Process Capability – Cp, Cpk, Pp and Ppk

Learning Objectives

By the end of this lesson, the learner will be able to:

Explain the purpose of process capability analysis in Six Sigma.
Distinguish between process specification limits and control limits.
Explain the difference between customer requirements and process performance.
Understand the concepts of Cp, Cpk, Pp, and Ppk.
Distinguish between process potential and actual process capability.
Explain the relationship between process mean, process variation, and capability.
Calculate and interpret Cp and Cpk.
Understand the importance of process centering.
Explain short-term and long-term process performance.
Understand the concept of Sigma Level.
Relate defect rates, DPMO, and Sigma Level.
Understand the concept of Defects Per Million Opportunities (DPMO).
Recognize the assumptions and limitations associated with capability analysis.
Understand the role of normality and data distribution in capability analysis.
Use Minitab to perform basic capability analysis.
Interpret capability results in the context of customer and process requirements.

1. Introduction

A Six Sigma project ultimately seeks to improve process performance.

After establishing that the measurement system is suitable, the Green Belt needs to determine how well the process is performing relative to customer or specification requirements.

For example, suppose a component has a required diameter of:

50.00 ± 0.20 mm

Therefore:

Lower Specification Limit (LSL) = 49.80 mm

Upper Specification Limit (USL) = 50.20 mm

The process may produce measurements that vary around a mean.

The important questions are:

Is the process variation small enough?
Is the process centered between the specification limits?
How much of the output falls outside specification?
How capable is the process of meeting customer requirements?
What is the current Sigma Level?

These questions are addressed through process capability and process performance analysis.

A fundamental Six Sigma principle is:

Capability compares process performance with customer or specification requirements.

2. Concept / Theory

2.1 What Is Process Capability?

Process capability is the ability of a stable process to consistently produce output within specified requirements.

It considers:

Process location
Process variation
Specification limits

A capable process has variation sufficiently small relative to the specification width and, where relevant, is appropriately centered.

2.2 Specification Limits

Specification limits are requirements established by:

Customers
Engineering
Product design
Regulatory requirements
Internal standards

The two primary specification limits are:

USL = Upper Specification Limit

LSL = Lower Specification Limit

Example:

Target = 50.00 mm

Tolerance = ±0.20 mm

Therefore:

LSL = 49.80 mm

USL = 50.20 mm

2.3 Specification Limits vs Control Limits

This distinction is extremely important.

Specification Limits

Specification limits describe what is acceptable to the customer, design, or requirement.

They answer:

What should the process produce?

Control Limits

Control limits are statistically calculated from process data and describe the expected behavior of a stable process.

They answer:

What is the process currently doing?

Specification limits and control limits are therefore not interchangeable.

2.4 Process Stability Before Capability

Capability analysis should normally be performed on a process that is reasonably stable and predictable.

If a process is unstable, its variation may change over time.

A capability calculation from an unstable process may therefore provide a misleading summary.

The Green Belt should generally consider:

Measurement System → Stability → Distribution → Capability

before drawing capability conclusions.

2.5 Process Mean

The process mean represents the center of the observed process measurements.

For a process with:

Mean = 50.00 mm

the process is centered exactly on the nominal target of 50.00 mm.

However, a process can have a desirable mean but excessive variation.

Therefore:

Centering alone does not make a process capable.

2.6 Process Variation

Process variation describes how widely observations are distributed around the process center.

A process with low variation produces measurements close together.

A process with high variation produces measurements spread over a wider range.

Capability therefore depends on both:

Process location

and

Process variation.

2.7 Cp — Process Capability Potential

The Cp index compares the specification width with the process’s short-term spread.

A commonly used formula is:

Cp = (USL − LSL) / (6σ)

Where:

USL = Upper Specification Limit
LSL = Lower Specification Limit
σ = estimated within-process standard deviation

Cp measures the potential capability assuming the process is centered between the specification limits.

Example

Suppose:

USL = 50.20

LSL = 49.80

and:

σ = 0.05

Then:

Cp = (50.20 − 49.80) / (6 × 0.05)

Cp = 0.40 / 0.30

Cp = 1.33

The process spread is therefore narrower than the specification width by the amount represented by the capability index.

However, Cp does not consider whether the process mean is centered.

2.8 Cpk — Capability Considering Centering

Cpk considers both:

Process variation
Process centering

A commonly used formula is:

Cpk = minimum [(USL − μ)/(3σ), (μ − LSL)/(3σ)]

Where:

μ = process mean
σ = within-process standard deviation

Cpk therefore reflects the distance from the process mean to the nearest specification limit.

Example

Suppose:

USL = 50.20

LSL = 49.80

Mean = 50.10

σ = 0.05

Then:

CPU = (50.20 − 50.10)/(3 × 0.05)

CPU = 0.10/0.15

CPU = 0.67

And:

CPL = (50.10 − 49.80)/(3 × 0.05)

CPL = 0.30/0.15

CPL = 2.00

Therefore:

Cpk = minimum(0.67, 2.00)

Cpk = 0.67

Although the process has a Cp of 1.33, its Cpk is only 0.67 because the process is shifted toward the upper specification limit.

This illustrates a critical concept:

A process can have good potential capability but poor actual capability because it is not centered.

2.9 Cp vs Cpk

Index	What It Considers
Cp	Process spread relative to specification width
Cpk	Process spread and process centering
Cp	Assumes centering
Cpk	Accounts for distance to nearest specification

A useful relationship is:

Cpk ≤ Cp

for the standard two-sided capability situation.

If Cp and Cpk are very different, the process may be substantially off-center.

2.10 Pp and Ppk

Pp and Ppk are commonly used as overall process performance indices.

They use overall variation rather than the within-process variation used in the conventional Cp/Cpk framework.

Common formulas are:

Pp = (USL − LSL) / (6s)

Ppk = minimum [(USL − μ)/(3s), (μ − LSL)/(3s)]

Where:

s = overall sample standard deviation

The distinction is therefore important:

Cp/Cpk → commonly based on within-process variation.
Pp/Ppk → commonly based on overall observed variation.

The exact methodology and estimates used by software should always be checked.

2.11 Cp, Cpk, Pp and Ppk — Comparison

Index	Variation Used	Centering Considered?
Cp	Within-process variation	No
Cpk	Within-process variation	Yes
Pp	Overall variation	No
Ppk	Overall variation	Yes

This distinction becomes particularly useful when comparing short-term capability with overall process performance.

2.12 Process Centering

Consider two processes with identical variation.

Process A

Mean is near the center of the specification range.

Process B

Mean is close to the USL.

Both may have similar Cp values.

However, Process B will have a lower Cpk because its mean is closer to the specification boundary.

Therefore, improvement can involve:

Reducing variation
Centering the process
Or both

2.13 Sigma Level

Sigma Level expresses process performance in relation to the number of standard deviations between the process center and the nearest specification limit, under the assumptions of the calculation being used.

For a centered normal process:

Sigma Level ≈ 3 × Cpk

when Cpk is calculated using the same within-process standard deviation and the process is centered appropriately.

For example:

Cpk = 1.33

would correspond to approximately:

4 Sigma

under this simple relationship.

However, Six Sigma literature also commonly uses a 1.5-sigma shift convention when translating long-term performance to traditional Sigma Level and DPMO tables.

Therefore, Green Belts must identify which Sigma Level convention is being used.

2.14 Defects Per Million Opportunities (DPMO)

DPMO means:

Defects Per Million Opportunities

The formula is:

DPMO = (Number of Defects / Total Opportunities) × 1,000,000

Where:

Total Opportunities = Number of Units × Opportunities per Unit

Example

Suppose:

1,000 units are produced.
Each unit has 4 possible defect opportunities.
20 defects are found.

Total opportunities:

1,000 × 4 = 4,000

Therefore:

DPMO = (20 / 4,000) × 1,000,000

DPMO = 5,000

The process has:

5,000 DPMO

2.15 Defect vs Defective Unit

These terms should not be confused.

Defective Unit

A unit that fails one or more requirements.

Defect

A specific nonconformance or failure.

One unit can contain multiple defects.

Therefore:

Number of defects does not necessarily equal number of defective units.

This distinction is important when calculating DPMO.

2.16 DPU

Another useful measure is:

DPU = Defects Per Unit

Formula:

DPU = Number of Defects / Number of Units

DPU is different from DPMO because DPMO considers the number of opportunities per unit.

2.17 Normality and Capability

Many traditional capability calculations assume that the data follow a distribution for which the selected capability method is appropriate, often a normal distribution for the standard Cp/Cpk approach.

Therefore, the Green Belt should examine:

Histogram
Probability plot
Distribution characteristics
Outliers
Process stability

If the data are substantially non-normal, an appropriate non-normal capability method or transformation may be needed.

The correct response is not automatically to force every dataset into a normal distribution.

2.18 Capability Does Not Equal Control

A process may be:

Stable but incapable.
Stable and capable.
Unstable but apparently capable based on an inappropriate calculation.

Therefore:

Control and capability answer different questions.

Control asks whether the process is predictable.

Capability asks whether a predictable process can meet requirements.

3. Key Topics

3.1 Specification Limits

LSL
USL
Target
Tolerance
Customer requirements

3.2 Process Parameters

Mean
Standard deviation
Within-process variation
Overall variation
Process centering

3.3 Capability Indices

3.4 Performance Indices

Pp
Ppk
Overall process variation

3.5 Capability Interpretation

A general interpretation framework is:

Index	General Meaning
Below 1.00	Process spread or centering does not adequately meet the specification under the calculation
Around 1.00	Process performance is close to the specification requirement
Above 1.00	Process has increasing capability relative to specification width
Higher values	Greater separation between process variation and specification limits

The exact capability requirement should be determined by the organization, customer, industry, and risk level.

A Green Belt should therefore avoid treating one universal Cpk value as appropriate for every process.

3.6 Sigma Level

The Green Belt should understand:

Sigma as a measure of variation
Distance to specification
Cpk relationship
DPMO
Short-term versus long-term interpretation
The 1.5-sigma shift convention used in traditional Six Sigma reporting

3.7 DPMO

DPMO requires:

Number of defects
Number of units
Number of opportunities per unit

Formula:

DPMO = Defects / Opportunities × 1,000,000

3.8 Capability Improvement Strategies

If capability is poor, improvement may involve:

Reducing variation
Centering the process
Eliminating special causes
Improving equipment
Optimizing process settings
Improving materials
Standardizing operating methods
Reducing measurement variation
Improving process control

4. Tools / Methodology

Step 1 — Confirm the Measurement System

Before capability analysis:

Complete appropriate MSA.

Ensure the measurement system is suitable for the intended use.

Step 2 — Confirm Process Stability

Use appropriate control charts or other stability analysis.

Investigate special causes before calculating capability for a process intended to represent stable performance.

Step 3 — Define the Specifications

Identify:

LSL
USL
Target, if applicable
Customer requirements

Do not confuse specification limits with statistically calculated control limits.

Step 4 — Collect Representative Data

Data should represent the process and relevant operating conditions.

Consider:

Time period
Shifts
Machines
Operators
Materials
Process conditions

Avoid collecting data from only the best-performing period unless that is the intended scope of the analysis.

Step 5 — Examine the Distribution

Review:

Histogram
Probability plot
Summary statistics
Outliers
Distribution shape

Determine whether the selected capability method is appropriate.

Step 6 — Calculate Capability

For an appropriate normal capability analysis, calculate:

Cpk

and, where appropriate:

Ppk

Step 7 — Interpret the Results

Ask:

Is the process stable?
Is the measurement system adequate?
Is the distribution appropriate?
Is the process centered?
Is variation sufficiently small?
Which specification limit is closest to the process?
Is the capability index adequate for the intended requirement?
What improvement is required?

Step 8 — Determine Improvement Direction

If:

Cp is low

→ Variation reduction is likely required.

If:

Cp is adequate but Cpk is much lower

→ Process centering may be an important issue.

If:

Ppk is substantially lower than Cpk

→ Overall process variation may be greater than the within-process variation, potentially indicating time-to-time or other sources of variation that warrant investigation.

These are diagnostic clues, not automatic root-cause conclusions.

4.9 Minitab Application

Minitab can be used to perform capability analysis.

A typical workflow is:

Stat → Quality Tools → Capability Analysis

The exact menu structure depends on:

Data type
Distribution
Minitab version
Selected capability method

For normal capability analysis, the Green Belt generally specifies:

Measurement column
LSL
USL
Target, if applicable

The output may include:

Histogram
Specification limits
Normal curve
Cp
Cpk
Pp
Ppk
Percent defective
Expected performance
Capability plots

The Green Belt should interpret the numerical results together with the graphical output.

5. Worked Example / Case Study

Case Study — Component Diameter

A company manufactures a shaft with a required diameter of:

50.00 ± 0.20 mm

Therefore:

LSL = 49.80 mm

USL = 50.20 mm

After confirming that the measurement system is suitable and the process is reasonably stable, the team analyzes the data.

Suppose:

Process Mean = 50.10 mm

Within-process σ = 0.05 mm

Step 1 — Calculate Cp

Cp = (USL − LSL) / (6σ)

Substitute:

Cp = (50.20 − 49.80) / (6 × 0.05)

Cp = 0.40 / 0.30

Cp = 1.33

Step 2 — Calculate CPU

CPU = (USL − μ) / (3σ)

CPU = (50.20 − 50.10) / (3 × 0.05)

CPU = 0.10 / 0.15

CPU = 0.67

Step 3 — Calculate CPL

CPL = (μ − LSL) / (3σ)

CPL = (50.10 − 49.80) / (3 × 0.05)

CPL = 0.30 / 0.15

CPL = 2.00

Step 4 — Calculate Cpk

Cpk = minimum(CPU, CPL)

Therefore:

Cpk = minimum(0.67, 2.00)

Cpk = 0.67

Step 5 — Interpret

The Cp value indicates that the process spread has potential to fit within the specification width if centered appropriately.

However, Cpk is much lower because the process mean is shifted toward the upper specification limit.

Therefore, the analysis suggests two separate considerations:

Variation: The process spread is relatively narrow compared with the specification width.

Centering: The process is not well centered.

Step 6 — Improvement Direction

The team should investigate why the process mean is at approximately:

50.10 mm

rather than near the target of:

50.00 mm

Possible investigation areas may include:

Machine setting
Tool offset
Material characteristics
Temperature
Process parameters
Calibration
Operator settings

The team should use appropriate data and process investigation rather than assuming any one factor is the cause.

Case Study — DPMO

A service process handles:

2,000 transactions

Each transaction has:

5 defect opportunities

During the study:

10 defects

are identified.

Total opportunities:

2,000 × 5 = 10,000

DPMO:

DPMO = (10 / 10,000) × 1,000,000

DPMO = 1,000

Therefore:

DPMO = 1,000

This means the process experienced 1,000 defects per million opportunities based on the defined opportunity structure and observed data.

Important Learning Point

Capability analysis should never be reduced to:

“Calculate Cpk and report the number.”

The Green Belt should understand:

Measurement → Stability → Distribution → Specifications → Capability → Interpretation → Improvement

6. Practical Application

Exercise 1 — Capability Calculation

A process has:

USL = 105

LSL = 95

Mean = 100

σ = 1.5

Calculate:

Then interpret whether the process is centered.

Exercise 2 — Centering

Two processes have the same specification limits and the same standard deviation.

Process A has a mean near the center of the specification range.

Process B has a mean close to the USL.

Ask:

Would Cp necessarily be different?
Which index better reflects the centering difference?
Why can Cpk be lower than Cp?

Exercise 3 — DPMO

A process produces:

5,000 units

Each unit has:

4 defect opportunities

The process produces:

25 defects

Calculate:

DPMO

Then explain why the number of opportunities must be included.

Exercise 4 — Capability Investigation

A process has:

Cp = 1.50

Cpk = 0.80

Ask:

What does the difference between Cp and Cpk suggest?
Is variation or centering likely to deserve investigation?
What additional process information should be examined?
What should the Green Belt avoid concluding from Cp and Cpk alone?

Exercise 5 — Green Belt Application

Select one measurable characteristic from a real or simulated process.

Define:

Y
Unit of measurement
LSL
USL
Target
Process mean
Standard deviation
Measurement system
Process stability

Then calculate and interpret:

where the data and selected methodology support those calculations.

Practical Checklist

Before reporting process capability, verify:

Has the measurement system been evaluated?
Is the process reasonably stable?
Are specification limits clearly defined?
Are specification limits being distinguished from control limits?
Is the data representative?
Is the distribution appropriate for the selected capability method?
Are outliers understood?
Has process centering been evaluated?
Has process variation been evaluated?
Are Cp and Cpk understood correctly?
Are Pp and Ppk being distinguished from Cp and Cpk?
Is the Sigma Level convention clearly identified?
Is DPMO based on correctly defined opportunities?
Are defects being distinguished from defective units?
Is the capability requirement appropriate for the application?
Are the results being interpreted together with process knowledge?

7. Lesson Summary

Process capability analysis determines how well a process can meet defined requirements.

The Green Belt must understand the difference between:

Specification limits

and

Control limits.

Specification limits represent customer, design, or other requirements.

Control limits describe statistically expected process behavior.

The major capability indices are:

Cp — potential capability based on process spread.

Cpk — capability considering both spread and centering.

Pp — overall process performance based on overall variation.

Ppk — overall process performance considering centering.

A key relationship is:

Cpk ≤ Cp

in the standard two-sided capability situation.

A large difference between Cp and Cpk may indicate that the process is not well centered.

The Green Belt should also understand:

Sigma Level
DPMO
DPU
Defects
Defective units
Normality
Process stability
Short-term versus long-term variation

Capability analysis is meaningful only when the measurement system, process stability, data distribution, and specifications have been properly considered.

8. Lesson Learnt / Conclusion

The central lesson is:

Process capability is the relationship between process performance and the requirements the process must satisfy.

A good capability study follows a disciplined sequence:

Verify the measurement system.

Confirm process stability.

Understand the data distribution.

Define the specifications.

Evaluate process variation.

Evaluate process centering.

Calculate the appropriate capability and performance indices.

Interpret the results in context.

Identify the appropriate improvement direction.

A Green Belt should not simply report a Cpk value.

The Green Belt should be able to explain:

Why the capability value is what it is.
Whether variation or centering is the main concern.
Whether the process is stable.
Whether the measurement system is trustworthy.
Whether the capability method is appropriate.
What the results mean for the customer and the process.

The ultimate objective is not merely to obtain a higher capability index.

It is to develop a stable, predictable process that consistently meets customer and specification requirements.

Measure correctly.
Stabilize the process.
Understand the variation.
Compare performance with requirements.
Improve where necessary.