Study guide
6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape
ASQ Certified Six Sigma Black Belt CSSBB
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1 Chapter 1: Introduction & CSSBB Exam Strategy
1.1 CSSBB Exam Format, Scoring, and Testing Windows 1.2 Eligibility, Project Affidavits, Application, and Recertification 1.3 Open-Book Strategy, Reference Materials, and a 16-Week Study Plan
2 Chapter 2: Organization-Wide Planning and Deployment
2.1 Fundamentals of Six Sigma and Lean Methodologies 2.2 Selecting the Right Methodology: DMAIC, DFSS, and Lean 2.3 Business Systems, Processes, and Stakeholder Relationships 2.4 Strategic Planning and Deployment of Six Sigma Initiatives 2.5 Six Sigma Roles and Responsibilities 2.6 Organizational Barriers and Common Causes of Six Sigma Failure 2.7 Change Management: Stakeholder Analysis, Readiness, and Communication
3 Chapter 3: Organizational Process Management and Measures
3.1 Impact of Six Sigma Projects on Customers, Suppliers, and Stakeholders 3.2 Benchmarking Types and Performance Goal Setting 3.3 Performance Measures: Balanced Scorecard, KPIs, KBIs, OKRs, and Line of Sight 3.4 Leading and Lagging Indicators and Performance Dashboards 3.5 Financial Measures: Revenue Growth, Margin, NPV, ROI, and Cost-Benefit Analysis 3.6 Cost of Poor Quality and Hard versus Soft Savings
4 Chapter 4: Team Management
4.1 Team Types and Constraints 4.2 Team Roles, Responsibilities, and Member Selection 4.3 Team Success Factors and Ground Rules 4.4 Motivational Techniques and Overcoming Demotivators 4.5 Stages of Team Development and Conflict Resolution 4.6 Team Communication Plans 4.7 Team Leadership Models: Direct, Coach, Support, and Delegate 4.8 Group Behaviors, Conflict Resolution, and Meeting Management 4.9 Facilitation Techniques and Team Decision-Making Methods 4.10 Team Training: Needs Assessment, Delivery, and Evaluation
5 Chapter 5: Define Phase
5.1 Customer Identification and Segmentation 5.2 Customer Data Collection: Methods, Validity, and Reliability 5.3 Customer Requirements: CTX, CTQ Trees, and Operational Definitions 5.4 The Kano Model and Quality Function Deployment 5.5 SIPOC and Process Mapping 5.6 Business Case Justification 5.7 The Project Charter and the Problem Statement 5.8 Project Scope and Boundaries 5.9 SMART Goals and Project Performance Measurements 5.10 Project Charter Reviews and Toll-Gate Governance 5.11 Project Management Tools: Gantt Charts, Toll-Gate Reviews, WBS, and RACI 5.12 Analytical Tools Used Throughout the DMAIC Cycle
6 Chapter 6: Measure Phase
6.1 Process Flow Metrics and the Hidden Factory 6.2 Process Analysis Tools and Value Stream Mapping 6.3 Types of Data and Measurement Scales 6.4 Sampling Concepts, Methods, and Sample Size for Estimation 6.5 Data Collection Plans and Common Pitfalls 6.6 Gage R&R for Variable Measurement Systems 6.7 Attribute Agreement Analysis and Audit MSA 6.8 Measurement Systems Across the Organization, and Metrology 6.9 Population Parameters, Sample Statistics, and Valid Statistical Conclusions 6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape 6.11 Graphical Methods for Exploratory Data Analysis 6.12 Probability Concepts, Common Distributions, and the Central Limit Theorem 6.13 Additional Distributions: Hypergeometric, Exponential, Lognormal, Weibull, and Bivariate 6.14 Process Capability and Performance Indices 6.15 Capability Studies, Attributes Capability, Yield Metrics, and Long-Term Capability
7 Chapter 7: Analyze Phase
7.1 The Correlation Coefficient, Its Confidence Interval, and Causation 7.2 Linear Regression and Residual Analysis 7.3 Multivariate Tools: Factor Analysis, Discriminant Analysis, and MANOVA 7.4 Hypothesis Testing Terminology, Errors, and Single-Sample Tests 7.5 Statistical Versus Practical Significance 7.6 Sample Size for Hypothesis Tests 7.7 Point Estimates, Confidence, Prediction, and Tolerance Intervals 7.8 Tests for Means, Variances, and Proportions 7.9 Analysis of Variance 7.10 Chi-Square Goodness-of-Fit and Tests of Independence 7.11 Types of Risk and Failure Mode and Effects Analysis 7.12 Gap Analysis and Root Cause Analysis 7.13 Waste Analysis: The Seven Classic Wastes and Resource Under-Utilization
8 Chapter 8: Improve Phase
8.1 Design of Experiments: Objectives, Terminology, and Effects 8.2 DOE Design Principles: Randomization, Blocking, Replication, and Resolution 8.3 Planning and Evaluating Designed Experiments 8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs 8.5 Full Factorial Experiments and Design Selection 8.6 Two-Level Fractional Factorial Experiments and Confounding 8.7 Response Surface Methodology and Optimization 8.8 Waste Elimination: Pull Systems, Kanban, 5S, and Poka-Yoke 8.9 Cycle-Time Reduction: Continuous Flow, SMED, and Heijunka 8.10 Kaizen, Kaizen Blitz, Theory of Constraints, and OEE 8.11 Implementation Planning, Piloting, and Solution Selection
9 Chapter 9: Control Phase
9.1 SPC Objectives, Variable Selection, and Rational Subgrouping 9.2 Control Charts for Variables Data 9.3 Control Charts for Attributes Data 9.4 Short-Run SPC and Moving Average Charts 9.5 Control Chart Analysis: Common Cause, Special Cause, and Run Rules 9.6 Total Productive Maintenance and Visual Controls 9.7 Measurement System Reanalysis 9.8 Control Plan Development and Process Owner Handoff 9.9 Lessons Learned, Documentation, and Training for Sustainment 9.10 Ongoing Evaluation of the Improved Process
10 Chapter 10: Design for Six Sigma (DFSS)
10.1 DFSS Frameworks: DMADV, DMADOV, and Design Transfer 10.2 Design for X (DFX) 10.3 Robust Design, Parameter Design, and Verification 10.4 Tolerance Design and Statistical Tolerancing
6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape
Key Takeaways Explain This Topic AI
Data is classified broadly into Continuous (Variable, infinite scale) and Discrete (Attribute, counted integers or categories), where continuous data provides higher statistical power per sample size.
Measurement scales follow the NOIR taxonomy: Nominal (unordered labels), Ordinal (ordered ranks with unequal intervals), Interval (equal scale intervals with arbitrary zero), and Ratio (equal scale intervals with absolute zero).
Sample variance uses Bessel's correction (n - 1 degrees of freedom) in the denominator to provide an unbiased estimator of population variance.
The median is robust against extreme outliers and skewed distributions, making it the preferred measure of central location for highly non-normal data.
Last updated: August 2026
In the Measure phase of a Six Sigma project, numerical data serves as the foundation for evaluating process performance, identifying sources of variation, and establishing baseline capability. A primary responsibility of a Black Belt is to classify data correctly according to its statistical characteristics and select appropriate descriptive statistics to summarize process performance without introducing distortion.
Measures of Central Tendency (Location)
Central tendency describes the central point or typical value around which process observations cluster:
1. Sample Mean (Arithmetic Average)
The sample mean x ˉ \bar{x} x ˉ is the sum of all individual observations divided by the sample size n n n : x ˉ = 1 n ∑ i = 1 n x i \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i x ˉ = n 1 i = 1 ∑ n x i
Properties : Includes every data point in its calculation; mathematically tractable. Highly sensitive to extreme outliers or skewed distributions, which pull the mean toward the long tail.
2. Sample Median ( x ~ \tilde{x} x ~ or Q 2 Q_2 Q 2 )
The median is the middle value of a dataset arranged in ascending numerical order ( 50 th 50^{\text{th}} 5 0 th percentile).
For an odd sample size n n n , the median is the exact center observation at position n + 1 2 \frac{n+1}{2} 2 n + 1 .
For an even sample size n n n , the median is the arithmetic mean of the two middle observations at positions n 2 \frac{n}{2} 2 n and n 2 + 1 \frac{n}{2} + 1 2 n + 1 .
Properties : Non-parametric metric. Highly robust against outliers and distribution skewness.
3. Mode
The mode is the most frequently occurring value in a dataset.
Properties : Can be used with nominal categorical data. A dataset may be unimodal, bimodal, multimodal, or have no mode.
- Distribution Skewness and Central Tendency Relationships
- Symmetric Distribution : Mean = Median = Mode \text{Mean} = \text{Median} = \text{Mode} Mean = Median = Mode
Right-Skewed (Positive Skew) : Tail extends toward larger values → Mode < Median < Mean \rightarrow \text{Mode} < \text{Median} < \text{Mean} → Mode < Median < Mean
Left-Skewed (Negative Skew) : Tail extends toward smaller values → Mean < Median < Mode \rightarrow \text{Mean} < \text{Median} < \text{Mode} → Mean < Median < Mode
Measures of Dispersion (Variability / Spread)
Dispersion quantifies the magnitude of variation or scatter within a process dataset:
1. Range ( R R R )
The difference between the maximum and minimum observed values: R = x max − x min R = x_{\max} - x_{\min} R = x m a x − x m i n . Range is simple to calculate but volatile, as it depends entirely on the two extreme observations.
2. Sample Variance ( s 2 s^2 s 2 ) & Bessel's Correction
Sample variance measures the average squared deviation of observations from the sample mean. The sample variance s 2 s^2 s 2 is defined as: s 2 = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} s 2 = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
Why Divide by ( n − 1 ) (n - 1) ( n − 1 ) Instead of n n n ?
When estimating population variance σ 2 \sigma^2 σ 2 from a sample, using n n n in the denominator results in a biased estimator that systematically underestimates population variance because deviations are calculated around the sample mean x ˉ \bar{x} x ˉ rather than the true population mean μ \mu μ . Dividing by n − 1 n - 1 n − 1 (known as Bessel's Correction ) adjusts for the loss of one degree of freedom, yielding an unbiased estimator such that the expected value E [ s 2 ] = σ 2 E[s^2] = \sigma^2 E [ s 2 ] = σ 2 .
3. Sample Standard Deviation ( s s s )
The positive square root of sample variance, returning the dispersion metric to the original units of measurement: s = s 2 = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}} s = s 2 = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
4. Interquartile Range (IQR)
The interquartile range measures the spread of the middle 50 % 50\% 50% of the data, calculated as the difference between the third quartile ( Q 3 Q_3 Q 3 , 75 th 75^{\text{th}} 7 5 th percentile) and first quartile ( Q 1 Q_1 Q 1 , 25 th 25^{\text{th}} 2 5 th percentile): IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 − Q 1
Like the median, IQR is a non-parametric measure robust against extreme outliers.
Distribution Shape: Skewness and Kurtosis
Central tendency and dispersion describe where a distribution sits and how wide it is. Shape describes how it is distributed within that width, and it determines whether normal-theory methods
apply at all.
Skewness
Skewness measures asymmetry: g 1 = 1 n ∑ ( x i − x ˉ ) 3 s 3 g_1 = \frac{\frac{1}{n}\sum (x_i - \bar{x})^3}{s^3} g 1 = s 3 n 1 ∑ ( x i − x ˉ ) 3
- Skewness Shape Relationship of centre measures Typical Six Sigma examples
- g 1 = 0 g_1 = 0 g 1 = 0 Symmetric mean = median = mode Machined dimensions under control
g 1 > 0 g_1 > 0 g 1 > 0 Right (positive) skew; long upper tail mode < median < mean Cycle times, repair times, waiting times, claim amounts
g 1 < 0 g_1 < 0 g 1 < 0 Left (negative) skew; long lower tail mean < median < mode Yields near an upper bound, exam scores near a ceiling
A rule of thumb: ∣ g 1 ∣ < 0.5 |g_1| < 0.5 ∣ g 1 ∣ < 0.5 is approximately symmetric; 0.5 0.5 0.5 to 1.0 1.0 1.0 is moderately skewed; above 1.0 1.0 1.0 is highly skewed and normal-theory capability calculations should not be applied without
transformation.
Right skew is by far the most common in transactional processes, because time-based measures are bounded
below by zero and unbounded above. That is also why the lognormal distribution fits them so often.
- Kurtosis
- Kurtosis measures tail weight relative to a normal distribution. Excess kurtosis is reported so that
a normal distribution scores zero: g 2 = 1 n ∑ ( x i − x ˉ ) 4 s 4 − 3 g_2 = \frac{\frac{1}{n}\sum (x_i - \bar{x})^4}{s^4} - 3 g 2 = s 4 n 1 ∑ ( x i − x ˉ ) 4 − 3
- Excess kurtosis Name Meaning
- g 2 = 0 g_2 = 0 g 2 = 0 Mesokurtic Normal-like tails
- g 2 > 0 g_2 > 0 g 2 > 0 Leptokurtic Heavier tails; more extreme values than normal predicts
- g 2 < 0 g_2 < 0 g 2 < 0 Platykurtic Lighter tails; often indicates a bounded or mixed process
- Leptokurtosis matters practically: heavy tails mean the process produces far more extreme values than a
- normal model predicts, so a capability index computed under the normality assumption understates the
true defect rate , sometimes by an order of magnitude.
- Neither statistic replaces a plot. A histogram plus a normal probability plot shows skew, tail weight,
- bimodality, and truncation at once, and bimodality -- which usually means two processes mixed together --
is invisible in the skewness and kurtosis values.
- Relative Dispersion: the Coefficient of Variation
- Standard deviation is expressed in the units of the measurement, so it cannot be compared across
characteristics with different units or very different magnitudes. The coefficient of variation normalizes it: C V = s x ˉ × 100 % CV = \frac{s}{\bar{x}} \times 100\% C V = x ˉ s × 100%
Two uses in Six Sigma work:
Comparing variability across dissimilar characteristics. A fill weight with s = 1.2 s = 1.2 s = 1.2 g on a mean
of 500 g ( C V = 0.24 % CV = 0.24\% C V = 0.24% ) is far better controlled than a cycle time with s = 1.2 s = 1.2 s = 1.2 minutes on a mean of
8 minutes ( C V = 15 % CV = 15\% C V = 15% ), even though the standard deviations are numerically identical.
Comparing the same characteristic across processes running at different levels , such as two lines
producing different sizes of the same part.
- Two cautions. CV is meaningful only on a ratio scale with a true zero -- it is undefined in any
- useful sense for Celsius temperature -- and it becomes unstable when the mean is near zero, because a
small denominator inflates the ratio without limit.