Statistics and Statistical Process Control

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Forging new generations of engineers
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Statistics andStatistical Process Control

• What is it and why do we use it?

It is the use of mathematics on collected data to
control manufacturing processes that aims to
maximize productivity, maximize material
utilization, and minimize defects, rejects, and
waste.
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Statistical Measures of a Sample

• MEAN ( X )
• a value that is obtained by adding the terms and
  then dividing their sum by the number of terms
• E.g..
• 4 4 5 7 10 30

30 5
Mean 6
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Statistical Measures of a Sample

• MODE
• The value of the terms that appears most
  frequently
• E.g..
• 4 4 5 7 10

Mode 4
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Statistical Measures of a Sample

• MEDIAN
• Once rank ordered (smallest to largest) it is the
  number in the middle
• e.g.
• 4 4 5 7 10

MEDIAN 5
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Statistical Measures of a Sample

• STANDARD DEVIATION
• a measure of the variation of the data from the
  mean symbolized by the Greek letter ? (sigma)
• The VARIANCE may be computed by
• subtract the mean from each term in the
  distribution to obtain a set of differences
• square each difference and add the squares
• divide the sum of the squared differences by the
  number of terms in the distribution less one

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Statistical Measures of a Sample

• Taking the square root of the quotient just
  obtained results in the positive square root,
  which is the standard deviation

?(4-6)2 (4-6)2 (5-6)2 (7-6)2
(10-6)2 5 - 1
2.55
S
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Statistical Measures of a Sample

• VARIANCE
• in any distribution(set of data from a study) the
  terms (data collected) may vary widely or they
  may be very close to each other.
• Is the mean of the squared differences from the
  mean of the distribution

• Spread - how far apart the terms are to each
  other, or how scattered the terms are

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STANDARD DEVIATION

• Data collected from a sample should statically
  fit this shape commonly referred to as the
  standard normal curve
• e.g.
• class test results
• Y-axis number of students
• X-axis grades

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STANDARD DEVIATIONNegative Skew

• If data shifts to the left of the standard normal
  curve we can conclude
• e.g.
• test results of class
• students are very dull
• questions are too hard
• teacher did not teach subject matter

Non-normally distributed data
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STANDARD DEVIATIONPositive Skew

• If data shifts to the right of the standard
  normal curve we can conclude
• e.g.
• test results of class
• students are very smart
• questions are too easy
• teacher taught subject matter very well

Non-normally Distributed Data
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STANDARD DEVIATION

• Normal Distribution
• once you have calculated the standard deviation ?
  (sigma)
• the area under the curve between -1 ? and 1 ?
  should contain 68.27 of the data or terms
• X is the mean

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STANDARD DEVIATION

• Normal Distribution
• the area under the curve between -2 ? and 2 ?
  should contain 95.45 of the data or terms

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STANDARD DEVIATION

• Normal Distribution
• the area under the curve between -3 ? and 3 ?
  should contain 99.73 of the data or terms
• This means that virtually all values of a
  population of data should fall between

X ? 3 ?
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PROCESS CONTROL

• If all your data falls inside your standard
  normal curve then your process is said to be
  under control

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PROCESS CONTROL

• If all your data falls scattered around your
  standard normal curve, your process is said to be
  not under control
• What are the causes?

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PROCESS CONTROLCAUSES OF POOR DATA

• unless all the special causes of variation are
  identified and corrected, they will continue to
  affect to effect the process output

• Non-assignable Causes
• (Special causes - random errors)
• factors that cannot be
• adequately explained by any single distribution
  of the process output

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PROCESS CONTROLCAUSES OF POOR DATA
e.g. backlash clearance tool ware operator
error fixture placement temperature control

• Assignable Causes
• (common causes)
• variations in the process caused by short-run
  piece to piece differences

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Assembly ClearanceStatistics

• Cylinder Normal Curve

C cylinder P piston K clearance
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Assembly ClearanceStatistics

• Piston Normal Curve

C cylinder P piston K clearance
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Assembly ClearanceStatistics

• Assembly Acceptable
• If you take the cylinder
• statistics and the piston statistics and look at
  them for assembly statistics all data appears to
  be in control and you get a normal curve.
  However, interchangeability is very poor

XC XP Xa
Xa
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Assembly Statistics

• Truncated Normal Distribution
• a method introduced by
• W. Edwards Deming, and
• used by the Japanese in
• the early 80s to increase interchangeability.
• Only the data between X ? 1? should be
  considered, effectively tightening machining
  capabilities

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1 Sigma Truncation

• Tight
• Cylinder Piston Clearance

? 3 1.32 ?12 ?22
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Statistical Process Control 6 Sigma Quality
Process

• The six sigma process was developed at the
  Motorola Company in the 80s.
• The dramatic improvements achieved in product
  quality since its introduction have earned
  Motorola both commercial success as well as the
  Malcolm Baldbrige Award in 1988.
• Since then, Motorola engineers have published
  many books and articles on Six Sigma.

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Statistical Process Control 6 Sigma Quality

• Capability Index

USL - LSL 6 ?
Cp
Where USL refers to the Upper Specification
Limit, LSL refers to the Lower Specification
Limit, and ? is the (population) standard
deviation of the process being studied. The
numerator is the customer functional limit
tolerance range for the design parameter in a
product or process. The denominator is a measure
of the manufacturing variability of that
parameter - a measure of how capable a machine
or process is to stay within the limits given
for that process or machine
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Statistical Process Control 3 Sigma Quality

• Three-sigma quality is achieved when Cp 1

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Statistical Process Control 6 Sigma Quality

• Six Sigma quality is achieved when Cp 2

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Statistical Process Control

• The capability index, Cp , is not an adequate
  description of quality by itself, because the
  process must be on target as well as capable.
  Thus, another capability index has been
  introduced Cp k
• Cp k Cp (1-k), k

? - T (USL - LSL) / 2
where
? is the standard deviation for the distribution
and T is the target. Typically, the target is
taken to be the center of the tolerance range
If the process is on target, then Cp k Cp and
if not, it is an indication of a
higher probability of defects
USL LSL 2
T
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References

• Creveling, Clyde M. Tolerance design, Addison
  Westley Longman, Inc. 1997 ISBN 0-201-63473-2.
• Creveling, Clyde M., Fowlkes, William Y.,
  Engineering methods for robust product design,
  Addison Westley Longman, Inc. 1995 ISBN
  0-201-63367-1.