NORMALLY DISTRIBUTED PROCESSES

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Production and OperationsManagementManufacturi
ng and Services
PowerPoint Presentation for Chapter 7
Supplement Statistical Quality Control
Chase Aquilano Jacobs

• The McGraw-Hill Companies, Inc., 1998 and (c)
  Stephen A. DeLurgio, 2000

Irwin/McGraw-Hill
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Chapter 7 Supplement - 1Statistical Quality
Control

• Process Control Procedures - 1
• Variable data
• Attribute data
• Process Capability - 2
• Acceptance Sampling - 3
• Operating Characteristic Curve

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Basic Forms of Statistical Sampling for Quality
Control

• Sampling to accept or reject the immediate lot of
  product at hand (Acceptance Sampling). Trying
  to Inspect Quality Into Product!
• Sampling to determine if the process is within
  acceptable limits (Statistical Process Control).
  Building Quality Into Product and Process!

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IMPORTANT UNDERLYING PRINCIPLE IT IS POSSIBLE TO
DESIGN A PROCESS SO THAT EVEN WHEN WE DETECT IT
AS BEING OUT OF CONTROL, NO DEFECTS ARE
PRODUCED. OUR GOAL REDUCE PROCESS VARIATION SO
MUCH THAT DEFECTS ARE NOT PRODUCED. WE DO THAT
BY CREATING CONTROL DEVICES, ELIMINATING THE
CAUSES OF LARGE, ASSIGNABLE PROCESS VARIATIONS,
AND COORDINATING PRODUCT DESIGN AND PROCESS
CAPABILITY.
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PRODUCTIVITY/QUALITY GAINS FROM SPC ARE TRULY
EXTRAORDINARY ! WE STUDY SCIENTIFIC METHODS OF
SPC TO Eliminate Causes of Defects Identify
Assignable Variations Adjust the Process
Reduce Risks of Defective Products ACHIEVE
VALUE FOR EVERYONE!
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UNDERSTANDING VARIABILITY
To understand variability, we need to understand
some basic statistics and random behavior.
These concepts apply to industrial processes,
how we perform at sports, how physical and
biological systems behave, and many other
occurrences. Well designed processes yield output
that is Normally Distributed. Your understanding
of the Normal Distribution(ND) is Essential -WHAT
IS AND WHAT CAUSES NORMALLY DISTRIBUED
VALUES? WHY IS THIS IMPORTANT?
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NORMALLY DISTRIBUTED
MEAN /- ONE STANDARD DEVIATION 68 MEAN /-
1.96 STANDAR DEVIATRIONS 95 MEAN /- 3.00
STANDARD DEVIATIONS 99.73 MEAN /- 4.00
STANDARD DEVIATIONS 99.994
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ND CHARACTERISTICS

• SYMMETRICAL - BELL SHAPED
• DISCOVERED BY K. F. GAUSS
• DEFINED COMPLETELY BY MEAN AND STANDARD DEVIATION
• GENERATED BY IN CONTROL RANDOM PROCESS
• CONTINUOUS DISTRIBUTION FROM -INFINITY TO
  INFINITY

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WHAT GENERATES ND OUTPUT? IF AN EVENT IS THE
RESULT OF A RELATIVELY LARGE NUMBER OF SMALL,
CHANCE, INDEPENDENT INFLUENCES, THEN ITS OUTPUT
WILL BE ND. MANY PROCESSES ARE ND BECAUSE WE
HAVE WORKED HARD TO ELIMINATE THE VERY LARGE
INFLUENCES, THUS ONLY A RELATIVELY LARGE NUMBER
OF SMALL, INDEPENDENT INFLUENCES REMAIN.
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FOR EXAMPLE THINK ABOUT THE PROCESS OF
PRODUCING GOLD COINS, IT IS IMPORTANT THAT EACH
WEIGHS 1.0 OZ. TO ACHIEVE A 1 OZ. WEIGHT WE
CONTROL THE SIZE OF GOLD STRIPS GOING INTO THE
PRESS. THE ADJUSTMENTS ON THE MACHINE. THE
TEMPERATURE OF THE MACHINE. THE HUMIDITY OF THE
ROOM. THE CLEANLINESS OF THE SET UP. THE
CONDITION OF THE TOOLS (DIES) USED. ALL OTHER
FACTORS THAT INFLUENCE WEIGHT.
1 OZ.
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COINING OUTPUT FOR n 600 NOTE SYMMETRY AND BELL
SHAPE
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HISTOGRAM OF COINING OUTPUT, n600 NOTE SYMMETRY
AND BELL SHAPE
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IN CONTROL PROCESS VARIATION
BY ELIMINATING ALL OF THE LARGE INFLUENCES WE ARE
LEFT WITH MANY SMALL INFLUENCES ACTING
SEPARETLY. THIS YIELDS A PROCESS WITH MEAN 1
OZ. STD. DEV. .001 OZS. AND IMPORTANTLY, THE
OUTPUT IS NORMALLY DISTRIBUTED CONSIDER THE
INTERVALS
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MEAN 1 OZ., STD DEV.001
1 /- .001 68 6,800 OF 10,000 IN THIS
RANGE 1 /- .00196 95 9,500 OF 10,000 IN
THIS RANGE 1 /- .003 99.73 9,973 OF
10,000 IN THIS RANGE 1 /- .004 99.994
9,999.4 OF 10,000 IN THIS RANGE
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DESCRIPTIVE STATISTICS

• MEAN CENTER OF DEVIATIONS
• POPULATION MEAN, ? ? X / N
• MEDIAN VALUE HAVING 50
  ABOVE, 50 BELOW
• MODE MOST FREQUENT VALUE
• FOR SYMMETRICAL DISTRIBUTION
• MEAN MEDIAN MODE

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COINING OUTPUT FOR n 600 NOTE SYMMETRY AND BELL
SHAPE
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STANDARD DEVIATION

• MEASURES VARIATION OR SCATTER
• SQUARE ROOT OF THE MEAN SQUARED ERROR
• ?????? ?x ? ?(X - ?)2 /N
  Population std.
  deviation of X with census.
• ???????? Sx ? ?(X -?X)2/(n-1)
  Sample standard deviation of X.
• Formulas may not yield much information, not as
  meaningful unless for known distribution.

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MEAN 1 OZ., STD DEV.001
1 /- .001 68 6,800 OF 10,000 IN THIS
RANGE 1 /- .00196 95 9,500 OF 10,000 IN
THIS RANGE 1 /- .003 99.73 9,973 OF
10,000 IN THIS RANGE 1 /- .004 99.994
9,999.4 OF 10,000 IN THIS RANGE
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OTHER ND INTERVALS
MEAN /- ONE STANDARD DEVIATION 68 MEAN
/- 1.96 STANDAR DEVIATRIONS 95 MEAN /-
3.00 STANDARD DEVIATIONS 99.73 MEAN /-
4.00 STANDARD DEVIATIONS 99.994 MEAN /-
5.00 STANDARD DEVIATIONS 99.99994 MEAN /-
6.00 STANDARD DEVIATIONS 99.99999
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THE CENTRAL LIMIT THEOREM
NOTE THAT SAMPLE MEANS ARE ND!
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THE CENTRAL LIMIT THEOREM
DISTRIBUTION OF SAMPLE MEANS IS ND FOR LARGE
SAMPLES FROM ANY GENERAL POPULATION!
MEAN OF MEANS ARE ND
__ __ X ? ? Z ?/? n
MEAN OF MEAN POP MEAN STD. DEV. OF MEANS POP
STD.DEV /n.5 __
___
X 1.0 ? Z .001/? 100
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Control Limits
Lets establish control limits at /- 3 standard
deviations, then We expect 99.7 of observations
to fall within these limits
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CONTROL CHARTS BASED ON ND
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TIME TO THE CONTROL CHART ADDS POWERFUL
INFERENCES!
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ALL POINTS IN CONTROL
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A IS OUT OF CONTROL, TWO PTS. IN B ARE OUT OF
CONTROL, TREND OF 7 OUT OF CONTROL
A B
7TREND
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X-BAR CHART FORMULAS When using known mean ? and
standard deviation ? _ __
X ? ? Z ?/? n When ? and ? are unknown,
they are estimated _
_ __ X X ? Z S/? n When using
measured Ranges _
_ X X ? A2 R
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THE RELATIONSHIP BETWEEN COOKBOOK FORMULAS AND
THEORY ?A2?R 3 ????
? n
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S-Charts and R-Charts The S-chart uses the
following formula S ? ? Z ?/?2n The R-Chart
uses the following formulas D4?R (UCL) R
D3?R (LCL) The results of
both will be the same in use, however, numerical
values using S and R will be different, the plots
will look nearly identical.
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A LITTLE MORE THEORY When small samples (nlt30)
are used, the assumption is that the sample comes
from a ND. When this is not true, then the above
formulas MAY NOT BE valid. If the process is
NOT ND, then large samples are necessary, or
other statistical tests called Nonparametric
methods must be used.
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Example x-Bar and R Charts
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Calculate sample means, sample ranges, mean of
means, and mean of ranges.
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Control Limit Formulas
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x-Bar Chart
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R-Chart
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Statistical Sampling--Data

• Attribute (Go no-go information)
• Defectives--refers to the acceptability of
  product across a range of characteristics.
• Defects--refers to the number of defects per
  unit--may be higher than the number of
  defectives.
• Variable (Continuous)
• Usually measured by the mean and the standard
  deviation.

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DISTRIBUTION OF SAMPLE PROPORTIONS
POP IS NOT ND ? .98
SAMPLE LOOKS LIKE POP, P .99
DIST. OF SAMPLE PS ARE ND
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P-CHARTS Require large samples n?30. When
population proportion is known
P ? ? Z? ?(1 - ?)/n When
population proportion is unknown _
_ _ P P ? Z? P(1 -
P)/n Where P-Bar is an estimate of ?
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Constructing a p-Chart
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Statistical Process Control--Attribute
Measurements (P-Charts)
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1. Calculate the sample proportion, p, for each
sample.
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2. Calculate the average of the sample
proportions.
3. Calculate the standard deviation of the
sample proportion
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4. Calculate the control limits.
UCL 0.093 LCL -0.0197 (or 0)
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• The McGraw-Hill Companies, Inc., 1998

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p-Chart (Continued)
5. Plot the individual sample proportions, the
average of the proportions, and the control
limits
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1. Calculate the sample proportion, p, for each
sample.
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