charts to monitor process
 Take  sample of size n at pre-specified time intervals
 Calculate the appropriate sample statistic for that
sample
 Plot sample values on appropriate charts
 Draw conclusion and take appropriate action
 Repeat process

What are you trying to accomplish
with SPC charts?
Which chart?
How is quality
assessed?
Measured                              Defects are
(variable data)                           counted
(weight, length,                       (attribute data)
Density, etc.)
X and R-charts                         Is the count out of some
known or set number?

Yes                                    No
(# of defects out of a                (# of defects each day or
sample of n units – so              # of defects in a windshield -
you can determine the                  so you do not know the
% defective)                       Sample size and can’t
Calculate the % defective)
P-chart                                  C-chart
Examples
   Telephone inquiries of IRS customers are monitored daily (5 days in total).
Incidents of incorrect information or other nonconformities (such as
impoliteness to customers) are recorded. The data for 5 sample days are
below. Construct the 3s control chart for nonconformities.

Day                           1       2       3    4      5

# of nonconformities          5       10      23   20     15

   Detroit Central Hospital is trying to improve its image by providing a positive
experience for its patients and their relatives. A 100-patient sample of
questionnaires that accompanied meals had the following results.
Construct 3s control limits.

Day                      1       2       3    4   5     6      7

# of unsatisfied
8       5       4    7   8     4      6
customers
Examples

   Telephone inquiries of IRS customers are monitored daily (5 days in total).
Incidents of incorrect information or other nonconformities (such as
impoliteness to customers) are recorded. The data for 5 sample days are
below. Construct the 3s control chart for nonconformities.

Day                         1      2      3      4      5

# of nonconformities        5     10     23     20     15

U , LCL  c  z c
 14.6  3 14.6
UCL  26.06
LCL  3.14
Examples
   Detroit Central Hospital is trying to improve its image by providing a positive
experience for its patients and their relatives. A 100-patient sample of
questionnaires that accompanied meals have the following results.
Construct 3s control limits.

Day                            1     2      3          4   5     6     7
# of unsatisfied customers     8     5      4          7   8     4     6
p                             .08   .05    .04     .07     .08   .04   .06

U , LCL  p  z

p 1 p   
n
.06.94 
 0.06  3
100
 .06  3(.0237 )
UCL  .13
LCL  .01  0
Control Charts for M&M’s

Two situations – what chart for each?
   monitor M&M processing based on # of defects per bag
   monitor mix of colors per sample of 55 M&M’s
Managerial Considerations

   where to use control charts
   where might process go “out of control”?
   where in the process is critical?
   type of chart
   variable or attribute data
   trade-off between them – ease vs richness of data
   when to stop process
   a single sample value outside limits
   trends or patterns
   sample size
   time and cost
   effectiveness
Sample Size
In-Class Demonstration

   No food this time
   Need 8-10 volunteers to come up front and bring
something with which to write

   Customer specifications: all rolls must be
between 4 and 10, inclusive
   Is the process capable of meeting customer
specifications essentially 100% of the time?
   What is needed to make it capable?
Number Production

   Process with only natural variation

2   3   4   5   6   7   8   9    10   11   12

   Cannot meet customer’s specifications of 4-10 – process
is not capable
Side note: this is what Deming meant by 85% of quality
problems have to do with materials and/or processes, not
employee performance
Number Production

   Change process to reduce natural variation

4   5   6   7   8    9   10

   Every roll will meet specifications – process is capable
Process Control vs Capability

   In control
 only   natural variation present

   Capable
 Process with natural variation is able to meet specs –
process variation is “small” enough such that output
meets specifications essentially 100% of the time
Process Variation

   Natural process variation – range for all units produced

process distribution

-3s                                                       +3s
6s
99.73% of area under curve falls within + 3s

   Measure as process width = + 3s (= 6s)
Specifications

   range of acceptable values established by design
engineer or customers

   Measure as specification width
   + x (= 2x)
   upper specification – lower specification

x                    x
Process Capability

   Natural variability compared to specifications– can we meet specs
given the inherent variability?

Lower spec                    2x                  Upper spec

99.74%
process distribution

-3s                                               +3s
Specification width           spec width
Cp =                            =
process width                   6s
Customer specs
Process is “not capable”

That means 4.6%
of the units produced
Only 95.4% of the area under         will not meet specs
process distribution               the curve is between specs

-3s         -2s       -1s                   +1s     +2s       +3s

C = 0.67
You make improvements to the process to
reduce the natural variation p
Process as it currently operates
(ex: manual welding)
Customer specs

Process is “capable”
That means 0.3%
Cp = 1.00

Customer specs

Process is “capable”
That means 0.27%
of the units produced
will not meet specs
Now 99.73% of the area under
process distribution       the curve is between specs

-3s   -2s   -1s          +1s    +2s    +3s

Make improvements to process
to reduce the natural variation
(ex: automated welding)
of the units produced
Now 99.7% of the area under          will not meet specs
process distribution            the curve is between specs

-3s   -2s    -1s    +1s            +2s       +3s
Cp = 2.00
You make more improvements to the process
to further reduce the natural variation

Customer specs
Process is “capable”
That means 0.0003%
and almost perfect
of the units produced
(Motorola’s goal)                     Now 99.9997% of the area under         will not meet specs
process distribution          the curve is between specs

-6s   -4s     -2s          +2s    +4s       +6s

Make further
Improvements to process
(ex: improved equipment for
automated process)
Process Control and Capability

Process Control                       Process Capability
   Based on sample values and           Based on individual observations
sampling distribution                 and process distribution
   Done on an on-going basis to         Done to evaluate a process before
monitor process                       production (e.g., new specification,
   Natural vs assignable variation       new technology available)
   Natural variation in process
compared to specs – is process
capable of meeting specs given
the natural variation of the process
Process Control and Capability
Control Chart
UCL

Sample
statistic

LCL

Each Xi                                     Design
specifications
Process Control and Capability

Process is …         Capable            Not capable

Try to reduce
In control          Good!
natural variation

Find assignable
Out of control                             Yuck!
cause of variation

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 views: 253 posted: 7/8/2011 language: English pages: 47