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# Understanding Statistics and Statistical Myths How to Become a Profound Learner

By

## Kicab Castaneda-Mendez

ISBN 9781498727457
Published November 18, 2015 by CRC Press
585 Pages 37 B/W Illustrations

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## Book Description

Addressing 30 statistical myths in the areas of data, estimation, measurement system analysis, capability, hypothesis testing, statistical inference, and control charts, this book explains how to understand statistics rather than how to do statistics. Every statistical myth listed in this book has been stated in course materials used by the author’s clients, by employers, or by experts in training thousands.

Each myth is an unconditional statement that, when taken literally and at face value, is false. All are false under some conditions while a few are not true under any condition. This book explores the conditions that render false the universality of the statements to help you understand why.

In the book, six characters discuss various topics taught in a fictional course intended to teach students how to apply statistics to improve processes. The reader follows along and learns as the students apply what they learn to a project in which they are team members.

Each discussion is like a Platonic dialogue. The purpose of a Platonic dialogue is to analyze a concept, statement, hypothesis, or theory through questions, applications, examples, and counterexamples, to see if it is true, when it is true, and why it is true when it is true. The dialogues will help readers understand why certain statements are not always true under all conditions, as well as when they contradict other myths.

Myth 1: Two Types of Data—Attribute/Discrete and Measurement/Continuous
Background
Measurement Requires Scale
Gauges or Instruments vs. No Gauges
Discrete, Categorical, Attribute versus Continuous, Variable: Degree of Information
Creating Continuous Measures by Changing the "Thing" Measured
Discrete versus Continuous: Half Test
Nominal, Ordinal, Interval, Ratio
Measurement to Compare
Scale Type versus Data Type
Scale Taxonomy
Purpose of Data Classification

Myth 2: Proportions and Percentages Are Discrete Data
Background
Denominator for Proportions and Percentages
Probabilities
Classification of Proportions, Percentages, and Probabilities

Myth 3: s = √[Σ(Xi- x)2/(n- 1)] The Correct Formula for Sample Standard Deviation
Background
Correctness of Estimations
Estimators and Estimates
Properties of Estimators

Myth 4: Sample Standard Deviation
√[Σ(Xi-x)2/(n- 1)] Is Unbiased
Background
Degrees of Freedom
t Distribution
Definition of Bias
Removing Bias and Control Charts

Myth 5: Variances Can Be Added but Not Standard Deviations
Background
Sums of Squares and Square Roots: Pythagorean Theorem
Functions and Operators
Random Variables
Independence of Factors
Other Properties

Myth 6: Parts and Operators for an MSA Do Not Have to Be Randomly Selected
Background
Types of Analyses of Variance
Making Measurement System Look Better than It Is: Selecting Parts to Cover the Range of Process Variation
Selecting Both Good and Bad Parts

Myth 7: % Study (% Contribution, Number of Distinct Categories) Is the Best Criterion for Evaluating a Measurement System for Process Improvement
Background
% Contribution versus % Study
P/T Ratio versus % Study
Distinguishing between Good and Bad Parts
Distinguishing Parts That Are Different

Myth 8: Only Sigma Can Compare Different Processes and Metrics
Background
Sigma and Specifications
Sigma as a Percentage

Myth 9: Capability Is Not Percent/Proportion of Good Units
Background
Capability Indices: Frequency Meeting Specifications
Capability: Actual versus Potential
Capability Indices
Process Capability Time-Dependent
Meaning of Capability: Short-Cut Calculations

Myth 10: p = Probability of Making an Error
Background
Only Two Types of Errors
Definition of an Error about Deciding What Is True
Calculation of p and Evidence for a Hypothesis
Probability of Making an Error for a Particular Case
Probability of Data Given Ho versus Probability of Ho Given Data
Non-probabilistic Decisions

Myth 11: Need More Data for Discrete Data than Continuous Data Analysis
Background
Discrete Examples When n = 1
Factors That Determine Sample Size
Relevancy of Data

Myth 12: Nonparametric Tests Are Less Powerful than Parametric Tests
Background
Distribution Free versus Nonparametric
Comparing Power for the Same Conditions
Different Formulas for Testing the Same Hypotheses
Assumptions of Tests
Comparing Power for the Same Characteristic
Converting Quantitative Data to Qualitative Data

Myth 13: Sample Size of 30 Is Acceptable (for Statistical Significance)
Background
A Rationale for n = 30
Contradictory Rules of Thumb
Uses of Data
Sample Size as a Function of Alpha, Beta, Delta, and Sigma
Sample Size for Practical Use
Sample Size and Statistical Significance

Myth 14: Can Only Fail to Reject Ho, Can Never Accept Ho
Background
Proving Theories: Sufficient versus Necessary
Prove versus Accept versus Fail to Reject: Actions
Innocent versus Guilty: Problems with Example
Two-Choice Testing
Significance Testing and Confidence Intervals
Hypothesis Testing and Power
Null Hypothesis of ≥ or ≤
Practical Cases
Which Hypothesis Has the Equal Sign?
Bayesian Statistics: Probability of Hypothesis

Myth 15: Control Limits Are ±3 Standard Deviations from the Center Line
Background
Standard Error versus Standard Deviation
Within- versus between-Subgroup Variation: How Control Charts Work
I Chart of Individuals

Myth 16: Control Chart Limits Are Empirical Limits
Background
Definition of Empirical
Empirical Limits versus Limits Justified Empirically
Shewhart’s Evidence of Limits Being Empirical
Wheeler’s Empirical Rule
Empirical Justification for a Purpose

Myth 17: Control Chart Limits Are Not Probability Limits
Background
Association of Probabilities and Control Chart Limits
Can Control Limits Be Probability Limits?
False Alarm Rates for All Special Cause Patterns
Wheeler Uses Probability Limits
Other Uses of Probability Limits

Myth 18: ±3 Sigma Limits Are the Most Economical Control Chart Limits
Background
Evidence for 3–Standard Error Limits Being Economically Best
Evidence against 3–Standard Error Limits Being the Best Economically
Counterexamples: Simple Cost Model Other Out-of-Control Rules—Assignable Causes Shewhart Didn’t Find but Exist
Small Changes Are Not Critical to Detect versus Taguchi’s Loss Function
Importance of Subgroup Size and Frequency on Economic Value of Control Chart Limits
Purpose to Detect Lack of Control—3–Standard Error Limits Misplaced

Myth 19: Statistical Inferences Are Inductive Inferences
Background
Reasoning: Validity and Soundness
Induction versus Deduction
Four Cases of Inductive Inferences
Statistical Inferences: Probability Distributions
Inferences about Population Parameters
Deductive Statistical Inferences: Hypothesis Testing
Deductive Statistical Inferences: Estimation
Real-World Cases of Statistical Inferences

Myth 20: There Is One Universe or Population If Data Are Homogeneous
Background
Definition of Homogeneous
Is Displaying Stability Required for Universes to Exist?
Are There Always Multiple Universes If Data Display Instability?
Is There Only One Universe If Data Appropriately Plotted Display Stability?
Control Chart Framework: Valid and Invalid Conclusions

Myth 21: Control Charts Are Analytic Studies
Background
Enumerative versus Analytic Distinguishing Characteristics
Enumerative Problem, Study, and Solution
Analytic Problem, Study, and Solution
Procedures for Enumerative and Analytic Studies
Are Control Charts Enumerative or Analytic Studies?
Cause–Effect Relationship
An Analytic Study Answers "When?"

Myth 22: Control Charts Are Not Tests of Hypotheses
Background
Definition and Structure of Hypothesis Test
Control Chart as a General Hypothesis Test
Statistical Hypothesis Testing: Alpha and p
Analysis of Means
Shewhart’s View on Control Charts as Tests of Hypotheses
Deming’s Argument: No Definable, Finite, Static Population
Woodall’s Two Phases of Control Chart Use
Finite, Static Universe
Control Charts as Nonparametric Tests of Hypotheses
Utility of Viewing Control Charts as Statistical Hypothesis Tests
Is the Process in Control? versus What Is the Probability the Process Changed?

Myth 23: Process Needs to Be Stable to Calculate Process Capability
Background
Stability and Capability: Dependent or Independent?
Actual Performance and Potential Capability versus Stability
Process Capability: Reliability of Estimates
Control Charts Are Fallible
Capable: 100% or Less than 100% Meeting Specifications
Process Capability: "Best" Performance versus Sustainability
Cp versus P/T
Random Sampling
Response Surface Studies

Myth 24: Specifications Don’t Belong on Control Charts
Background
Run Charts
Charts of Individual Values
Confusion Having Both Control and Specification Limits on Charts
Stability, Performance, and Capability
Specifications on Averages and Variation

Myth 25: Identify and Eliminate Assignable Causes of Variation
Background
Assignable Causes versus Process Change
Is Increase in Process Variation Always Bad?
Good Assignable Causes

Myth 26: Process Needs to Be Stable before You Can Improve It
Background
History of Improvement before the 1920s
Control Chart Fallibility
Stabilizing a Process and Improving It
Stability Required versus Four States of a Process
Shewhart’s Counterexample

Myth 27: Stability (Homogeneity) Is Required to Establish a Baseline
Background
Purpose of Baseline
Just-Do-It Projects
Natural Processes
Processes Whose Output We Want to Be "Out of Control"
Meaning of "Meaningless"
Daily Comparisons
"True" Process Average: Process, Outputs, Characteristics, and Measures
Ways to Compare
Universe or Population and Descriptive Statistics
Random Sampling
When Is Homogeneity/Stability Not Required or Unimportant?

Myth 28: A Process Must Be Stable to Be Predictable
Background
Types of Predictions: Interpolation and Extrapolation
Interpolation: Stability versus Instability
Conditional Predictions
Extrapolation: Stability versus Instability
Fallibility of Control Chart Stability
Control Charts in Daily Life
Statistical versus Causal Control

Myth 29: Adjusting a Process Based on a Single Defect Is Tampering, Causing Increased Process Variation
Background
Definition of Tampering Zero versus One versus Multiple Defects to Define Tampering
Role of Theory and Understanding When Adjusting
Defects Arise from Special Causes: Anomalies
Control Limits versus Specification Limits
Actions for Common Cause Signals versus Special Cause Signals
Is Reducing Common Cause Variation Always Good?
Fundamental Change versus Tampering
Funnel Exercise: Counterexample

Myth 30: No Assumptions Required When the Data Speak for Themselves
Background
Math and Descriptive Statistics: Adding versus Aggregating
Inferences versus Facts: Conditions for Paradoxes
Assumptions for Modeling
Assumptions for Causal Inferences
Assumptions for Inferences from Reasons

Epilogue

References

Index

...

## Author(s)

### Biography

Kicab Castaneda-Mendez, founder of Process Excellence Consultants, Chapel Hill, NC, provides consulting and training on operational excellence using lean Six Sigma methodologies, balanced scorecard and Baldrige framework.