ENGM 720 - Lecture 03

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ENGM 720 - Lecture 03

• Describing Using Distributions, SPC Process

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Assignment

• Reading
• Chapter 2
• Finish reading
• Chapter 3
• Start reading
• Assignment 2
• Obtain access to MS Excel
• Verify access to the Data Analysis Add-In
• Access the class website
• Download of Normal Plot data spreadsheet
• Download Assignment 2 Instructions (Materials
  page)

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What is Quality

• Many definitions
• Better performance
• Better service
• Better value
• Whatever the customer says it is
• For SPC, quality means better
• Understanding of process variation,
• Control of the variation in the process, and
• Improvement in the process variation.

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Understanding Process Variation

• Three Aspects
• Location
• Spread
• Shape
• Basic Statistics
• Quantify
• Communicate

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Location Mode

• The mode is the value (or values) that occurs
  most frequently in a distribution.
• To find the mode
• Sort the values into order (with no repeats),
• Tally up how many times each value appears in the
  original distribution.
• The mode (or modes) has the largest tally
• Dist. 1 has two modes 20 and 15 (four times,
  ea.)
• Dist. 2 has one mode 15 (appearing seven times)

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Location Median

• Half of the values will fall above and half of
  the values will fall below the median value.
• To estimate the median
• Sort the values (keeping the duplicates in the
  list), and then count from one end until you get
  to one half (rounding down) of the total number
  of values.
• For an odd number of values, the median is the
  next value.
• For an even number of values, the median value is
  half of the sum of the current value and the next
  sorted value.
• Dist. 1 median is 19.5
• Dist. 2 median is 15

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Location Mean

• The mean has a special notation x for a sample
  (? for the entire population)
• To calculate the mean
• add up all of the values
• divide the sum by the number of values
• Dist. 1 mean is 18.6,
• Dist. 2 mean is 15.0 Mean is influenced by
  outliers

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Spread Range

• Range is the difference between the maximum and
  the minimum values, denoted R.
• This value gives us the extreme limits of the
  distribution spread.
• Much easier to calculate than other measures
• Very sensitive to outliers
• Range of Dist. 1 is 11
• Range of Dist. 2 is 4

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Spread Variance

• Variance has the symbol ?2 when referring to the
  entire population (s2 for a sample variance)
• The formula for the variance is
• Measures the dispersion with less emphasis on
  outliers
• Units for variance arent very intuitive
• Manual calculation is unpleasant
• (calculating equation could be used)
• The variance for Dist. 1 is 10.58, for Dist. 2 it
  is 1.63

If population is known, use n in denominator!
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Spread Standard Deviation

• The standard deviation (? for the population, or
  s for a sample) is the square root of the
  variance.
• Defn. Special calculating formula
• Not as easily influenced by outliers
• Has the same units as measure of location.
• Std deviation for Dist. 1 is 3.25
• Std deviation for Dist. 2 is 1.28

If population is known, use n in denominator!
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Shape Prob. Density Functions

• The shape of a distribution is a function that
  maps each potential x-value to the likelihood
  that it would appear if we sampled at random from
  the distribution. This is the probability density
  function (PDF).

? ? 1? 68.26 of the total area ? ? 2? 95.46
of the total area ? ? 3? 99.73 of the total area
Area Under the Normal Curve
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Shape Stem-and-Leaf Plot
48 53 49 52 51 52 63 60 53 64
59 54 47 49 45 64 79 65 62 60

• Divide each number into
• Stem one or more of the leading digits
• Leaf remaining digits (may be ordered)
• Choose between 4 and 20 stems

• Example
• 4
• 5
• 5
• 6
• 6
• 7
• 7

8
9
7
9
5
3
2
1
2
3
4
9
3
0
4
4
2
0
5
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Done!
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Shape Box (and Whisker) Plot
Max value
Third quartile
Mean
Median
First quartile

• Visual display of
• central tendency, variability, symmetry, outliers

Min value
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Shape Histogram

• A histogram is a vertical bar chart that takes
  the shape of the distribution of the data. The
  process for creating a histogram depends on the
  purpose for making the histogram.
• One purpose of a histogram is to see the shape of
  a distribution. To do this, we would like to
  have as much data as possible, and use a fine
  resolution.
• A second purpose of a histogram is to observe the
  frequency with which a class of problems occurs.
  The resolution is controlled by the number of
  problem classes.

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Histogram Example (Excel)
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Goals of Statistical Quality Improvement

• Find special causes
• Head off shifts in process
• Obtain predictable output
• Continually improve the process

Statistical Quality Control and Improvement
Improving Process Capability and Performance
Continually Improve the System
Characterize Stable Process Capability
Head Off Shifts in Location, Spread
Time
Identify Special Causes - Bad (Remove)
Identify Special Causes - Good (Incorporate)
Reduce Variability
Center the Process
LSL ?0 USL
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Distributions

• Distributions quantify the probability of an
  event
• Events near the mean are most likely to occur,
  events further away are less likely to be
  observed

35.0 ? 2.5
43.6 (?3?)
30.4 (?-3?)
39.2 (??)
34.8 (?-?)
37 (?)
41.4 (?2?)
32.6 (?-2?)
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Normal Distribution

• Notation r.v.
• This is read x is normally distributed with
  mean m and standard deviation s.
• Standard Normal Distribution r.v.
• (z represents a Standard Normal r.v.)

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Simple Interpretation of Standard Deviation of
Normal Distribution
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Standard Normal Distribution

• The Standard Normal Distribution has a mean (?)
  of 0 and a standard deviation (?) of 1
• Total area under the curve, ?(z), from z ? to
  z ? is exactly 1
• The curve is symmetric about the mean
• Half of the total area lays on either side, so
• ?( z) 1 ?(z)

?(z)
z
?
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Standard Normal Distribution

• How likely is it that we would observe a data
  point more than 2.57 standard deviations beyond
  the mean?
• Area under the curve from ? to z 2.5 ? is
  found by using the table on pp. 716-717, looking
  up the cumulative area for z 2.57, and then
  subtracting the cumulative area from 1.

?(z)
z
?
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Standard Normal Distribution

• How likely is it that we would observe a data
  point more than 2.57 standard deviations beyond
  the mean?
• Area under the curve from ? to z 2.5 ? is
  found by using the table on pp. 716-717, looking
  up the cumulative area for z 2.57, and then
  subtracting the cumulative area from 1.
• Answer 1 .99492 .00508, or about 5 times in
  1000

?(z)
z
?
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What if the distribution isnt a Standard Normal
Distribution?

• If it is from any Normal Distribution, we can
  express the difference from an observation to the
  mean in units of the standard deviation, and this
  converts it to a Standard Normal Distribution.
• Conversion formula is
• where
• x is the point in the interval,
• ? is the population mean, and
• ? is the population standard deviation.

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What if the distribution isnt even a Normal
Distribution?

• The Central Limit Theorem allows us to take the
  sum of several means, regardless of their
  distribution, and approximate this sum using the
  Normal Distribution if the number of observations
  is large enough.
• Most assemblies are the result of adding together
  components, so if we take the sum of the means
  for each component as an estimate for the entire
  assembly, we meet the CLT criteria.
• If we take the mean of a sample from a
  distribution, we meet the CLT criteria (think of
  how the mean is computed).

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Example Process Yield

• Specifications are often set irrespective of
  process distribution, but if we understand our
  process we can estimate yield / defects.
• Assume a specification calls for a value of 35.0
  ? 2.5.
• Assume the process has a distribution that is
  Normally distributed, with a mean of 37.0 and a
  standard deviation of 2.20.
• Estimate the proportion of the process output
  that will meet specifications.

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Continuous Discrete Distributions

• Continuous
• Probability of a range of outcomes is the area
  under the PDF (integration)

• Discrete
• Probability of a range of outcomes is the area
  under the PDF
• (sum discrete outcomes)

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Discrete Distribution Example

• Sum of two six-sided dice
• Outcomes range from 2 to 12.
• Count the possible ways to obtain each individual
  sum - forms a histogram
• What is the most frequently occurring sum that
  you could roll?
• Most likely outcome is a sum of 7 (there are 6
  ways to obtain it)
• What is the probability of obtaining the most
  likely sum in a single roll of the dice?
• 6 ? 36 .167
• What is the probability of obtaining a sum
  greater than 2 and less than 11?
• 32 ? 36 .889

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How do we know what the distribution is when all
we have is a sample?

• Theory CLT applies to measurements taken
  consisting of many assemblies
• Experience past use of a distribution has
  generated very good results
• Testing combination of the above in this
  case, anyway!
• If we know the generating function for a
  distribution, we can construct a grid
  (probability paper) that will allow us to observe
  a straight line when sufficient data from that
  distribution are plotted on the grid
• Easiest grid to create is the Standard Normal
  Distribution
• because it is an easy transformation to
  standard parameters

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Normal Probability Plots

• Take raw data and count observations (n)
• Set up a column of j values (1 to j)
• Compute ?(zj) for each j value
• ?(zj) (j - 0.5)/n
• Get zj value for each ?(zj) in Standard Normal
  Table
• Find table entry(?(zj)), then read index value
  (zj)
• Set up a column of sorted, observed data
• Sorted in increasing value
• Plot zj values versus sorted data values
• Approximate with sketched line at 25 and 75
  points

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Interpreting Normal Plots

• Assess Equal-Variance and Normality assumptions
• Data from a Normal sample should tend to fall
  along the line, so if a fat pencil covers
  almost all of the points, then a normality
  assumption is supported
• The slope of the line reflects the variance of
  the sample, so equal slopes support the equal
  variance assumption
• Theoretically
• Sketched line should intercept the zj 0 axis at
  the mean value
• Practically
• Close is good enough for comparing means
• Closer is better for comparing variances
• If the slopes differ much for two samples, use a
  test that assumes the variances are not the same

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Relationship with Hypothesis Tests

• Assuming that our process is Normally Distributed
  and centered at the mean, how far apart should
  our specification limits be to obtain 99. 5
  yield?
• Proportion defective will be 1 .995 .005, and
  if the process is centered, half of those
  defectives will occur on the right tail (.0025),
  and half on the left tail.
• To get 1 .0025 99.75 yield before the right
  tail requires the upper specification limit to be
  set at
• ? 2.81?.

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Relationship with Hypothesis Tests

• Assuming that our process is Normally Distributed
  and centered at the mean, how far apart should
  our specification limits be to obtain 99. 5
  yield?
• Proportion defective will be 1 .995 .005, and
  if the process is centered, half of those
  defectives will occur on the right tail (.0025),
  and half on the left tail.
• To get 1 .0025 99.75 yield before the right
  tail requires the upper specification limit to be
  set at ? 2.81?.
• By symmetry, the remaining .25 defective should
  occur at the left side, with the lower
  specification limit set at ? 2.81?
• If we specify our process in this manner and made
  a lot of parts, we would only produce bad parts
  .5 of the time.

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Questions Issues