Descriptive Statistics and Graphical Analysis
Learning Objective
By the end of this lesson, the learner will be able to:
- Explain the role of probability in Six Sigma.
- Understand the purpose of probability distributions.
- Recognize the Normal, Binomial, and Poisson distributions.
- Understand how distributions help describe process behaviour and uncertainty.
- Select an appropriate distribution for a process situation.
- Use probability and distribution analysis to support further Six Sigma investigation.
1. Introduction
In Six Sigma, process data contains uncertainty and variation.
Probability provides a way to understand uncertainty, while probability distributions describe how data behaves.
Understanding distributions helps a Black Belt analyse process behaviour, predict outcomes, and establish appropriate process limits.
The Black Belt programme specifically identifies three distributions for this lesson:
- Normal Distribution
- Binomial Distribution
- Poisson Distribution
2. Understanding Probability
Probability describes the likelihood of an event occurring.
In a process, many outcomes are uncertain.
For example:
- How many customers may arrive in an hour?
- How many defects may occur?
- How many items may pass inspection?
- How frequently might a particular event occur?
Probability provides a structured way of analysing such uncertainty.
3. Understanding Probability Distributions
A probability distribution describes how possible outcomes are distributed.
For a Black Belt, understanding the distribution of process data helps answer questions such as:
- What type of behaviour does the data show?
- What outcomes occur more frequently?
- How much variation is present?
- What outcomes may reasonably be expected?
The appropriate distribution depends on the type and behaviour of the data.
4. Normal Distribution
The Normal Distribution is used to describe continuous data that follows an approximately normal pattern.
Examples of continuous process measurements may include:
- Length
- Weight
- Temperature
- Processing time
A Black Belt may examine the data distribution before selecting appropriate statistical analysis.
The Normal Distribution is therefore important when understanding the behaviour of continuous process data.
5. Binomial Distribution
The Binomial Distribution is used for situations involving a defined number of trials where outcomes can be classified into categories such as:
- Pass / Fail
- Defective / Non-defective
- Yes / No
For example, a quality team may examine the number of defective items among a defined number of inspected products.
The distribution helps describe the probability of different numbers of outcomes occurring.
6. Poisson Distribution
The Poisson Distribution is useful for analysing the occurrence of events over a defined interval.
Examples include:
- Customer arrivals per hour
- Number of defects
- Number of service requests
- Number of equipment failures
Application Example
A call centre records the number of customer calls received each hour.
A Poisson model can be considered to understand the pattern of arrivals and support staffing decisions.
7. Selecting an Appropriate Distribution
A Black Belt should consider the nature of the data and the process being studied before selecting a probability distribution.
A simplified guide is:
| Distribution | Typical Application |
|---|---|
| Normal | Continuous measurements |
| Binomial | Defined trials with categorical outcomes |
| Poisson | Counts of events occurring over an interval |
Distribution selection should be based on the characteristics of the data and the process question being investigated.
8. Tools & Techniques
Relevant tools and techniques include:
- Probability analysis
- Probability distributions
- Probability plots
- Goodness-of-fit analysis
- Normality tests
- Minitab
- Excel
These tools help the Black Belt evaluate whether a distribution is appropriate for the data and process being studied.
9. Application Example — Call Centre
A call centre wants to improve staffing decisions.
The organization records the number of customer calls received during each hour.
The Black Belt examines the call-arrival data and considers a Poisson distribution to describe the number of arrivals per hour.
The analysis can help the organization understand the expected pattern of arrivals and support appropriate staffing decisions.
10. Case Study — Hospital Emergency Department
A hospital emergency department experiences varying numbers of patient arrivals throughout the day.
The team analyses patient-arrival patterns using a Poisson model.
The analysis supports improved nurse scheduling.
The case study reports a 15% reduction in patient waiting time following the scheduling improvement.
The important lesson is that understanding the probability distribution of process events can support operational decisions.
11. Black Belt Perspective
Probability and distribution analysis should not be treated simply as mathematical exercises.
The Black Belt should understand:
What type of data do we have?
What process behaviour does the data show?
Which distribution may describe that behaviour?
What analysis can be performed after understanding the distribution?
This provides a stronger foundation for subsequent statistical analysis.
12. Lesson Practice
Scenario
A service organization records the number of customer requests received every hour.
The Black Belt wants to understand the arrival pattern before making staffing decisions.
Questions
- What type of data is being collected?
- Which probability distribution might be considered?
- Why is distribution selection important?
- What information could the analysis provide?
- What evidence should the Black Belt examine before deciding that the selected distribution is appropriate?
13. Key Learning Points
After completing this lesson, the learner should be able to:
- Explain the meaning of probability.
- Explain the purpose of probability distributions.
- Recognize the Normal Distribution.
- Recognize the Binomial Distribution.
- Recognize the Poisson Distribution.
- Understand how distributions describe process behaviour.
- Select a distribution based on the nature of the data and process.
- Understand the use of probability plots and goodness-of-fit analysis.
- Use Minitab or Excel to support probability and distribution analysis.
- Use distribution analysis as a foundation for further statistical analysis.
14. Lesson Conclusion
Probability provides a structured way to understand uncertainty, while probability distributions help describe process behaviour.
The key distributions covered in this lesson are:
Normal → Binomial → Poisson
A Black Belt should understand the type of data, examine its behaviour, and select an appropriate distribution before proceeding to further statistical analysis.
Understanding the distribution helps the Black Belt understand the process.