Process Capability: Cp, Cpk, Pp, and Ppk
Calculate and read Cp, Cpk, Pp and Ppk, tell within-subgroup sigma from overall sigma, and check that a process is stable before you trust the numbers.
Table of contents
A control chart tells you whether a process is stable. A capability index tells you whether that stable process fits the tolerance. These numbers show up in customer approvals, inspection reports and quality meetings, and they are easy to misread. Each one compares the width of the spec with the width of the process, so a single number can sum up how much room a process has.
What You Will Be Able to Do
- Say why stability comes before capability.
- Read a histogram for shape, center and spread.
- Calculate Cp, Cpk, Pp and Ppk from real data.
- Explain the difference between Cpk and Ppk, and what each pair of numbers tells you.
- Turn an index into an expected defect rate, and state common targets correctly.
Stability Comes First
A capability index predicts the future from the past. That only works if the process behaves the same way tomorrow. If a process is out of control, its mean or spread is moving, and the number describes a moment that has already passed.
Do a capability study in this order:
- Prove the measurement system. A gauge that is too noisy makes the process look worse than it is.
- Run a control chart and clear any special causes.
- Collect enough data. About 100 readings or more is a common floor, such as 25 subgroups of 4 or 5. Fewer than that gives a shaky number.
- Plot a histogram and check its shape.
- Calculate the indices and read them together.
Reading a Histogram
A histogram sorts readings into bars, so you can see the shape, center and spread of the data at a glance. Draw the spec limits on it. Unlike a control chart, a histogram can carry spec limits, because it compares the process to the requirement.
| Shape | What it suggests |
|---|---|
| Single bell, centered between the limits | A healthy process. The indices will mean what they say |
| Bell shifted toward one limit | Good spread, poor centering |
| Two humps | Two sources mixed, such as two machines or two lots |
| Cut off sharply at a limit | Parts were sorted before you got the data |
| Long tail on one side | A feature that cannot go below zero, such as flatness or runout |
The indices below assume a roughly bell-shaped process. If the shape is far from that, the numbers and defect rates are off. Ask a quality engineer how to handle it.
The Four Indices
All four use the same logic. The difference is which sigma they use.
| Index | Sigma used | What it asks |
|---|---|---|
| Cp | Within-subgroup | Could the spread fit, if it were centered? |
| Cpk | Within-subgroup | Does it fit, given where it is centered? |
| Pp | Overall | Did the spread fit over the whole run? |
| Ppk | Overall | Did it fit, given the center, over the whole run? |
Within-subgroup sigma comes from short-term variation, usually R-bar / d2 from the control chart. It leaves out shifts between subgroups, so Cp and Cpk are short-term measures of what the process can do. Overall sigma is the plain standard deviation of every reading in the study. It includes shifts and drift over time, so Pp and Ppk are long-term measures of what the process really did.
Cpk and Ppk are not the same number, and they are not interchangeable. Always check which one a report or a customer is asking for.
The formulas use the upper and lower specification limits, USL and LSL:
- Cp = (USL - LSL) / (6 x within sigma)
- Cpk = the smaller of (USL - mean) / (3 x within sigma) and (mean - LSL) / (3 x within sigma)
- Pp = (USL - LSL) / (6 x overall sigma)
- Ppk = the smaller of (USL - mean) / (3 x overall sigma) and (mean - LSL) / (3 x overall sigma)
Cp and Pp measure spread only, as a ratio of the spec width to the process width. Cpk and Ppk measure the distance from the mean to the nearest limit, so they account for centering. Cpk can never be higher than Cp. When they are equal, the process is perfectly centered. A negative Cpk means the mean itself is outside the spec.
Worked Example
A spec of 0.500 +/- 0.006 gives a USL of 0.506, an LSL of 0.494 and a width of 0.012. The study covers 25 subgroups of 5 parts.
- Mean = 0.5018
- R-bar = 0.0028, so within sigma = 0.0028 / 2.326 = 0.0012
- Overall sigma, from all 125 readings = 0.0016
Cp = 0.012 / (6 x 0.0012) = 0.012 / 0.0072 = 1.67
Cpk:
- Upper side: (0.506 - 0.5018) / (3 x 0.0012) = 0.0042 / 0.0036 = 1.17
- Lower side: (0.5018 - 0.494) / 0.0036 = 0.0078 / 0.0036 = 2.17
- Take the smaller: Cpk = 1.17
Pp = 0.012 / (6 x 0.0016) = 0.012 / 0.0096 = 1.25
Ppk:
- Upper side: 0.0042 / (3 x 0.0016) = 0.0042 / 0.0048 = 0.88
- Lower side: 0.0078 / 0.0048 = 1.63
- Take the smaller: Ppk = 0.88
Here is what those four numbers say. Cp is strong, so the machine can hold the spec. Cpk is lower, so the process is running 0.0018 high. Centering it would lift Cpk toward 1.67. Pp is well below Cp, so the process also moves around between subgroups, and that is worth a look on the control chart. Ppk of 0.88 is what the customer really got over this run.
Reading the Numbers Together
| What you see | What it means | What helps |
|---|---|---|
| Cp high, Cpk low | Good spread, off center | Recenter the process |
| Cp and Cpk both low, and close together | Centered, but too wide | Reduce variation. This takes a process change |
| Cp and Cpk high, Pp and Ppk much lower | Shifts or drift over time | Find what changes between subgroups |
| All four high and close together | Stable, centered and capable | Keep doing what you are doing |
A Cp of 1.6 with a Cpk of 0.7 is the first row. The spread is fine, the process is off center, and an adjustment to the mean would fix most of it. Moving the mean is usually much easier than narrowing the spread.
What the Numbers Mean in Defects
If a process is centered and normal, each index maps to an expected defect rate. Parts per million (PPM) here counts both tails.
| Index | Sigma to nearest limit | Expected PPM out of spec |
|---|---|---|
| 1.00 | 3 | About 2,700 |
| 1.33 | 4 | About 63 |
| 1.67 | 5 | About 0.6 |
| 2.00 | 6 | About 0.002 |
A Cp of 1.0 means the process spread exactly equals the spec width. It has no margin, and 0.27 percent of parts fall out. A value of 1.33 means the spread uses 75 percent of the spec width. That leaves comfortable room for drift, which is why it is such a common goal. A value of 1.67 uses 60 percent, and 2.0 uses 50 percent.
An off-center process is different, because nearly all the risk sits on one side. Use the one-sided figure for the near limit. A Cpk of 0.80 means about 8,200 PPM past that limit. In the worked example, a Ppk of 0.88 means about 4,300 PPM past the upper limit. These are estimates that assume a bell shape, not counts.
Targets
No single target fits every job. These are common starting points, and your customer's requirements win.
| Situation | Usual target |
|---|---|
| Ongoing production, general manufacturing | Cpk of at least 1.33 |
| New process at automotive part approval | Preliminary Ppk of at least 1.67 |
| Ongoing automotive production after approval | Cpk of at least 1.33 |
| Safety or critical characteristics | Often higher, as the customer sets |
The automotive figures come from the Production Part Approval Process (PPAP) manual published by the Automotive Industry Action Group (AIAG). In the usual PPAP reading, a preliminary index above 1.67 meets the criteria, 1.33 to 1.67 may be acceptable with customer agreement, and below 1.33 does not meet them.
IATF 16949, the automotive quality standard, requires process studies and capability results, but it does not set a number of its own. It points to the customer's part approval requirements. So "IATF requires 1.67" is a loose way to say it. The number comes from PPAP and the customer, and a customer may set a different one.
Common Mistakes
| Mistake | What it costs |
|---|---|
| Calculating capability on an unstable process | A number that predicts nothing |
| Using Cpk and Ppk as if they were the same | Reports that overstate how the process really ran |
| Reporting only Cp | Hides a process running off center |
| Using sorted data | The process looks better than it is |
| Using too few readings | A number that can swing widely from study to study |
| Trusting the index on a skewed feature | Defect estimates that are wrong |
| Fixing spread when the problem is centering | Time and money spent on the harder fix |
Check Your Understanding
1. A spec is 10.00 +/- 0.03, the mean is 10.00, and within sigma is 0.005. What are Cp and Cpk?
Cp = 0.06 / 0.03 = 2.0. The mean is on center, so Cpk is also 2.0.
2. The same process drifts to a mean of 10.02. What is Cpk now, and did Cp change?
Upper side: (10.03 - 10.02) / 0.015 = 0.67. Cp stays 2.0, because spread did not change. Recentering is the fix.
3. A report shows Cpk 1.5 and Ppk 0.9. What does the gap suggest?
Short-term the process is good, but over the whole run it moved. Look for shifts between subgroups, such as setups, lots or tool changes. Check the control chart before trusting the Cpk.
4. A quality tech calculated a Cpk of 1.6 from 15 parts, on a chart with several out-of-control points. Should you report it?
No. The process is not stable, and the sample is too small. Clear the special causes, collect more data, and redo the study.
Key Takeaways
- Stability comes first, because capability only predicts a stable process.
- Cp and Cpk use within-subgroup sigma. Pp and Ppk use overall sigma.
- Cp and Pp show spread only. Cpk and Ppk also show centering.
- A Cpk of 1.33 is a common target and means comfortable margin, about 63 PPM when centered.
- PPAP's usual figures are a preliminary Ppk of 1.67 and an ongoing Cpk of 1.33. IATF 16949 sets no number, and the customer's requirement wins.
- Read all the numbers together, and look at the histogram too.