1. Six Sigma Green Belt Training Presented by Harry H Holdorf

2. Analyze Basic Stats Intro to Hypothesis Testing Lean Sigma
Multi-Vari Studies
Identifying Root Cause

3. Analyze Lean Sigma Basic Stats Intro to Hypothesis Testing
Multi-Vari Studies
Identifying Root Cause

4. Basic Statistics Objectives
Purpose of Statistics
Types of Data
Normal Distribution
Z Values
Measures of Locations
Graphing Data

5. 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!

6. What is statistics? Numbers, data and statistics
100, 140, 213, 230, 180, 211, 120, 160, 200, 110, 260, 235, 280, 180, 300
NUMBERS
Human Weight (pound):
100, 140, 213, 230, 180, 211, 120, 160, 200, 110, 260, 235, 280, 180, 300
Statistics
Mean Value = pound
Minimum = 100 pound
Maximum = 300 pound
Range = 200 pounds

7. Statistical Notation – Cheat Sheet
Summation
The Standard Deviation of sample data
The Standard Deviation of population data
The variance of sample data
The variance of population data
The range of data
The average range of data
Multi-purpose notation, i.e. # of subgroups, # of classes
The absolute value of some term
Greater than, less than
Greater than or equal to, less than or equal to
An individual value, an observation
A particular (1st) individual value
For each, all, individual values
The Mean, average of sample data
The grand Mean, grand average
The Mean of population data
A proportion of sample data
A proportion of population data
Sample size
Population size

8. Parameters vs. Statistics
Population: All the items that have the “property of interest” under study.
Sample: A significantly smaller subset of the population used to make an inference.
Population
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

9. The Performance Standard can be expressed by either type of data
Two Types of Data
Discrete Data
Continuous Data
Values can be measured to any degree of precision...
Values can vary only by whole units…
Examples:
Binary - pass/fail, on-time/late
Count - number of students, number of
questions on a test
Attribute - light is red / yellow / green
Examples:
Miles
Time
Weight
Dollar amount on invoice
The Performance Standard can be expressed by either type of data

10. Possible Values for the Variable
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)

11. Possible Values for the Variable
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

12. 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.

13. Possible nominal level data values for the variable
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

14. Possible Ordinal level data values
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

15. Interval Scale Interval Variable Possible Scores
IQ scores of students in Black Belt Training
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)
Weight of a Person
100 lbs
115 lbs
150 lbs
175 lbs
210 lbs

16. Ratio Scale Ratio Variable Possible Scores
Grams of fat consumed per adult in the United States
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.)

17. Converting Attribute Data to Continuous Data
Continuous Data is always more desirable for statistical analysis
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

18. 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?

19. 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.

20. The Normal (Z) Distribution
Characteristics of Normal Distribution (Gaussian curve) are:
It is considered to be the most important distribution in statistics.
The total area under the curve is equal to 1.
The distribution is mounded and symmetric; it extends indefinitely in both directions, approaching but never touching the horizontal axis.
All processes will exhibit a normal curve shape if you have pure random variation (white noise).
The Z distribution has a Mean of 0 and a Standard Deviation of 1.
The Mean divides the area in half, 50%
on one side and 50% on
the other side.
The Mean, Median and Mode are at the same data point. +6 -1 -3 -4 -5 -6 -2 +4 +3 +2 +1 +5

21. Standard 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.

22. Distribution For a normal distribution we care about the mean m
Measurements Vary From Each Other
But They Form a Pattern That,
If Stable ,
Can Be Described as a Distribution
Distributions Can Differ In:
Location
Spread
Shape
For a normal distribution we care about the
mean m
, and the standard deviation s .

23. Non-Normal Distributions
1 Skewed
2 Kurtosis
3 Multi-Modal
4 Granularity

24. The Statistical Problem
From A Statistical Perspective, There are only Two Problems (imperfections)
Target
LSL
USL
Target
LSL
USL
Centering
– the process is
not on target (mean shift)
Spread
– the process
variation is too large (s)
Target
LSL
USL
Center Process
Reduce Spread

25. Levels of Standard Deviation
= 0.04 s
= 0.41 s
= 0.81
Process A
Process B
Process C
Which process is the BEST? s
= standard deviation
WHY ?

26. As the standard deviation increases
Distribution & DPMO
DPMO = defects per million units .
= Proportion of observations outside spec * 1,000,000
Lower spec
Upper spec.
B distribution
A distribution
C distribution
Defects
Defects
As the standard deviation increases
DPMO increases.

27. More on Normal Distribution
Normal Distribution and Defects
(1 Sigma process, Z=1)
Defects
Defects 1s
Mean 1s
Upper Customer Specification Limit
USL
Lower Customer Specification Limit
LSL
1s( pronounced one sigma) = one standard deviation
What percentage of defects would you expect?

28. Z predicts the defect levelO u t p
68%
95% 4 3 2 1 - . N o r m a l C v e n d P b i y A s
99.73%
Spec Limit =
USL
LSL Z SL - s
Targeting, Mean
Spread, standard deviation
Long or
short term X
General Formula for Z:
Between
Percent of area under
normal (Z) curve m
- 3 s
and
+ 3
99.73
99.7
- 2
+ 2
95.44 95 -1
+ 1
68.26 68

29. Six Sigma Capability & Mean Shifts
USL
LSL
A 1.5 Sigma Shift is typical, over time. With a 6 Sigma Process Capability - performance is always acceptable.
In the Real World “SHIFT Happens”

30. The Empirical Rule

31. 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.

32. 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.
So how do we assess for normality?
Watch that curve!

33. if the P-value is less than .05, your data are non-normal.
Goodness-of-Fit
The Anderson-Darling test measures 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.
P-value
if the P-value is less than .05, your data are non-normal.

34. 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!

35. Normality Exercise
Exercise objective: To demonstrate how to test for Normality.
Generate the graphical summary using the “Descriptive Statistics.MTW” file.
Use only the columns Dist A and Dist D.

36. Descriptive Statistics
Measures of Location (central tendency)
Mean
Trimmed Mean
Median
Mode
Measures of Variation (dispersion)
Range
Interquartile Range
Standard deviation
Variance

37. Descriptive Statistics
Open the MINITAB™ Project “Measure Data Sets.mpj” and select the worksheet “basicstatistics.mtw”

38. Non-Normal Right (Positive) Skewed
Moment coefficient of Skewness will be close to zero for symmetric distributions, negative for left Skewed and positive for right Skewed.

39. Bimodal Distributions
2 Different Distributions
2 different machines
2 different operators
2 different administrators

40. Extreme Bi-Modal (Outliers)

41. Bi-Modal – Multiple Outliers

42. Measures of location

43. Measures of Location Mean is: Commonly referred to as the average.
The arithmetic balance point of a distribution of data.
Population
Sample

44. Measures of Location
Calculating Mean
in Minitab

45. 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

46. 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
Variable Q3 Maximum
Data

47. Measures of Location Mode is:
The most frequently occurring value in a distribution of data.
Mode = 5

48. Measure of variation

49. Use Range or Interquartile Range when the data distribution is Skewed.
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.
Interquartile Range is the:
Difference between the 75th percentile and the 25th percentile.
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
Data
Variable Maximum
Data
Use Range or Interquartile Range when the data distribution is Skewed.

50. 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.
Population
Sample
Cannot calculate population Standard Deviation because this is sample data.

51. Measures of Variation Variance is the:
Average squared deviation of each individual data point from the Mean.
Sample
Population

52. Graphing Data

53. 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…
Box Plots
Scatter Plots
Dot Plots
Time Series Plots
Histograms

54. 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
3. Operator
Training
Experience
Skill
Adherence to procedures
2. Location/position
Facility
Region
Office
4. Any other sources?

55. Graphical Concepts The characteristics of a good graph include:
Variety of data
Selection of
Variables
Graph
Range
Information to interpret relationships
Explore quantitative relationships

56. 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.
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).

57. 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.

58. Variation on a Histogram
Using the worksheet “Graphing Data.mtw” create a simple Histogram for the data column called granular.

59. 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.

60. 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.
Middle
50% of
Data
50th Percentile (Median)
25th Percentile
75th Percentile
min(1.5 x Interquartile Range
or minimum value)
Outliers
Maximum Value
Mean

61. Box Plot Anatomy * Box Outlier Upper Limit: Q3+1.5(Q3-Q1)
Median
Upper Whisker
Lower Whisker
Upper Limit: Q3+1.5(Q3-Q1)
Lower Limit: Q1+1.5(Q3-Q1)
Q3: 75th Percentile
Q1: 25th Percentile
Q2: Median 50th Percentile
Box *
Outlier

62. Box Plot Examples
What can you tell about the data expressed in a Box Plots?
Eat this – then check the Box Plot!

63. Box Plot Example

64. Box Plot Example

65. Individual Value Plot Enhancement

66. 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

67. Attribute Y Box Plot

68. 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

69. 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.

70. 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...
Use
Analysis
Continuous data
Find shifts, trends, outliers
Logical time order
See natural spread of process

71. Time Series Example
Looking at the Time Series Plot below, the response appears to be very dynamic.
What other characteristic is present?

72. Summary Understand the different types of data
At this point, you should be able to:
Understand the different types of data
Describe a normal distribution and check for normality in Minitab
Check for measures of location and variation in Minitab
Create different graphs in Minitab

73. FIN