Quantitative

1
STATISTICS
Quantitative Graphical Analysis
DEPARTMENT OF STATISTICS
REDGEMAN_at_UIDAHO.EDU
OFFICE 1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR CHAIR SIX SIGMA
BLACK BELT
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Measurement a numerical assignment to
something, usually a non-numerical
element. Scales of Measurement Nominal Ordinal I
nterval Ratio
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Data Reliability Validity Validity An item
measures what it is intended to
measure. Reliability A re-measurement would
order individual responses in the same way. Bias
The difference between the average measured
value and a reference value is called bias.
Bias is controlled via calibration.
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Repeatability Reproducibility Repeatability is
the variation in measurements obtained with one
measurement instrument when used several times by
one appraiser, while measuring the identical
characteristic on the same part. Variation
obtained when the measurement system is applied
repeatedly under the same conditions and is
often caused by conditions inherent in the
measurement system. Precision is the closeness
of agreement between randomly selected individual
measurement or test results. Reproducibility is
the variation in the average of the measurements
made by different appraisers using the
same measuring instrument when measuring the
identical characteristic on the same item.
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Stability is the total variation in the
measurements obtained with a measurement system
on the same master or parts when measuring a
single characteristic over an extended
time period. A system is said to be stable if the
results are the same at different points in
time. Linearity is the difference in the bias
values through the expected operating range of
the gage.
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Inside Data
Pareto Diagrams
Process Flow Diagrams
Cause-and-Effect Diagrams
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Cause-and-Effect Diagram Also called a fishbone
or Ishikawa diagram
Materials Environment Methods
Branch
Stem
Effect
Major Cause Limb
Manpower-HR Machinery Measurement
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• This is a team tool - assemble the right team!
• After the team has finished the diagram, post
  it prominently in a department that either
  affects or is impacted by the process.
• Leave a blue marker by the diagram so that
  annotations can be made - give it a few days.
• Move from department to department, changing the
  color of the marker in each department.
• After this rotation is complete have the team
  update the diagram.
• State effects in the positive - youll get
  better results.
• Time-Sequencing may be possible.
• Try balancing enablers (solutions) on top of the
  diagram and obstacles on the bottom of the
  diagram - similar to Force Field Analysis.

Cause Effect Diagram Thoughts
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Pareto Analysis

• Graphically indicates which diseases or problems
  most infect the process. Pareto diagrams may be
  regarded as a special use form of the histogram.
• This aids organizations in focusing efforts
  resources and is a commonly used quality
  improvement tool.

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Pareto Analysis for Process Improvement
f or

• A Useful Team Diagnostic and Improvement Tool

Policy More Pages
Toner Other Unclear Copies
Copied Marks
Reasons on Than That
Personal Needed Arent
Usage Needed
Cause of Excessive Photocopier Use
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Process Flow Diagrams

• A visual portrayal of the sequential and parallel
  stages steps and the relationships between
  these steps that make up a process or a portion
  of a process. Such diagrams can function as
  aids to understanding.

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Process Flow Diagrams
Start
Recommendation Current Ideal!
Finish
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Analyzing Data
Boxplots
Histograms
Scatter Diagrams
Time Series Plots
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The Voice of the ProcessA Statistical
Perspective
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• Statistics as a vehicle to search for, quantify,
  understand and respond to truth is explored.
• Foundational concepts are presented and
  illustrated, including process characterization
  and guidance.
• Measures of process centrality, variation,
  position, shape are introduced and illustrated.

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Application of Material

• Material from this segment is useful in most
  fields of endeavor including business. Areas of
  application include finance, marketing,
  management, real estate, information systems, and
  accounting.
• READ Journal of Financial Quantitative
  Analysis
• Journal of Accounting
  Research
• Management Information Systems
  Quarterly
• Decision Sciences
• Journal of Marketing Research
• Quality Progress.

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Presentation Content

• The Quest for Truth
• Sources of and Paths to Truth
• The Scientific Method of Inquiry
• Process Proactivity Reactivity
• Wiretapping the Process
• Signal Processing
• Process Centrality, Consistency, Shape and the
  Unusual.

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The Quest for Truth

• Eternal / Immutable Truth
• Changing and Changeable Truth
• Situational Truth - Process Control
• Sources Paths to Truth
• Revelation / Providence
• Serendipity
• Tenacity / Persistence / Investigation
• Historical Research Scientific Method

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Correlation Causality

• Correlation typically implies a relationship
  between traits and MAY indicate of causality.
  If there is a relationship, then behavior of one
  trait allows prediction of the behavior of the
  other trait.
• Causality implies both the ability to predict and
  (often) to control the behavior of a second
  trait.

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Process Reactivity Proactivity

• Statistical Methods are Used Both to Analyze and
  Respond to What has Occurred in the Past ---
  Reactive Use --- and to Plan and Study Process
  Interventions for the Purpose of Process
  Improvement --- Proactive Use.
• Even if youre on the right track, youll get
  run over if you just sit there. Will
  Rogers

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WIRETAPPING THE PROCESSWhat is Our Interest in
the Process?

• Understanding
• Prediction of Process Behavior
• Guidance Control of Process Behavior

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The Voice of the Process

• What Do We Measure?
• Process Performance Measures (PPM)
• These Pickup the Heartbeat of the Process and May
  be DIRECT or SURROGATE.
• Why Do We Measure?
• How Do We Measure?
• Where When in the Process Do We Measure?

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Signal Processing

• Having addressed issues related to PPMs, we may
  now gather process data.
• Data should provide an image of the process from
  which the data originates.
• We will want to know
• Where does the process live?
• How consistent is the process?
• What is the shape of the process?
• What is unusual about the process?
• Does process behavior vary over time?

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Signal Processing Data Analysis

• Signal processing, more commonly called data
  analysis, is often conducted with the aid of
  canned spreadsheet or statistical software
  packages such as
• MINITAB, SAS, SPSS, BMDP, or SYSTAT.

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Copier Abuses
Firm in multilevel building, each level has its
own photocopier Free usage the tenth level is
selected and each person is given a key to the
machine on their level and will have their usage
tracked. Third level employee usage is tracked
in aggregate but those employees are unaware of
monitoring. Data for 50 days follows.
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50 Days of Photocopier Usage
Day 10th 3rd Day 10th 3rd
Day 10th 3rd --------------------------
--------------------------------------------------
--------------------------------------------------
--------------- 1 500 440 18
360 20 35 150 370 2 420
220 19 310 250 36 140
405 3 440 360 20 320
350 37 130 130 4 480
110 21 290 150 38 150 120 5
450 240 22 290 250 39
130 70 6 460 360 23
270 230 40 110 240 7 450
80 24 250 90 41 90
20 8 420 420 25 240
50 42 80 450 9 410
310 26 250 320 43 90
20 10 405 30 27 250
360 44 70 40 11 380
290 28 230 450 45 20
320 12 360 410 29 240
270 46 50 140 13 360
460 30 220 380 47 40
90 14 370 420 31 190
190 48 20 130 15 350
150 32 150 500 49 30
480 16 320 170 33 170
290 50 30 350 17 350
250 34 120 150
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Where Does the Process Live?Measures of
Centrality

• In a sense we seek an address for the process -
  forced to provide exactly one number which best
  represents the location of the process, what
  should we use? Traditional measures have
  included
• The Mean or Average
• The Median
• The Midrange
• The Mode.

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Determination of the Mean

• The true average of values for the process is
  symbolized by the Greek lower case letter mu,
  that is
• True Process Mean ?
• This value tends to be both unknown and
  unknowable and is generally estimated from the
  sample data. If we represent the PPM (shelf life
  in days) as X, then the sample mean is X-bar or X.

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Sample Mean for 10th Level

• The sample mean is determined as the simple
• arithmetic average of the sample data values.
• That is
• X ?Xi/n (500 420 ... 30)/50 or
• X 248.1 copies/day
• Average number of copies per day is about 248

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Mode, Midrange Median

• Three other common measures of process address
  (or centrality as this concept is called in
  statistics) are the
• MODE, or the most frequently occurring value
  (150, 250, and 360 each occur 3 times).
• MIDRANGE or the average of the minimum and
  maximum process PPM values. This value is (20
  500)/2 260
• MEDIAN or Q2, the value for which half of the
  process values are larger in value and half of
  values are smaller. Q2 is the (n1)th/2 ordered
  value, averaging the two surrounding ordered
  values if there is a fractional part. For the
  10th level data (n1)/2 25.5 so that Q2 is
  the average of the 25th 26th ordered values or
  (250 270)/2 260 copies/day

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Process Consistency

• This issue is like a coin --- it is two-sided.
• On one side, the side generally emphasized in
  statistics, is the side labeled variability
  with statisticians discussing measures of
  variability - variability is an enemy!
• On the other side is process consistency where
  consistency is a desirable trait!

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Consistent With What?

• When measuring process consistency we generally
  are referring to how close to (e.g. consistent
  with) or how far away from (e.g. variable) is the
  process with respect to some point of reference
  or an anchor,
• Typically the mean serves as this anchor.
• Though there are many measures of variability /
  consistency, only the most commonly applied
  measures are examined
• range, interquartile range, variance standard
  deviation.

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Range Interquartile Range

• The range is simply the distance between the
  largest and smallest values in the process and is
  estimated from the sample values. For the 10th
  level data
• Range max - min 500 - 20 480 copies/day
• The interquartile range (IQR) is given by IQR
  (Q3 - Q1) where Q3 and Q1 are the values at the
  75th and 25th percentiles, respectively. That
  is
• Q3 value with 75 of the sample smaller
• Q1 value with 25 of the sample smaller.

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Interquartile Range (IQR)

• Q3 the 3(n1)th/4 ordered value where 3(n1)/4
  3(51)/4 38.25 so Q3 is the number that is
  one-fourth of the way from the 38th ordered value
  to the 39th ordered value. These values are 360
  and 370 so that Q3 362.5
• Q1 the (n1)th/4 ordered value where (n1)/4
  51/4 12.75 so Q1 is three-fourths of the way
  from the 12th ordered value to the 13th ordered
  value. These are 120 and 130 for the 10th Level
  data so that Q1 127.5.
• Thus IQR (Q3-Q1) (362.5 - 127.5) 235 copies
  / day

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The Sample Variance

• The true variance of the process is represented
  by ?2 and can be thought of as the average
  squared distance of observations in the process
  from the true process mean, ?.
• Generally, the values of both ? and ?2 are
  unknown and must be estimated, ? by X and ?2 by
  S2. This latter value, S2, is called the
  sample variance.

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Sample Variance - Calculation

• S2 is defined as S2 ?(Xi - X)2/(n-1) but is
  better calculated using
• S2 (?Xi2 - nX2)/(n-1)
• For the 10th Level data we have
• S????????????

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Sample Standard Deviation

• The sample variance, S2, has desirable properties
  but is difficult to interpret since it is defined
  in terms of the square of the original PPM units.
• More understandable is the sample standard
  deviation, S, which is simply the positive square
  root of S2 and can be thought of as
  representative (standard) of the distance
  (deviation) between values in the data set and
  the sample mean.
• This value estimates the true process standard
  deviation, ?.

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Sample Standard Deviation

• The value of the sample standard deviation for
  the 10th Leveldata is
• S ????????? 143.3 copies /day
• That is, it is representative of the values in
  the 10th Level data set that they vary from the
  average by about 143.3 copies / day, more or
  less.

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The Empirical Rule

• The Empirical Rule is applicable for
  approximately mound-shaped distributions and
  relates X and S as follows (figures are rule
  of thumb)
• 67 of all values within X /- S
• 95 of all values within X /- 2S
• 99 of all values within X /- 3S
• Values outside this latter range are often
  considered unusual.

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Process Shape

• It is often important to know not just how much
  the PPM varies with regard to some anchor point,
  say the mean, but also the particular pattern of
  variation followed by the PPM is of value. This
  concept is often referred to as shape.
• Shape is often represented graphically through
  use of histograms or box-and-whisker plots
  (boxplots).

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Process Histograms

• Such plots generally show the number of data
  values in successive categories with categories
  being
• mutually exclusive
• and exhaustive.
• Construction of a histogram is both art and
  science.

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Histogram Construction

• Determine the range, R max -min,
• for 10th Level data R 500 - 20 480.
• Determine the number of categories to be used, k.
  A useful rule of thumb is k log2(n) where n
  is the sample size. This gives k 5 (4 to 6)
• Determine W R/k, this is the minimum width that
  a category should be, and usually W will be
  rounded to a convenient value. That is W is
  between 480/4 120 and 480/6 80
• Construct categories classify data.

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Histogram Categories

• Given these guidelines, the number of histogram
  categories, k, for a sample of n items is
  approximately
• n k
• 1-7 dont bother
• 8-15 3 /- 1 These are rule-of-thumb
• 16-31 4 /- 1 We should not under- or
• 32-63 5 /- 1 over- resolve the data.
• 64-127 6 /- 2
• 128-255 7 /- 2

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Process Shape Via the Histogram
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Box-and-Whisker Plots

• Most commonly called boxplots.
• Consist of a five-number summary and four
  outlier points
• min, Q1, Q2, Q3, max
• Inner LOP Q1-1.5(IQR) Outer LOP
  Q1-3(IQR)
• Inner UOP Q31.5(IQR) Outer UOP Q33(IQR)
• min 20 max 500
• Q1 127.5 Q2 250 Q3 362.5
• Inner LOP 127.5 - 1.5(235) -225.5 copies
  (N/A) Outer LOP 127.5 -
  3.0(235) -577.5 copies (N/A)
• Inner UOP 362.5 1.5(235) 615 days
  Outer UOP 362.5
  3.0(235) 1,067.5 copies
• NO OUTLIERS BY THESE MEASURES!

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Shape - The Rest of the Story?
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Correlation Causality

• PATTERNS
• The Graph at Left Indicates, in Part, a
  Relationship Between Two Variables With One
  Increasing in Value as the Other Increases in
  Value. Cycling is also Evident.

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Scatter Diagrams Correlation Causality

PATTERNS This graph shows
an indirect relation with one
trait
increasing in value as the other
trait decreases.
Correlation
sometimes indicates causality. Prediction
and possibly process guidance may be
possible.
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Time Series Plot
250
50
A Case Report - What Might We Conclude?
Let the punishment fit the crime!
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Shelf Life of a Perishable Food Product - An
Exercise

• Consider the shelf life of a perishable food
  product. A sample of n 25 such products, taken
  under essentially uniform conditions provided the
  following results, with days being the unit of
  measurement

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Data Analysis Assignment

• Determine mean, median, mode and midrange.
• Determine range, variance, standard deviation,
  and inter-quartile range.
• Apply the empirical rule.
• Construct histogram, and box-plot that includes
  all outlier point criteria.
• Construct time series plot (the data should be
  read left-to-right).
• INTERPRET ALL RESULTS what does all of this
  mean? see the following slide

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Written Case Guidelines
GOOD ADVICE! A. One or
two page written summary - concise Key Issues
Data Results Recommendation(s)
Limitations. B. Only vitally important
statistical / graphical output should be
included in these two pages C. Otherwise
important output should be placed in an
appendix and should be clearly summarized D.
Spell- and grammar-checked
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STATISTICS
Quantiative Graphical Analysis
End of Session
DEPARTMENT OF STATISTICS
REDGEMAN_at_UIDAHO.EDU
OFFICE 1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR CHAIR SIX SIGMA
BLACK BELT