Tailed Hypothesis Tests

A tailed hypothesis tests is an assumption about a population parameter. The assumption may or may not be true.  Hypothesis testing is a key procedure in inferential statistics used to make statistical decisions using experimental data.  It is basically an assumption that we make about the population parameter.

Tailed Hypothesis Tests
Tailed Hypothesis Tests. Photo by Bernal Saborio

Null and alternative hypothesis

Null hypothesis (H0): A statistical hypothesis assumes that the observation is due to the chance factor. In other words, the null hypothesis states that there is no (statistical significance) difference or effect.

Alternative hypothesis (H1): The complementary hypothesis of the null hypothesis is an alternative hypothesis. In other words, the alternative hypothesis shows that observations are the results of a real effect.

What are the tails in a hypothesis test?

A tail in hypothesis testing refers to the tail at either end of a distribution curve. Generally, in hypothesis tests, test statistic means to obtain all of the sample data and convert it to a single value. For example, Z-test calculates Z statistics, t-test calculates t-test statistic, and F-test calculates F values etc., are the test statistics. Test statistics need to compare to an appropriate critical value. A decision can then be made to reject or not reject the null hypothesis.

In probability distribution plots, the shaded area in the plot (one side in one-tailed hypothesis and two sides in a two-tailed hypothesis) indicates the probability of a value falls within that range.

Critical region: In a hypothesis test, critical regions are ranges of the distributions where the values represent statistically significant results. If the test statistic falls in the critical region, reject the null hypothesis.

Tailed Hypothesis Tests

Types of tailed hypothesis tests

There are three basic types of ‘tails’ that hypothesis tests can have:

  • One-tailed test
    • Left-tailed test: where the alternative hypothesis includes a ‘<’ symbol.
  • Two-tailed test: where the alternative hypothesis includes a ≠.

One-tailed hypothesis tests

A test of hypothesis where the area of rejection is only in one direction. In other words, when change is expected to have occurred in one direction, i.e expecting output either increase or to decrease.

If the level of significance is 0.05, a one-tail test allots the entire alpha (α) in the one direction to test the statistical significance. Since the statistical significance in the one direction of interest, it is also known as a directional hypothesis.

Reject the null hypothesis; If the test statistic falls in the critical region, that means the test statistic has a greater value than the critical value (for the right-tailed test) and the test statistic has a lesser value than the critical value (for the left tailed test).

Generally, one-tailed tests are more powerful than two-tailed tests; because of that, one-tailed tests are preferred.

The basic disadvantage of a one-tailed test is it considers effects in one direction only. There is a chance that an important effect may miss in another direction. For example, a new material used in the production and checking whether the yield improved over the existing material. There is a possibility that new material may give less yield than the current material.

One tailed tests are further divided into

  • Right-tailed test
  • Left-tailed test

Right-tailed test

Right tailed test is also called the upper tail test. A hypothesis test is performed if the population parameter is suspected to be greater than the assumed parameter of the null hypothesis.

  • H0: The population mean (µ) is less than or equal to the hypothesized value (µ0).
  • H1: The population mean (µ) is greater than µ0.
Tailed Hypothesis Tests

Example: The average weight of an iron bar population is 90lbs. Supervisor believes that the average weight might be higher. Random samples of 5 iron bars are measured, and the average weight is 110lbs and a standard deviation of 18lbs. With a 95% confidence level, is there enough evidence to suggest the average weight is higher?

  • Population average score (µ) = 90
  • Sample average (x̅) = 110
  • s =18
  • number of samples (n) = 5
  • Level of significance α=0.05

H0: The average weight is equal to 90, µ=90.

H1: The average score is higher than 90, µ>90

Since supervisor is keen to check the average weight is higher, it is a right-tailed test.

α=0.05

Compute the critical value: For 95% confidence level t value with a degrees of freedom n-1= 2.132

Critical value =2.132

Calculate the test statistics t = x̅-µ/(s/√n) = 110-90/(18/√5)=2.484.

Tailed Hypothesis Tests

Conclusion: Test statistic is greater than the critical value, and it is in the rejection region. Hence, we can reject the null hypothesis. So the average weight of the iron bar is may be higher than the 90lbs.

Left-tailed test

Left-tailed test is also known as a lower tail test. A hypothesis test is performed if the population parameter is suspected to be less than the assumed parameter of the null hypothesis.

  • H0: The population mean (µ) is greater than or equal to the hypothesized value (µ0).
  • H1: The population mean (µ) is less than µ0.

Example: The average weight of an iron bar population is 90lbs. Supervisor believes that the average weight might be lower. Random samples of 6 iron bars are measured, and the average weight is 82lbs and a standard deviation of 18lbs. With a 95% confidence level, is there enough evidence to suggest the average weight is lower?

Population average score (µ) = 90

  • Sample average (x̅) = 82
  • s =18
  • number of samples (n) = 6
  • Level of significance α=0.05

H0: The average weight is equal to 90, µ=90.

H1: The average score is less than 90, µ<90

Since the supervisor is keen to check the average weight is lower, hence it is a left-tailed test.

α=0.05

Compute the critical value: For 95% confidence level t value with a degrees of freedom n-1= -2.015

Critical value =-2.015

Calculate the test statistics t = x̅-µ/(s/√n) = 82-90/(18/√6)=-1.088

Conclusion: Test statistic is not in the rejection region. Hence, we failed to reject the null hypothesis. So the average weight of the iron bar is 90lbs.

Two-tailed hypothesis tests

A test of hypothesis where the area of rejection is on both sides of the sampling distribution.

If level of significance is 0.05 of a two-tailed test, it distributes the alpha (α) into two equal parts (α/2 & α/2) on both sides to test the statistical significance. 

  • H0: The population mean (µ) is equal to the hypothesized value (µ0).
  • H1: The population mean (µ) is not equal to µ0.
Tailed Hypothesis Tests

Two-tailed tests also known as two-sided or non-directional test, as it tests the effects on both sides. In a two-tailed test, extreme values above or below are evidence against the null hypothesis.

Reject the null hypothesis if test statistics fall on either side of the critical region.

Example: The average score for the mean population is 80, with a standard deviation of 10. With a new training method, the professor believes that the score might change. Professor tested randomly 36 students’ scores. The average score of the sample is 88. With a 95% confidence level, is there enough evidence to suggest the average score changed?

Population average score (µ) = 80

  • Sample average (x̅) = 88
  • σ =10
  • number of samples (n) = 36
  • Level of significance α=0.05

H0: The average score is equal to 80, µ=80.

H1: The average score is not equal to 80, µ≠80

Since the professor is keen to check the change in average score, it is a two tail test

α/2=0.025

Compute the critical value: For 95% confidence level Z value = 1.96

Critical value =±1.96

Calculate the test statistics Z = x̅-µ/(σ/√n) = 88-80/(10/√36)=4.8

Conclusion: The test statistic is greater than the critical value, which means the test statistic is in the rejection region. So, we can reject the null hypothesis.

Few more Example

For Example, consider a null hypothesis that states that cars traveling on a particular road have a mean velocity of 40 miles/hour:

  • A right-tailed test would state that cars traveling on a particular road have a mean velocity greater than 40 miles/hour.
  • A left-tailed test would state that cars traveling on a particular road have a mean velocity less than 40 miles/hour.
  • A two-tailed test would state that cars traveling on a particular road have a mean velocity greater than or less than 40 miles/hour.

Additional Resources:

Important Videos

ASQ Six Sigma Green Belt Study Guide

Preparing for the ASQ Six Sigma Green Belt exam can be overwhelming. This ASQ Six Sigma Green Belt Study Guide lists all key concepts (and links to learn more) for the ASQ Six Sigma Green Belt Certification. This list comes from the 2022 update to the official ASQ Green Belt BOK. And it’s totally free to use.

What is ASQ Six Sigma Green Belt BOK

The ASQ Six Sigma Green Belt BOK is created by the American Society of Quality (ASQ). It’s periodically reviewed and updated by leading members and Six Sigma professionals to ensure it’s relevant to today’s world.

The ASQ Six Sigma Green Belt BOK has six sections geared around Six Sigma Fundamentals and the different phases of DMAIC.

You’ll need to master various topics in order to pass the exam.

How to use the ASQ Six Sigma Green Belt Study Guide

Some ASQ Six Sigma Green Belt candidates will have taken Six Sigma training. Others will be self-taught. No matter your path, you will have to know what’s on the standard BOK – Body of Knowledge.

Step 1: Scan over the material below.

Scan through the rest of this page before proceeding further. Take a look at what will be asked. Do you feel comfortable with this material? Is anything new to you? Are there any surprises?

Make a mental note of that and proceed.

Step 2: Review (or learn!) the material.

Not every training program covers every topic. Depending on the quality of the training you took, your instructor may have skipped a section entirely! Click one of the links to dive into any topic that you don’t feel as strongly as you’d like and read up.

Step 3: Leave a comment on any page where you have questions.

Each topic below is linked to a detailed article where I thoroughly review the concept. If you have any questions, or if any topic is less clear than you’d like it to be, leave a comment, and I’ll get back to you ASAP.

Step 4: Practice as many questions as possible.

Find an excellent question bank covering each section, and take as many practice tests as possible.

The ASQ Six Sigma Green Belt BOK

What follows below is the ASQ Six Sigma Green Belt BOK. The top-level sections I through VI also list the number of questions you can expect in that section. This will help you see the relative weight of those topics compared to others.

I Overview: Six Sigma and the Organization (11 Questions)

I.A Six Sigma and organizational goals

I.A.1 Value of Six Sigma

Recognize why organizations use Six Sigma, how they apply their philosophy and goals, and the evolution of Six Sigma from quality leaders such as (Juran, Deming, Shewhart, Ishikawa, and others.).

I.A.2 Organizational goals and Six Sigma projects

Identify the linkages and supports that need to be established between a selected Six Sigma project and the organization’s goals, including SMART goals, and describe how process inputs, outputs, and feedback at all levels can influence the organization as a whole. (Understand)

I.A.3 Organizational drivers and metrics

Recognize key drivers for business (profit, market share, customer satisfaction, efficiency, product differentiation, key performance indicators (KPIs)) for all types of organizations. Understand how key metrics and scorecards are developed and how they impact the entire organization. (Understand)

I.B Lean principles in the organization

I.B.1 Lean Concepts

Define and describe lean concepts such as the theory of constraints, value chain, flow, takt time,
just-in-time (JIT), Gemba, spaghetti diagrams, and perfection. (Apply)

I.B.2 Value-stream mapping

Use value-stream mapping to identify value-added processes and steps or processes that
produce waste, including excess inventory, unused space, test inspection, rework,
transportation, and storage. (Understand)

I.C Design for Six Sigma (DFSS) methodologies

I.C.1 Road maps for DFSS

Describe and distinguish between DMADV (define, measure, analyze, design, verify) and IDOV (identify, design, optimize, verify), and identify how they relate to DMAIC. Describe how these methodologies are used for improving the end product or process during the design (DFSS) phase. Understand how verification and validation are used to compare results against stated goals.

I.C.2 Basic failure mode and effects analysis (FMEA)

Use FMEA to evaluate a process or product and determine what might cause it to fail and the effects that failure could have. Identify and use scale criteria, calculate the risk priority number (RPN), and analyze the results. (Analyze)

I.C.3 Design FMEA and process FMEA

Define and distinguish between these two uses of FMEA. (Apply)

II Define Phase (20 Questions)

II.A Project identification

II.A.1 Project selection

Describe the project selection process and what factors should be considered in deciding whether to use the Six Sigma DMAIC methodology or another problem-solving process. (Understand)

II.A.2 Process elements

Define and describe process components and boundaries. Recognize how processes cross various functional areas and the challenges that result from process improvement efforts. (Analyze)

II.A.3 Benchmarking

Understand various types of benchmarking, including competitive, collaborative, and best practices. (Understand)

II.A.4 Process inputs and outputs

Identify process input variables and process output variables and document their relationships using the supplier, inputs, process, output, and customer (SIPOC) model. (Apply)

II.A.5 Owners and stakeholders

Identify process owners and other stakeholders in a project. (Apply)

II.B Voice of the Customer (VOC)

II.B.1 Customer Identification

Identify the internal and external customers of a project, and what effect the project will have on them. (Apply)

II.B.2 Customer data

Collect feedback from customers using surveys, focus groups, interviews, and various forms of observation. Identify the key elements that make these tools effective. Review data collection questions to eliminate vagueness, ambiguity, and any unintended bias. (Apply)

II.B.3 Customer requirements

Use quality function deployment (QFD), Critical to X (CTX when ‘X’ can be quality, cost, safety, etc.), Critical to Quality tree (CTQ), and Kano model to translate customer requirements statements into product features, performance measures, or opportunities for improvement. Use weighting methods as needed to amplify the importance and urgency of different kinds of input: telephone call vs. survey response, product complaint vs. expedited service request. (Apply)

II.C Project management basics

II.C.1 Project Methodology

Define and apply agile and top-down management methods. (Apply)

II.C.2 Project charter

Define and describe elements of a project charter and develop a problem statement, including baseline or current status to be improved and the project’s goals. (Apply)

II.C.3 Project scope

Help define the scope of the project using process maps, Pareto charts, and other quality tools. (Apply)

II.C.4 Project metrics

Help develop primary metrics (reduce defect levels by x amount) and consequential metrics (the negative effects that making the planned improvement might cause). (Apply)

II.C.5 Project planning tools

Use work breakdown structures (WBS), Gantt charts, critical path method (CPM), program evaluation and review technique (PERT) charts, and toll-gate reviews to plan projects and monitor their progress.

II.C.6 Project documentation

Describe the types of data and input needed to document a project. Identify and help develop appropriate presentation tools (storyboards, spreadsheet summary of results) for phase reviews and management updates. (Apply)

II.C.7 Project risk analysis and management

Describe the elements of project risk analysis, including feasibility, potential impact, risk priority number (RPN), and risk management. Identify the potential effect risk can have on project goals and schedule, resources (materials and personnel), business continuity planning, costs and other financial measures, and stakeholders.

II.C.8 Project closure

Review the project objectives achieved in relation to the project charter with team members and sponsors and ensure that documentation is completed and stored appropriately. Identify lessons learned and inform other parts of the organization about opportunities for improvement. (Apply)

II.D Management and planning tools

Define, select, and use 1) affinity diagrams, 2) interrelationship digraphs, 3) tree diagrams, 4) prioritization matrices, 5) matrix diagrams, 6) process decision program (PDPC) charts, 7) activity network diagrams, and SWOT analysis. (Apply)

II.E Business results for projects

II.E.1 Process performance

Calculate process performance metrics such as defects per unit (DPU), rolled throughput yield (RTY), cost of poor quality (COPQ), defects per million opportunities (DPMO), sigma levels, and process capability indices. Track process performance measures to drive project decisions. (Analyze)

II.E.2 Communication

Define and describe communication techniques used in organizations: top-down, bottom-up, and horizontal. (Apply)

II.F Team dynamics and performance

II.F.1 Team stages and dynamics

Define and describe the stages of team evolution, including forming, storming, norming, performing, adjourning, and recognition. Identify and help resolve negative dynamics such as overbearing, dominant, or reluctant participants, the unquestioned acceptance of opinions as facts, groupthink, feuding, floundering, the rush to accomplishment, attribution, discounts, plops, digressions, and tangents. (Understand)

II.F.2 Team roles and responsibilities

Use tools such as RACI to describe and define the roles and responsibilities of participants on Six Sigma and other teams, including black belt, master black belt, green belt, champion, executive, coach, facilitator, team member, sponsor, process owner, etc. (Apply)

II.F.3 Team tools

Define and apply team tools such as brainstorming and decision-making concepts such as nominal group technique and multi-voting. (Apply)

II.F.4 Team Communication

Identify and use appropriate communication methods (both within the team and from the team to various stakeholders) to report progress, conduct reviews, and support the overall success of the project. (Apply)

III Measure Phase (20 Questions)

III.A Process analysis and documentation

Develop process maps and review written procedures, work instructions, and flowcharts to identify any gaps or areas of the process that are misaligned. (Create)

III.B Probability and statistics

Rusty on the math behind these topics? Join the Math Confidence for Six Sigma waitlist and I’ll let you know when the course opens.

III.B.1 Basic probability concepts

Describe and interpret basic probability concepts: independent events, mutually exclusive events, multiplication rules, permutations, and combinations. (Understand)

III.B.2 Central limit theorem

Define the central limit theorem and describe its significance in relation to confidence intervals, hypothesis testing, control charts, etc. (Apply)

III.C Statistical distributions

Define and describe various distributions as they apply to statistical process control and probability: normal, binomial, Poisson, chi-square, Student’s t, and F distributions. (Apply)

III.D Collecting and summarizing data

III.D.1 Types of data and measurement scales

Identify and classify continuous (variables) and discrete (attributes) data. Describe and define
nominal, ordinal, interval, and ratio measurement scales.
(Analyze)

III.D.2 Sampling and data collection plans and methods

Define and apply various sampling methods (random and stratified) and data collection methods (check sheets and data coding). Prepare data collection plans that include gathering data and performing quality checks (e.g., minimum/maximum values, erroneous data, null values). (Apply)

III.D.3 Descriptive statistics

Define, calculate, and interpret measures of dispersion and central tendency. Develop and interpret frequency distributions and cumulative frequency distributions. (Evaluate) Also, see Kinds of Statistics.

III.D.4 Graphical methods

Depict relationships by constructing, applying, and interpreting diagrams and charts such as stem-and-leaf plots, box-and-whisker plots, run charts, scatter diagrams, Pareto charts, etc. Depict distributions by constructing, applying, and interpreting diagrams such as histograms, normal probability plots, etc. (Create)

III.E Measurement system analysis (MSA)

Calculate, analyze, and interpret measurement system capability using gauge repeatability and reproducibility (GR&R) studies, measurement correlation, bias, linearity, percent agreement, and precision/tolerance (P/T). (Evaluate)

III.F Process and Performance Capability

III.F.1 Process performance vs. specification

Define and distinguish between natural process limits and specification limits, and calculate process performance metrics. (Evaluate)

III.F.2 Process Capability Studies (aka Capability Analysis)

Define, describe, and conduct process capability studies, including identifying characteristics, specifications, and tolerances and verifying stability and normality. (Evaluate)

III.F.3 Process capability (Cp, Cpk) and process performance (Pp, Ppk) indices

Describe the relationship between these types of indices. Define, select, and calculate process capability and process performance. Describe when Cpm measures can be used. Calculate the sigma level of a process. (Evaluate)

III.F.4 Short-term vs. long-term capability

Describe the assumptions and conventions that are appropriate when only short-term data are collected and when only attribute data are available. Describe the changes in relationships that occur when long-term data are used, and interpret the relationship between long- and short-term capability as it relates to a 1.5 sigma shift. (Evaluate)

IV Analyze Phase (18 Questions)

IV.A Exploratory data analysis

IV.A.1 Multi-vari studies

Select appropriate sampling plans to create multi-vari study charts and interpret the results for
positional, cyclical, and temporal variation. (Create)

IV.A.2 Correlation and linear regression

Describe the difference between correlation and causation. Calculate the correlation
coefficient and linear regression and interpret the results in terms of statistical significance (p-value). Use regression models for estimation and prediction. (Evaluate)

IV.B Hypothesis testing

IV.B.1 Basics

Define and distinguish between statistical and practical significance and apply tests for significance level, power, type I, and type II errors. Determine the appropriate sample size for various tests. (Apply).

IV.B.2 Tests for means, variances, and proportions

Conduct hypothesis tests to compare means, variances, and proportions (paired-comparison t-test, F-test, analysis of variance (ANOVA), Chi-square) and interpret the results. (Analyze)

IV.C Additional Analysis Methods

IV.C.1 Gap analysis

Analyze scenarios to identify performance gaps and compare current and future states using predefined metrics.

IV.C.2 Root cause analysis

Use cause and effect diagrams, relational matrices, 5 Whys, fault tree analysis, and other problem-solving tools to identify the true cause of a problem. (Analyze)

V Improve Phase (16 Questions)

V.A Design of experiments (DOE)

V.A.1 Basic terms

Define and describe terms such as independent and dependent variables, factors and levels, response, treatment, error, repetition, and replication. (Understand)

V.A.2 DOE graphs and plots

Interpret main effects analysis and interaction plots. (Apply)

V.B Implementation planning

Apply implementation planning by using proof of concept, try-storming, simulations, and conducting pilot tests. (Apply)

V.C Lean tools

V.C.1 Waste elimination

Select and apply tools and techniques for eliminating or preventing waste, including pull systems, kanban, 5S, standard work, and poka-yoke. (Apply)

V.C.2 Cycle-time reduction

Use various techniques to reduce cycle time (continuous flow, setup reduction), single-minute
exchange of dies (SMED)
. (Analyze)

V.C.3 Kaizen and kaizen blitz

Define and distinguish between these two methods and apply them in various situations. (Apply)

VI Control Phase (15 questions)

VI.A Statistical process control (SPC)

VI.A.1 SPC Basics

Describe the theory and objectives of SPC, including measuring and monitoring process
performance for both continuous and discrete data. Define and distinguish between common
and special cause variation and how these conditions can be deduced from control chart
analysis.
(Analyze)

VI.A.2 Rational subgrouping

Define and describe how rational subgrouping is used. (Understand)

VI.A.3 Control Charts

Identify, select, construct, and use control charts: X Bar−R, X Bar−s, individuals and moving range (ImR / XmR), median, p, np, c, and u. (Apply)

VI.B Sustain improvements

VI.B.1 Control plan

Assist in developing and implementing a control plan to document and monitor the process. (Apply)

VI.B.2 Document control

Understand document control and its role in controlling and sustaining improvements. (Understand)

VI.B.3 Training plans

Develop training plans to implement and sustain improvements. (Apply)

VI.B.4 Audits

Define first-, second-, and third-party audits. (Remember)

VI.B.5 Plan-do-check-act (PDCA)

Apply and distinguish between the steps of plan-do-check-act (PDCA). (Apply)

Six Levels of Cognition based on Bloom’s Taxonomy (Revised)

In addition to content specifics, the subtext detail also indicates the intended complexity level of the test questions for that topic. These levels are based on the Revised “Levels of Cognition” (from Bloom’s Taxonomy, 2001) and are presented below in rank order, from least complex to most complex.

Remember
Be able to remember or recognize terminology, definitions, facts, ideas, materials, patterns, sequences, methodologies, principles, etc. (Also commonly referred to as recognition, recall, or rote knowledge)

Understand
Be able to read and understand descriptions, communications, reports, tables, diagrams, directions, regulations, etc.

Apply
Be able to apply ideas, procedures, methods, formulas, principles, theories, etc., in job-related situations.

Analyze
Be able to break down information into its constituent parts and recognize the parts’ relationship to one another and how they are organized; identify sublevel factors or salient data from a complex scenario.

Evaluate
Be able to make judgments regarding the value of proposed ideas, solutions, methodologies, etc., by using appropriate criteria or standards to estimate accuracy, effectiveness, economic benefits, etc.

Create
Be able to put parts or elements together in such a way as to show a pattern or structure not clearly there before; able to identify which data or information from a complex set is appropriate to examine further or from which supported conclusions can be drawn.

ASQ Six Sigma Green Belt Study Guide Helpful Links:

https://asq.org/cert/resource/pdf/certification/2022-SSGB-BoK-Map.pdf

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Frequency Plots

Frequency plots are helpful for monitoring the spread of continuous data. A frequency plot is a graph showing a variable’s distributional information. In other words, it summarizes the data by plotting how frequently a particular value occurs.

Frequency plots allow you to summarize lots of data in a graphical manner making it easy to see the distribution of that data and process capability, especially when compared to specifications. For example, is our data normally distributed or not?

Frequency Plots can tell us a lot about a data set or a process. For instance, the quantity of occurrences of the response variable is computed for each interval. Now that we have this information, we can create a graph that consists of frequencies on the vertical axis and response variables on the horizontal axis.

frequency plot
Google Ngram Viewer provides an interesting frequency plot. Photo by woodlywonderworks

Why Frequency Plots

Frequency plots reveal the spread, kinds of variation, and center of the data without making us do calculations. They also help detect overlapping processes and sometimes measurement errors.

You generally need 50 data points of continuous, non-time-ordered data. In other words, should NOT use time-ordered data in frequency plots.

Types of Frequency Plots

Histogram

A Histogram is the graphical representation of a frequency distribution. It is in the form of a rectangle with class intervals as bases and the corresponding frequencies as heights. Particularly, there is no gap between any two successive rectangles.

In a Histogram, each bar or column represents the frequency of occurrence of continuous quantitative variables.

Frequency Plots

The basic advantage of Histograms is to display a large amount of data graphically that are difficult to interpret in a tabular form. In addition, it shows the process’s central tendency.

How to Make a Box and Whisker Plot?

  • Firstly, collect the data
  • Secondly, arrange the data points in order and assign categories
  • Thirdly, find the lowest and highest value
  • Fourthly, find the number of intervals
  • You could use 2^k = N where k = # of categories (smallest positive integer) and N = # of data.

The number of categories to use in the histogram depends on the sample size.

  • Now, compute the intervals by covering the entire range of the data set
  • Then, find the frequency of occurrence of each data point as per the interval
  • Finally, draw the bar chart preserving counts and categories

Example: Following are the 15 student’s marks in a science subject. Compute the Histogram

94,72,78,73,82,42,62,56,85,64,75,66,68,76,58

  • Arrange the data in order

42,56,58,62,64,66,68,72,73,75,76,78,82,85,94

  • Find the intervals

  • Finally, draw the bar chart preserving counts and categories

Frequency Plots

Histogram Excel Template


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Dot Plot

A dot plot, also known as a dot matrix, is the most basic frequency plot. It shows the number of times a value occurred in a data set, where every data value is represented by a dot in the graph.

In other words, the number of times a value is repeated corresponds to the number of dots above that particular value or item.

How to Make a Dot Plot?

  • Firstly, collect the data
  • Secondly, organize the data by arranging the numbers in ascending order
  • Then, draw a horizontal number line that corresponds to the range of the data values that covers all the data points
  • Finally, place a dot above the number line to represent each data against the particular value

Example: Below is data showing the number of swimming sessions 25 high school students attended last summer. Draw a dot plot and also identify the session that was attended by the most number of students.

Organize data in ascending order.

Summarize the data values.

Finally, place a dot above the number line to represent each data against the particular value.

Frequency Plots

From the above graph, it is clear that most numbers of students attended the 5th session.

Dot Plot Template

Download a dot plot template here.


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Benefits of Frequency Plots

  • Easy to review lots of data in a graphical manner.
  • It provides the shape, center, and range of the data.
  • Frequency plots also help to identify where the data is located relative to the process specifications or targets.
  • It helps to predict the future performance of the process.

Frequency Plot Videos

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If you’re looking to pass your Six Sigma Green Belt or Black Belt exams, I’d recommend starting with my affordable study guide:

1)→ 🟢Pass Your Six Sigma Green Belt: a study guide organized by the exam Body of Knowledge, 1,000+ practice questions with detailed answer explanations, and 8 full-length mock exams

2)→ ⚫Pass Your Six Sigma Black Belt: the full Black Belt Body of Knowledge, 2,000+ practice questions with detailed answer explanations, and 8 full-length mock exams

You’ve spent so much effort learning Lean Six Sigma. Why leave passing your certification exam up to chance? This comprehensive study guide offers 1,000+ exam-like questions for Green Belts (2,000+ for Black Belts) with full answer walkthroughs, access to instructors, detailed study material, and more.

​ Thousands have used this system to prepare. Join them here. 

Free Six Sigma Green Belt and Black Belt Practice Exams

You want a free Six Sigma Green Belt practice test (or the Black Belt version) so you can see where you stand before you spend months studying. That’s a good instinct. So here’s what I offer for free, what’s in it, and how to use it well.

Create a free account to take the practice exams. Pick the belt you’re studying for:

No credit card needed. Signing up also adds you to my email list, and you can unsubscribe at any time.

What’s in the free Six Sigma Green Belt practice test and Black Belt exams

Free Green Belt account Free Black Belt account
ASQ-style practice exam 11 questions, 45-minute timer 75 questions, 90-minute timer
IASSC-style practice exam 14 questions across Define, Measure, Analyze, Improve and Control 15 questions across Define, Measure, Analyze, Improve and Control
Topic quizzes None 9 short ASQ topic quizzes (Define through Control, DFSS, team management and more)
How-to-study lessons 5 short lessons 5 short lessons
Printable Bodies of Knowledge ASQ, IASSC and Villanova Green Belt None

Every question is scored when you submit. You then see your answer, the right answer and an explanation of why it’s right. Your results are also broken down by topic, so you know where to focus. Finally, you can retake each Six Sigma Green Belt practice test (and each Black Belt exam) as often as you like.

To be clear about what these are: they’re short benchmarks. Most are automated versions of the free sample exams ASQ and IASSC publish, with my explanations added. In other words, they tell you where you stand. However, they are not full-length mock exams. (The paid study guides have those, and there’s more on that at the bottom of this page.)

Empty exam room, the setting a Six Sigma Green Belt practice test prepares you for

Which free account should you pick?

First, pick the belt you’re sitting. For example, Green Belt candidates should start with the Green Belt account. Black Belt candidates, on the other hand, get more from the Black Belt account, because its 75-question ASQ exam covers the most ground of anything in the free set.

Not sure which belt you need yet? Then start with Green Belt. Black Belt exams assume you already know the Green Belt material, so it’s the right baseline either way. Also note one small difference: the Black Belt account asks you to confirm your email address before you start.

How the practice exams line up with ASQ, IASSC and other exams

Each certifying body publishes a Body of Knowledge (BoK). It’s the list of topics its exam can test. The ASQ and IASSC practice exams follow their own BoKs, so the topic breakdown on your results maps straight onto what you’ll study.

Whatever exam you’re sitting, I suggest you also take the ASQ exam for your belt, because it’s the broadest of the free sets.

How to get the most out of a Six Sigma Green Belt practice test

Most people use a practice exam as a final check. However, it’s far more useful at the start. The same steps work for the Black Belt exams too.

  1. First, take one cold, before you study. Your first score is a baseline, not a grade. A low number on day one is normal, and it’s useful.
  2. Next, study by topic, weakest first. The results page splits your score by topic. As a result, that list becomes your study plan for the next few weeks.
  3. Then read every explanation, including the ones you got right. After all, a lucky guess and real understanding look the same on a scorecard.
  4. Finally, retake it later, under the clock. If your topic scores move, your studying is working. If they don’t, change how you study, not just how long.

I call this benchmarking, and it’s how I prepared for my own exams. For more detail, read how the benchmarking approach works, how I practiced for my Six Sigma exams and how to study for Six Sigma certification.

Try two Six Sigma Green Belt practice test questions

For instance, here are two questions from the free IASSC-style Green Belt exam. Below each one, you’ll find the explanation you’d see after you submit.

1. A shoe manufacturer learned through a Lean Six Sigma project that its boot soles could be made from a different material, which removes two steps from the process. Removing those steps saves $7,500 per month. Reported on an annual basis, the financial savings for this project are:

  1. $45,000
  2. $75,000
  3. $90,000
  4. $120,000
Show the answer

C, $90,000. Savings are usually reported per year so they can be compared with project costs and with other projects. $7,500 x 12 months = $90,000. The other options are six, ten and sixteen months of savings. More on this in Financial Measures in Six Sigma.

2. When analyzing sample data, a Belt sees a bimodal distribution in which each mode is roughly normal. The most likely explanation is that the data:

  1. come from one stable process with only common-cause variation
  2. mix output from two distinct sources, such as two machines, shifts or suppliers
  3. were collected from a sample that was too small
  4. are the result of measurement resolution only
Show the answer

B. Two peaks usually mean two processes are mixed in one data set, for example two machines, two shifts or two suppliers’ parts. Stratify the data to separate them. See Bimodal Distribution.

Want the full free set? Create your free account and take the practice exams for your belt.

How other students used practice exams

A few students wrote up how they studied and passed. Each story is different. So find the one that sounds like your situation.

Ready for the full Green Belt or Black Belt study guide?

The free exams show you where you stand. If you want a complete program to close the gaps, however, that’s what my two study guides are for. Both cover ASQ, IASSC, Villanova and other exams, with full answer walkthroughs on the practice questions. Thousands have used this system to prepare.

Pass Your Six Sigma Green Belt Pass Your Six Sigma Black Belt
Practice questions 1,000+ 2,000+
Full-length mock exams 8 8
Price $49 a month, or $247 for a year $74 a month, or $347 for a year
Details See the Green Belt study guide See the Black Belt study guide

Regional pricing: prices adjust to where you live. That’s because checkout applies a regional discount automatically, so the price you pay may be lower than the price above.

Two guarantees: if it’s not the right fit in the first 14 days, you get your money back. And if you complete every lesson for your certification body’s exam, score 90% or better on each of its quizzes, score 90% or better on at least one mock exam on your first try, and still don’t pass your exam, send me your score report within 30 days of the exam and I’ll refund you: the full price on an annual plan, or your last payment on a monthly plan.

Or compare every plan side by side on the pricing page.

Not ready to buy? Start with the free practice exams, because they’ll tell you a lot about what to study next.

EWMA – Exponentially Weighted Moving Average Chart

The EWMA – Exponentially Weighted Moving Average chart is used in statistical process control to monitor variables (or attributes that act like variables) that use the entire history of a given output. This differs from other control charts that treat each data point individually.

EWMA Exponentially Weighted Moving Average Chart
EWMA Chart image by Minitab

The user gives each output (previous sample mean) a weighting. The most recent samples are weighted the highest. This means you give the oldest data the least amount of weight. The chart displays the data geometrically. This gives the advantage of the chart not being greatly affected when a small or large value enters the calculation.

The EWMA chart will detect shifts of .5 sigma to 2 sigmas much faster than Shewhart charts with the same sample size. They are, however, slower in detecting large shifts in the process mean.

Another advantage is that each data point plotted on the chart represents a moving average of points. Thus you can use the Central Limit Theorem to say that the plotted points (the moving average of the subgroups) are normally distributed, and the control limits are clearly defined.

Use EWMA Charts When:

  • When you have continuous data from the entire life of a process.
  • You want to detect small shifts in the process. For larger shifts, use Shewart style charts like the X Bar R and the X Bar S charts.
  • When you want to measure the mean, monitoring the process variability requires the use of some other technique.
  • The subgroup sample size should be > 1. If the sample size in the subgroup is 1, try using an Individual X chart.
  • When you want to smooth out the effect of uncontrollable noise in the data.

How to Make an EWMA Control Chart

  • Decide the weightings
    • Use smaller weightings to discern smaller shifts.
    • Set between 0 and 1.
      • If you pick a weighting of 1, you have an xbar chart.
    • Based on user experience and preference.
  • where 0 < l ≤ 1
  • We also need to define a starting value of Z0 before the first sample is taken. If a target value µ is specified, then Z0 = µ. Otherwise, it is typical to use the average of some preliminary data. That is, Z0 = Xbar
  • Create the control limits
    • Generally, default to 3 standard deviations for Six Sigma quality purposes and to match what other charts generally do.
    • May need to change the control limits to something smaller if the weightings are very small.
  • UCL =µ+Lσ√λ/(2-λ)
  • LCL =µ-Lσ√λ/(2-λ)
  • Where L is the length: The control limits are often set at 3, 3 standard deviations from the mean. or use page 5 Lucas and Saccucci
  • Plot the points
    • Can be either subgroups or individual observations.
      • When plotting a subgroup, use the mean of that subgroup.
  • See if the points are within the control limits.
  • Look for trends or patterns.

Examples of Uses of EWMA Control Charts

  • Detecting drift that is caused by tool wear
    • Ex. Manufacturer produces a widget of a certain diameter. If the diameters of the widget are off, there are consequences. Measuring using an EWMA Chart helps understand the manufacturing machine wear and its impact on the creation of the widgets.
  • Accounting processes
    • Ex. Day-to-day fluctuations in accounting processes may be large but may not necessarily mean the process is unstable. The choice of lambda can be determined to make the chart more or less sensitive to these daily fluctuations.
  • Chemical processes
  • Website visitors fluctuate depending on the day of the week.
    • Ex. This website gets far more visitors when people are at work Monday through Thursday compared to Friday, the weekends, or holidays.

Important Notes on EWMA Charts

  • Your data must be time-ordered.
  • Consecutive points have the highest chance of being alike – so default to a range of 2 when possible.
  • Created for normal data but is robust enough for non-normal data sets.

Example of Using an EWMA – Exponentially Weighted Moving Average – chart in DMAIC

Generate an Exponentially Weighted Moving Average (EWMA) chart based on the cycle time data, depicting the processing duration for customer orders in a car showroom over a specified period. The chart should have a target mean value of 30, a standard deviation of 2.13, and a smoothing factor (λ) set to 0.3.

Calculate EWMA (Zi) values

EWMA - Exponentially Weighted Moving Average Chart
  • First value Z0 = µ , Z0 = 30
  • Second value (31.6) Z1 = (0.3*31.6)+(1-0.3)*30= 30.48
  • Third value (27.9) Z2 = (0.3*27.9)+(1-0.3)*30.48= 29.71
  • Similarly, calculate EWMA values for the rest of the duration

Calculate UCL and LCL

  • Where µ=30
  • L =3
  • σ =2.13
  • λ=0.3
  • UCL =µ+Lσ√λ/(2-λ) = 30+3*2.13*Sqrt(0.3/1.7) = 32.684
  • LCL =µ-Lσ√λ/(2-λ) = 30-3*2.13*Sqrt(0.3/1.7) = 27.316

Compute the EWMA chart using Excel or any statistical software such as Minitab

Conclusion: No parameter in the EWMA chart falls outside the control limits. Therefore, based on the established control limits, the process is considered stable during the observed time period. This means that the variation in cycle time is within the expected range, given the selected smoothing parameter (λ=0.3) and control limits.

In the Control phase, it’s essential to continue monitoring the process over time and take corrective actions if any new patterns or points outside the control limits emerge. If the process remains stable and meets performance expectations, it can be considered successfully controlled. If further improvements are necessary, further analysis and improvement efforts may be required.

EWMA Excel sheet

What You Need to Know for Your Exam

Green Belts

The IASSC Green Belt BOK requires you to know EWMAs as part of Statistical Process Control.

Black Belts

The Villanova Black Belt BOK requires you to know EWMAs as part of Advanced Statistical Process Control; “Understand appropriate uses of short-run SPC, EWMA, CuSum, and moving average.“

The IASSC Black Belt BOK requires you to know EWMAs as part of Statistical Process Control.

EWMA – Exponentially Weighted Moving Average – Videos

ASQ Six Sigma Black Belt Moving Average Questions

Question:  Which of the following charts plots the mean of a set of values and recalculates the mean with each new value?

(A) Moving range
(B) Moving average
(C) X Bar and s
(D) c

Answer:


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