Probability Distributions

Learning Objectives

Explain the purpose of probability distributions in Six Sigma.
Understand the relationship between probability and process variation.
Distinguish between discrete and continuous probability distributions.
Understand the characteristics of the Normal distribution.
Understand the Binomial distribution and its applications.
Understand the Poisson distribution and its applications.
Identify when Normal, Binomial, and Poisson distributions may be appropriate.
Understand the assumptions associated with commonly used distributions.
Apply probability distributions to practical Six Sigma situations.
Interpret probability-based process information for decision-making.
Recognize the importance of selecting an appropriate probability model.

1. Introduction

In Six Sigma, processes produce outcomes that vary.

For example:

The time required to complete a transaction may vary.
The diameter of a manufactured component may vary.
The number of defects in a unit may vary.
The number of customer complaints received during a day may vary.

Probability distributions provide a mathematical way of describing how possible outcomes are distributed.

They help the Green Belt answer questions such as:

What outcomes are likely?
How frequently might an event occur?
How much variation can be expected?
What is the probability of observing a particular result?
Which statistical method is appropriate for the data?

Probability distributions are therefore an important foundation for statistical analysis.

A key principle is:

The probability model should reflect the type and behavior of the data.

2. Concept / Theory

2.1 What Is Probability?

Probability describes the likelihood that an event will occur.

Probability values range from:

0 to 1

where:

0 represents an impossible event.
1 represents a certain event.

Probability can also be expressed as a percentage.

For example:

0.25 = 25%

2.2 Random Variables

A random variable is a variable whose observed value depends on the outcome of a random process.

Examples include:

Number of defects on a product.
Number of customer complaints received in one day.
Processing time for a transaction.
Diameter of a manufactured component.

Random variables can be broadly classified as:

Discrete.
Continuous.

2.3 Discrete Random Variables

A discrete random variable takes countable values.

Examples include:

Number of defects.
Number of rejected units.
Number of complaints.
Number of failures.

For example:

0, 1, 2, 3, 4…

These values represent counts.

2.4 Continuous Random Variables

A continuous random variable can take any value within a meaningful interval.

Examples include:

Weight.
Length.
Temperature.
Cycle time.
Diameter.
Pressure.

For example, a cycle time could be:

10.1 minutes, 10.15 minutes, 10.153 minutes, depending on measurement precision.

3. Key Topics

3.1 Probability Distribution

A probability distribution describes the probabilities associated with possible values of a random variable.

Different probability distributions are appropriate for different types of data and process situations.

Three distributions particularly relevant to Green Belt work are:

Normal distribution.
Binomial distribution.
Poisson distribution.

3.2 Normal Distribution

The Normal distribution is a continuous probability distribution with a characteristic bell-shaped form.

It is symmetric about its mean when represented as an ideal Normal distribution.

Its key parameters are:

Mean, μ.
Standard deviation, σ.

The mean determines the center.

The standard deviation determines the spread.

3.3 Characteristics of the Normal Distribution

A Normal distribution has the following important characteristics:

It is continuous.
It is symmetric around its mean.
Mean, median, and mode coincide for the ideal Normal distribution.
The total area under the probability curve equals 1.
The standard deviation describes the spread of the distribution.

The Normal distribution is widely used in statistical analysis and process improvement.

3.4 The Empirical Rule

For data that follows a Normal distribution, the approximate empirical-rule percentages are:

About 68% of observations fall within ±1 standard deviation of the mean.
About 95% fall within ±2 standard deviations.
About 99.7% fall within ±3 standard deviations.

This is commonly known as the 68–95–99.7 rule.

For example, if a process has:

Mean = 100 units

and:

Standard deviation = 5 units

then, under a Normal model:

Approximately 68% of observations are expected to fall between:

95 and 105 units

Approximately 95% are expected to fall between:

90 and 110 units

Approximately 99.7% are expected to fall between:

85 and 115 units

These are model-based expectations, not guarantees for every individual dataset.

3.5 Standard Normal Distribution and Z-Score

The standard Normal distribution has:

Mean = 0

and:

Standard deviation = 1

A measurement can be standardized using a z-score:

z = (x − μ) / σ

where:

x = observed value.
μ = population mean.
σ = population standard deviation.

The z-score indicates how many standard deviations an observation is from the mean.

For example, if:

μ = 50

σ = 5

and:

x = 60

then:

z = (60 − 50) / 5 = 2

The observation is therefore 2 standard deviations above the mean.

3.6 Binomial Distribution

The Binomial distribution is a discrete probability distribution used for situations involving a fixed number of trials where each trial has two possible outcome categories, often described as:

Success / failure.
Defective / non-defective.
Pass / fail.
Yes / no.

Important conditions for a basic Binomial model include:

A fixed number of trials.
Two possible outcomes for each trial.
A consistent probability of success for each trial.
Independence of trials under the model.

3.7 Binomial Example

Suppose a quality inspection classifies each product as:

Defective or Non-defective.

If a fixed number of products are independently inspected under conditions where the probability of a defect remains constant, the number of defective products can be modeled using a Binomial distribution.

The Green Belt may then investigate questions such as:

What is the probability of obtaining a specified number of defective units?
What is the probability of observing at least a specified number of defects?
How does the defect probability affect expected results?

3.8 Binomial Probability Formula

For a Binomial random variable:

P(X = x) = C(n,x) pˣ (1 − p)ⁿ⁻ˣ

where:

n = number of trials.
x = number of successes.
p = probability of success.
1 − p = probability of failure.
C(n,x) = number of combinations of n items taken x at a time.

The formula is useful for understanding the mathematical structure of the distribution.

Statistical software can be used for practical probability calculations.

3.9 Poisson Distribution

The Poisson distribution is a discrete probability distribution commonly used for counts of events occurring within a specified interval of time, area, length, volume, or another defined exposure.

Examples include:

Number of customer complaints per day.
Number of defects per unit of exposure.
Number of equipment failures per month.
Number of calls received per hour.

The Poisson distribution is particularly useful when the data represents the number of occurrences of an event within a defined interval or opportunity.

3.10 Poisson Parameter

The Poisson distribution has a parameter commonly denoted by:

λ (lambda)

Lambda represents the expected number of events within the defined interval or exposure under the model.

The probability of observing exactly x events can be expressed as:

P(X = x) = e⁻λ λˣ / x!

where:

λ = expected event count.
x = observed event count.
e = mathematical constant.
x! = factorial of x.

3.11 Poisson Example

Suppose a call center historically receives an average of:

4 complaints per day

If the conditions are appropriate for a Poisson model, the distribution can be used to estimate the probability of receiving:

Exactly 2 complaints.
Exactly 4 complaints.
More than a specified number of complaints.

The important point is that the model describes counts of events over a defined exposure.

4. Tools / Methodology

4.1 Selecting a Probability Distribution

A Green Belt should begin by asking:

What type of outcome am I studying?

If the outcome is a continuous measurement such as:

Length.
Weight.
Temperature.
Cycle time.

a continuous probability model may be appropriate.

If the outcome is a count of successes or failures across a fixed number of trials, a Binomial model may be appropriate.

If the outcome is a count of events occurring within a defined interval or exposure, a Poisson model may be appropriate.

4.2 Distribution Selection Guide

Situation	Potential Distribution
Continuous measurement with approximately bell-shaped behavior	Normal
Number of successes in a fixed number of trials	Binomial
Count of events within a defined interval or exposure	Poisson

The table is a starting point rather than an automatic decision rule.

The actual process conditions and assumptions should always be examined.

4.3 Normal Distribution in Six Sigma

The Normal distribution can be useful for:

Understanding continuous process variation.
Estimating probabilities.
Standardizing measurements.
Understanding z-scores.
Supporting statistical inference.
Interpreting process behavior.

However, the Green Belt should not automatically assume that every continuous dataset is Normal.

The actual distribution should be examined.

4.4 Binomial Distribution in Six Sigma

Binomial models can be useful for:

Defective/non-defective outcomes.
Pass/fail inspections.
Success/failure trials.
Attribute-type quality outcomes.

The Green Belt should verify that the conditions required for the model are reasonably appropriate.

4.5 Poisson Distribution in Six Sigma

Poisson models can be useful for:

Defect counts.
Complaint counts.
Failure counts.
Event counts.
Opportunities occurring over time or another exposure.

Again, the assumptions of the model should be considered before applying it.

4.6 Probability and Risk Assessment

Probability distributions allow the Green Belt to move beyond simply describing what has happened.

They can support questions about what may happen under specified assumptions.

For example:

Historical question:

How many customer complaints were received last month?

Probability question:

Given an appropriate probability model, how likely is it that the number of complaints in a future day exceeds a specified level?

This can support risk assessment and operational planning.

5. Worked Example / Case Study

Case Study — Manufacturing Defects

A manufacturing process produces units that are classified as:

Defective or Non-defective.

Suppose the process has a defect probability of:

p = 0.02

and the quality team examines a fixed sample of:

n = 100 units

The number of defective units is a count of successes/failures across a fixed number of trials.

A Binomial distribution may therefore be an appropriate model if the required assumptions are reasonably satisfied.

5.1 Interpretation

The Green Belt can use the model to investigate probabilities such as:

Exactly 0 defective units.
Exactly 1 defective unit.
Exactly 2 defective units.
More than a specified number of defective units.

This provides a probabilistic view of expected quality performance.

5.2 Different Situation — Complaint Counts

Now consider a customer-service center that receives complaints.

Suppose the center historically receives an average of:

3 complaints per day

The outcome is now a count of events occurring within a defined time interval.

A Poisson distribution may therefore be appropriate if its assumptions are reasonably satisfied.

The Green Belt can investigate the probability of receiving a specified number of complaints on a future day.

5.3 Different Situation — Process Measurement

Consider a manufacturing process measuring component diameter.

Suppose diameter is measured continuously and the observed data is approximately bell-shaped.

A Normal distribution may provide a useful model for understanding the probability of observations occurring within particular ranges.

This illustrates an important principle:

The nature of the outcome helps determine the appropriate probability model.

6. Practical Application

Exercise 1 — Product Weight

A manufacturing process measures the weight of every sampled product.

The measurement can take decimal values and is treated as a continuous variable.

Questions:

Is the variable discrete or continuous?
Which probability distribution might be considered if the observed data is approximately bell-shaped?
What process characteristics should be examined before using the model?

Exercise 2 — Defective Units

A quality inspector checks a fixed number of products and records each as:

Defective / Non-defective

Questions:

What type of outcome is being recorded?
Is the number of trials fixed?
Could a Binomial model be appropriate?
What assumptions should be considered?

Exercise 3 — Equipment Failures

A maintenance department records the number of equipment failures occurring during each month.

Questions:

What type of variable is being studied?
Is the outcome a count?
What is the defined exposure period?
Could a Poisson model be considered?
What assumptions should be checked?

6.1 Common Errors in Distribution Selection

Assuming Every Continuous Dataset Is Normal

Continuous data does not automatically follow a Normal distribution.

The distribution should be examined.

Treating Categories as Continuous Measurements

Pass/fail data should not automatically be treated as continuous numerical measurements.

The nature of the outcome matters.

Ignoring the Sampling or Process Conditions

Probability models depend on assumptions.

The Green Belt should consider whether the actual process reasonably matches the conditions required by the selected model.

Confusing Binomial and Poisson Applications

The two distributions both involve counts, but their structures are different.

Binomial:

Number of successes in a fixed number of trials.

Poisson:

Number of events within a defined interval or exposure.

6.2 Probability Distribution and Six Sigma Decision-Making

Probability distributions can support:

Risk estimation.
Capacity planning.
Quality assessment.
Defect analysis.
Reliability-related studies.
Statistical inference.
Process-performance interpretation.

They provide a bridge between observed process data and probability-based reasoning.

7. Lesson Summary

Probability distributions provide mathematical models for describing the behavior of random variables.

A Green Belt should understand the difference between:

Discrete random variables.
Continuous random variables.

Three important distributions are:

Normal Distribution

A continuous, bell-shaped distribution characterized by its mean and standard deviation.

Binomial Distribution

A discrete distribution used for the number of successes in a fixed number of trials under appropriate assumptions.

Poisson Distribution

A discrete distribution used for counts of events within a defined interval or exposure under appropriate assumptions.

Distribution selection should be based on:

Type of data.
Nature of the outcome.
Sampling or process conditions.
Model assumptions.
Purpose of the analysis.

Probability distributions can help Six Sigma teams understand variation, estimate risk, and support evidence-based decisions.

8. Lesson Learnt / Conclusion

Probability distributions provide a structured way to understand how process outcomes behave.

The Green Belt should not select a distribution simply because it is familiar or commonly used.

Instead, ask:

What type of data do I have?
What does each observation represent?
What conditions generated the data?
Which probability model is appropriate?
Are the model assumptions reasonably satisfied?

Correct distribution selection supports reliable statistical analysis and interpretation.

Green Belt Takeaway:

Understanding probability distributions helps the Green Belt select appropriate statistical methods and interpret process risk more effectively.