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Measure Phase
Six Sigma Statistics
Six Sigma Statistics
Welcome to Measure
Process Discovery
Six Sigma Statistics
Basic Statistics
Descriptive Statistics
Normal Distribution
Assessing Normality
Special Cause / Common Cause
Graphing Techniques
Measurement System Analysis
Process Capability
Wrap Up & Action Items
2
Purpose of Basic Statistics
The purpose of Basic Statistics is to:
•
Provide a numerical summary of the data being analyzed.
–
Data (n)
•
Factual information organized for analysis.
•
Numerical or other information represented in a form suitable for processing by
computer
•
Values from scientific experiments.
•
Provide the basis for making inferences about the future.
•
Provide the foundation for assessing process capability.
•
Provide a common language to be used throughout an organization to
describe processes.
Relax….it won’t be
that bad!
3
Statistical Notation – Cheat Sheet
Summation
An individual value, an observation
The Standard Deviation of sample data
A particular (1st) individual value
The Standard Deviation of population data
For each, all, individual values
The variance of sample data
The Mean, average of sample data
The variance of population data
The grand Mean, grand average
The range of data
The Mean of population data
The average range of data
Multi-purpose notation, i.e. # of subgroups, #
of classes
A proportion of sample data
A proportion of population data
The absolute value of some term
Sample size
Greater than, less than
Greater than or equal to, less than or equal to
Population size
4
Parameters vs. Statistics
Population:
All the items that have the “property of interest” under study.
Frame:
An identifiable subset of the population.
Sample:
A significantly smaller subset of the population used to make an inference.
Population
Sample
Sample
Sample
Population Parameters:
–
–
Arithmetic descriptions of a population
µ,  , P, 2, N
Sample Statistics:
–
–
Arithmetic descriptions of a
sample
X-bar , s, p, s2, n
5
Types of Data
Attribute Data (Qualitative)
– Is always binary, there are only two possible values (0, 1)
• Yes, No
• Go, No go
• Pass/Fail
Variable Data (Quantitative)
– Discrete (Count) Data
• Can be categorized in a classification and is based on counts.
– Number of defects
– Number of defective units
– Number of customer returns
– Continuous Data
• Can be measured on a continuum, it has decimal subdivisions that are
meaningful
– Time, Pressure, Conveyor Speed, Material feed rate
– Money
– Pressure
– Conveyor Speed
– Material feed rate
6
Discrete Variables
Discrete Variable
Possible Values for the Variable
The number of defective needles in boxes of 100
diabetic syringes
0,1,2, …, 100
The number of individuals in groups of 30 with a
Type A personality
0,1,2, …, 30
The number of surveys returned out of 300
mailed in a customer satisfaction study.
0,1,2, … 300
The number of employees in 100 having finished
high school or obtained a GED
0,1,2, … 100
The number of times you need to flip a coin
before a head appears for the first time
1,2,3, …
(note, there is no upper limit because you might
need to flip forever before the first head appears)
7
Continuous Variables
Continuous Variable
Possible Values for the Variable
The length of prison time served for individuals
convicted of first degree murder
All the real numbers between a and b, where a is
the smallest amount of time served and b is the
largest.
The household income for households with
incomes less than or equal to $30,000
All the real numbers between a and $30,000,
where a is the smallest household income in the
population
The blood glucose reading for those individuals
having glucose readings equal to or greater than
200
All real numbers between 200 and b, where b is
the largest glucose reading in all such individuals
8
Definitions of Scaled Data
Understanding the nature of data and how to represent it can
affect the types of statistical tests possible.
•Nominal Scale – data consists of names, labels, or categories. Cannot be
arranged in an ordering scheme. No arithmetic operations are performed for
nominal data.
•Ordinal Scale – data is arranged in some order, but differences between data
values either cannot be determined or are meaningless.
•Interval Scale – data can be arranged in some order and for which differences
in data values are meaningful. The data can be arranged in an ordering scheme
and differences can be interpreted.
•Ratio Scale – data that can be ranked and for which all arithmetic operations
including division can be performed. (division by zero is of course excluded)
Ratio level data has an absolute zero and a value of zero indicates a complete
absence of the characteristic of interest.
9
Nominal Scale
Qualitative Variable
Possible nominal level data values for
the variable
Blood Types
A, B, AB, O
State of Residence
Alabama, …, Wyoming
Country of Birth
United States, China, other
Time to weigh in!
10
Ordinal Scale
Qualitative Variable
Possible Ordinal level data
values
Automobile Sizes
Subcompact, compact,
intermediate, full size, luxury
Product rating
Poor, good, excellent
Baseball team classification
Class A, Class AA, Class AAA,
Major League
11
Interval Scale
Interval Variable
IQ scores of students in
BlackBelt Training
Possible Scores
100…
(the difference between scores
is measurable and has
meaning but a difference of 20
points between 100 and 120
does not indicate that one
student is 1.2 times more
intelligent )
12
Ratio Scale
Ratio Variable
Grams of fat consumed per adult in the
United States
Possible Scores
0…
(If person A consumes 25 grams of fat and
person B consumes 50 grams, we can say
that person B consumes twice as much fat
as person A. If a person C consumes zero
grams of fat per day, we can say there is a
complete absence of fat consumed on that
day. Note that a ratio is interpretable and
an absolute zero exists.)
13
Converting Attribute Data to Continuous Data
Continuous Data is always more desirable
In many cases Attribute Data can be converted to Continuous
Which is more useful?
– 15 scratches or Total scratch length of 9.25”
– 22 foreign materials or 2.5 fm/square inch
– 200 defects or 25 defects/hour
Is this data continuous?
14
Descriptive Statistics
Measures of Location (central
tendency)
– Mean
– Median
– Mode
Measures of Variation (dispersion)
–
–
–
–
Range
Interquartile Range
Standard deviation
Variance
15
Descriptive Statistics
Open the MINITAB™ Project “Measure Data Sets.mpj” and
select the worksheet “basicstatistics.mtw”
16
Measures of Location
Mean is:
• Commonly referred to as the average.
• The arithmetic balance point of a distribution of data.
Stat>Basic Statistics>Display Descriptive Statistics…>Graphs…
>Histogram of data, with normal curve
Sample
Histogram (with Normal Curve) of Data
Mean
StDev
N
80
70
Population
5.000
0.01007
200
Frequency
60
50
40
Descriptive Statistics: Data
30
Variable N N* Mean SE Mean StDev Minimum
Q1
Median
Q3
Data
200 0 4.9999 0.000712 0.0101 4.9700 4.9900
5.0000 5.0100
20
10
0
4.97
4.98
5.00
4.99
Data
5.01
5.02
Variable Maximum
Data
5.0200
17
Measures of Location
Median is:
• The mid-point, or 50th percentile, of a distribution of data.
• Arrange the data from low to high, or high to low.
– It is the single middle value in the ordered list if there is an odd
number of observations
– It is the average of the two middle values in the ordered list if there
are an even number of observations
Histogram (with Normal Curve) of Data
Mean
StDev
N
80
70
5.000
0.01007
200
Frequency
60
50
Descriptive Statistics: Data
40
Variable N N* Mean SE Mean StDev Minimum Q1 Median
Q3
Data
200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100
30
20
Variable Maximum
Data
5.0200
10
0
4.97
4.98
4.99
5.00
5.01
5.02
Data
18
Measures of Location
Trimmed Mean is a:
Compromise between the Mean and Median.
• The Trimmed Mean is calculated by eliminating a specified percentage
of the smallest and largest observations from the data set and then
calculating the average of the remaining observations
• Useful for data with potential extreme values.
Stat>Basic Statistics>Display Descriptive Statistics…>Statistics…> Trimmed Mean
Descriptive Statistics: Data
Variable N N* Mean SE Mean TrMean StDev Minimum
Q1 Median
Data
200 0 4.9999 0.000712 4.9999 0.0101 4.9700 4.9900 5.0000
Variable
Q3 Maximum
Data
5.0100 5.0200
19
Measures of Location
Mode is:
The most frequently occurring value in a distribution of data.
Mode = 5
Histogram (with Normal Curve) of Data
Mean
StDev
N
80
70
5.000
0.01007
200
Frequency
60
50
40
30
20
10
0
4.97
4.98
4.99
5.00
5.01
5.02
Data
20
Measures of Variation
Range is the:
Difference between the largest observation and the smallest
observation in the data set.
• A small range would indicate a small amount of variability and a large
range a large amount of variability.
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum
Q1 Median
Q3
Data
200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100
Variable Maximum
Data
5.0200
Interquartile Range is the:
Difference between the 75th percentile and the 25th percentile.
Use Range or Interquartile Range when the data distribution is Skewed.
21
Measures of Variation
Standard Deviation is:
Equivalent of the average deviation of values from the Mean for a
distribution of data.
A “unit of measure” for distances from the Mean.
Use when data are symmetrical.
Sample
Population
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum
Q1 Median
Q3
Data
200 0 4.9999 0.000712 0.0101 4.9700 4.9900 5.0000 5.0100
Variable Maximum
Data
5.0200
Cannot calculate population Standard Deviation because this is sample data.
22
Measures of Variation
Variance is the:
Average squared deviation of each individual data point from the
Mean.
Sample
Population
23
Normal Distribution
The Normal Distribution is the most recognized distribution in
statistics.
What are the characteristics of a Normal Distribution?
– Only random error is present
– Process free of assignable cause
– Process free of drifts and shifts
So what is present when the data is Non-normal?
24
The Normal Curve
The Normal Curve is a smooth, symmetrical, bell-shaped
curve, generated by the density function.
It is the most useful continuous probability model as
many naturally occurring measurements such as
heights, weights, etc. are approximately Normally
Distributed.
25
Normal Distribution
Each combination of Mean and Standard Deviation generates a
unique Normal curve:
“Standard” Normal Distribution:
– Has a μ = 0, and σ = 1
– Data from any Normal Distribution can be made to
fit the standard Normal by converting raw scores
to standard scores.
– Z-scores measure how many Standard Deviations from the
mean a particular data-value lies.
26
Normal Distribution
The area under the curve between any 2 points represents the
proportion of the distribution between those points.
The area between the
Mean and any other
point depends upon the
Standard Deviation.
m
x
Convert any raw score to a Z-score using the formula:
Refer to a set of Standard Normal Tables to find the proportion
between μ and x.
27
The Empirical Rule
The Empirical Rule…
-6
-5
-4
-3
-2
-1
+1
+2
+3
+4
+5
+6
28
The Empirical Rule (cont.)
No matter what the shape of your distribution is, as you travel 3 Standard
Deviations from the Mean, the probability of occurrence beyond that point
begins to converge to a very low number.
29
Why Assess Normality?
While many processes in nature behave according to the Normal
Distribution, many processes in business, particularly in the areas of
service and transactions, do not.
There are many types of distributions:
There are many statistical tools that assume Normal Distribution
properties in their calculations.
So understanding just how “Normal” the data are will impact how we
look at the data.
30
Tools for Assessing Normality
The shape of any Normal curve can be calculated based on the
Normal Probability density function.
Tests for Normality basically compare the shape of the calculated
curve to the actual distribution of your data points.
For the purposes of this training, we will focus on 2 ways in
MINITAB™ to assess Normality:
– The Anderson-Darling test
– Normal probability test
Watch that curve!
31
Goodness-of-Fit
The Anderson-Darling test uses an empirical density function.
Departure of the
actual data from the
expected Normal
Distribution. The
Anderson-Darling
Goodness-of-Fit test
assesses the
magnitude of these
departures using an
Observed minus
Expected formula.
100
Expected for Normal Distribution
Actual Data
20%
80
C
u
m
u
l
a 60
t
i
v
e
P
e 40
r
c
e
n
t
20
20%
0
3.0
3.5
4.0
4.5
5.0
5.5
Raw Data Scale
32
The Normal Probability Plot
Probability Plot of Amount
Normal
99.9
Mean
StDev
N
AD
P-Value
99
Percent
95
90
84.69
7.913
70
0.265
0.684
80
70
60
50
40
30
20
10
5
1
0.1
60
70
80
90
100
110
Amount
P-value
0.684
The Anderson-Darling test is a good litmus
test for normality: if the P-value is more
than .05, your data are normal enough for
most purposes.
33
Descriptive Statistics
The Anderson-Darling test also appears in this output. Again,
if the P-value is greater than .05, assume the data are Normal.
P-value
0.921
34
Anderson-Darling Caveat
Use the Anderson Darling column to generate these graphs.
Summary for Anderson Darling
Probability Plot of Anderson Darling
A nderson-Darling N ormality Test
Normal
99.9
Mean
StDev
N
AD
P-Value
99
Percent
95
90
50.03
4.951
500
0.177
0.921
80
70
60
50
40
30
20
36
40
44
48
52
56
A -S quared
P -V alue
0.18
0.921
M ean
S tDev
V ariance
S kew ness
Kurtosis
N
50.031
4.951
24.511
-0.061788
-0.180064
500
M inimum
1st Q uartile
M edian
3rd Q uartile
M aximum
60
35.727
46.800
50.006
53.218
62.823
95% C onfidence Interv al for M ean
49.596
10
50.466
95% C onfidence Interv al for M edian
5
49.663
50.500
95% C onfidence Interv al for S tDev
1
9 5 % C onfidence Inter vals
4.662
5.278
Mean
0.1
35
40
45
50
55
Anderson Darling
60
65
Median
49.50
49.75
50.00
50.25
50.50
In this case, both the Histogram and the Normality Plot look very “normal”. However,
because the sample size is so large, the Anderson-Darling test is very sensitive and any
slight deviation from Normal will cause the P-value to be very low. Again, the topic of
sensitivity will be covered in greater detail in the Analyze Phase.
For now, just assume that if N > 100 and the data
look Normal, then they probably are.
35
If the Data Are Not Normal, Don’t Panic!
• Normal Data are not common in the transactional world.
• There are lots of meaningful statistical tools you can use to
analyze your data (more on that later).
• It just means you may have to think about your data in a
slightly different way.
Don’t touch that button!
36
Normality Exercise
Exercise objective: To demonstrate how to test
for Normality.
1. Generate Normal Probability Plots and the
graphical summary using the “Descriptive
Statistics.MTW” file.
2. Use only the columns Dist A and Dist D.
3. Answer the following quiz questions based on
your analysis of this data set.
37
Isolating Special Causes from Common Causes
Special Cause: Variation is caused by known factors that result in
a non-random distribution of output. Also referred to as
“Assignable Cause”.
Common Cause: Variation caused by unknown factors resulting in
a steady but random distribution of output around the average of the
data. It is the variation left over after Special Cause variation has
been removed and typically (not always) follows a Normal
Distribution.
If we know that the basic structure of the data should follow a
Normal Distribution, but plots from our data shows otherwise; we
know the data contain Special Causes.
Special Causes = Opportunity
38
Introduction to Graphing
The purpose of Graphing is to:
•
•
•
•
Identify potential relationships between variables.
Identify risk in meeting the critical needs of the
Customer, Business and People.
Provide insight into the nature of the X’s which may or
may not control Y.
Show the results of passive data collection.
In this section we will cover…
1. Box Plots
2. Scatter Plots
3. Dot Plots
4. Time Series Plots
5. Histograms
39
Data Sources
Data sources are suggested by many of the tools that have been
covered so far:
–
–
–
–
Process Map
X-Y Matrix
FMEA
Fishbone Diagrams
Examples are:
1. Time
Shift
Day of the week
Week of the month
Season of the year
2. Location/position
3. Operator
Training
Experience
Skill
Adherence to procedures
4. Any other sources?
Facility
Region
Office
40
Graphical Concepts
The characteristics of a good graph include:
• Variety of data
• Selection of
– Variables
– Graph
– Range
Information to interpret relationships
Explore quantitative relationships
41
The Histogram
A Histogram displays data that have been summarized into intervals. It
can be used to assess the symmetry or Skewness of the data.
Histogram of Histogram
40
Frequency
30
20
10
0
98
99
100
101
Histogram
102
103
To construct a Histogram, the horizontal axis is divided into equal
intervals and a vertical bar is drawn at each interval to represent its
frequency (the number of values that fall within the interval).
42
Histogram Caveat
All the Histograms below were generated using random samples of
the data from the worksheet “Graphing Data.mtw”.
Be careful not to determine Normality simply from a Histogram plot, if
the sample size is low the data may not look very Normal.
43
Variation on a Histogram
Using the worksheet “Graphing Data.mtw” create a simple
Histogram for the data column called granular.
Histogram of Granular
25
Frequency
20
15
10
5
0
44
46
48
50
Granular
52
54
56
44
Dot Plot
The Dot Plot can be a useful alternative to the Histogram especially if
you want to see individual values or you want to brush the data.
Dotplot of Granular
44
46
48
50
Granular
52
54
56
45
Box Plot
Box Plots summarize data about the shape, dispersion and center of the data
and also help spot outliers.
Box Plots require that one of the variables, X or Y, be categorical or Discrete
and the other be Continuous.
A minimum of 10 observations should be included in generating the Box Plot.
Maximum Value
75th Percentile
Middle
50% of
Data
50th Percentile (Median)
Mean
25th Percentile
min(1.5 x Interquartile Range
or minimum value)
Outliers
46
Box Plot Anatomy
*
Outlier
Upper Limit: Q3+1.5(Q3-Q1)
Upper Whisker
Q3: 75th Percentile
Box
Median
Q2: Median 50th Percentile
Q1: 25th Percentile
Lower Whisker
Lower Limit: Q1+1.5(Q3-Q1)
47
Box Plot Examples
Boxplot of Glucoselevel vs SubjectID
225
What can you tell
about the data
expressed in a
Box Plots?
200
150
125
100
75
Cholesterol Levels
50
1
2
3
4
5
SubjectID
6
7
8350
9
300
Eat this –
then check
the Box
Plot!
Data
Glucoselevel
175
250
200
150
100
2-Day
4-Day
14-Day
48
Box Plot Example
49
Box Plot Example
Setup Cycle Time for "Lockout - Tagout"
20.0
17.5
Data
15.0
12.5
10.0
7.5
5.0
Brian
Greg
Shree
50
Individual Value Plot Enhancement
51
Attribute Y Box Plot
Box Plot with an Attribute Y (pass/fail) and a Continuous X
Graph> Box Plot…One Y, With Groups…Scale…Transpose value and category scales
52
Attribute Y Box Plot
Boxplot of Hydrogen Content vs Pass/Fail
Pass/Fail
1
2
215.0
217.5
220.0
222.5
225.0
Hydrogen Content
227.5
230.0
232.5
53
Individual Value Plot
The Individual Value Plot when used with a Categorical X or Y enhances
the information provided in the Box Plot:
– Recall the inherent problem with the Box Plot when a bimodal
distribution exists (Box Plot looks perfectly symmetrical)
– The Individual Value Plot will highlight the problem
Stat>ANOVA> One-Way (Unstacked )>Graphs…Individual value plot, Box Plots of data
Individual Value Plot of Weibull, Normal, Bi Modal
30
30
25
25
20
20
Data
Data
Boxplot of Weibull, Normal, Bi Modal
15
15
10
10
5
5
0
0
Weibull
Normal
Bi Modal
Weibull
Normal
Bi Modal
54
Jitter Example
Once your graph is created, click once on any of the data points (that action should
select all the data points).
Then go to MINITAB™ menu path: “Editor> Edit Individual Symbols>Identical
Points>Jitter…”
Increase the Jitter in the x-direction to .075, click OK, then click anywhere on the
graph except on the data points to see the results of the change.
Individual Value Plot of Weibull, Normal, Bi Modal
30
25
Data
20
15
10
5
0
Weibull
Normal
Bi Modal
55
Time Series Plot
Time Series Plots allow you to examine data over time.
Depending on the shape and frequency of patterns in the plot, several
X’s can be found as critical or eliminated.
Graph> Time Series Plot> Simple...
Time Series Plot of Time 1
602
Time 1
601
600
599
598
597
1
10
20
30
40
50
Index
60
70
80
90
100
56
Time Series Example
Looking at the Time Series Plot below, the response appears to be
very dynamic.
Time Series Plot of Time 1
602
Time 1
601
600
599
598
597
1
10
20
30
40
50
Index
60
70
80
90
100
What other characteristic is present?
57
Time Series Example (Cont.)
Let’s look at some other Time Series Plots.
What is happening within each plot?
What is different between the two plots?
Graph> Time Series Plot> Multiple...(use variables Time 2 and Time 3)
Time Series Plot of Time 2, Time 3
605
Variable
Time 2
Time 3
604
603
Data
602
601
600
599
598
597
596
1
10
20
30
40
50
60
Index
70
80
90
100
58
Curve Fitting Time Series
MINITAB™ allows you to add a smoothed line to your time series
based on a smoothing technique called Lowess.
Lowess means Locally Weighted Scatterplot Smoother.
Graph> Time Series Plot> Simple…(select variable Time 3)…Data View…Smoother…Lowess
Time Series Plot of Time 3
605
604
603
Time 3
602
601
600
599
598
597
596
1
10
20
30
40
50
Index
60
70
80
90
100
59
Summary
At this point, you should be able to:
• Explain the various statistics used to express
location and spread of data
• Describe characteristics of a Normal
Distribution
• Explain Special Cause variation
• Use data to generate various graphs and make
interpretations based on their output
60
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The Certified Lean Six Sigma Yellow Belt (CLSSYB) tests are
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Sigma. The CLSSYB can be used in preparation for the ASQ
or IASSC Certified Six Sigma Yellow Belt exam or for any
number of other certifications, including private company
certifications.
The Lean Six Sigma Yellow Belt Course Manual
Open Source Six Sigma Course Manuals are professionally
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