What Is Attribute Capability (Binomial/Poisson) and How Do You Calculate It?

Attribute capability is a set of process capability indices used when your quality data are counts or proportions — like defect rates, number of nonconformities per unit, or error counts — rather than continuous measurements. It answers the question: Is our defect-generating process stable and capable of meeting a customer-specified defect limit? The method follows the ISO 22514 series (Part 2 and related parts) for process capability and performance, and it is based on binomial or Poisson distribution assumptions.

What It Is

Attribute capability indices evaluate a process that produces discrete data. Unlike Cp/Cpk for continuous data, you do not have a normal distribution with a mean and standard deviation. Instead, you model the process using:

• Binomial model – when you count the number of defective items in a sample of size n (each item is either good or bad). The key parameter is the proportion defective, p.
• Poisson model – when you count the number of nonconformities (defects) per unit, with an average rate of occurrence, u (or c per fixed area/volume/time).

The most common index is Cpu (capability upper) because attribute data typically have only an upper specification limit (USL) — e.g., "maximum 1% defective" or "maximum 5 defects per 100 units." A lower limit is rarely meaningful for counts.

How It Works: Formula and Steps

The procedure follows the logic of the ISO 22514 series, which requires that you first verify process stability (using a p-chart or u-chart) before calculating capability.

Step 1 – Collect data and check stability.
Take k subgroups of size n (binomial) or k subgroups of inspection units (Poisson). Plot a p-chart or u-chart. If the process is out of control, remove assignable causes and re-collect data. Capability indices are meaningless for an unstable process.

Step 2 – Estimate the process average.

• For binomial (proportion defective):

\( \bar{p} = \frac{\text{total defects}}{\text{total items inspected}} \)

• For Poisson (defects per unit):

\( \bar{u} = \frac{\text{total nonconformities}}{\text{total units inspected}} \)

Step 3 – Define the upper specification limit (USL).
This is the maximum acceptable proportion (e.g., 0.01 = 1%) or maximum rate (e.g., 3 defects per 100 units).

Step 4 – Calculate the attribute capability index.

For a binomial process, the standard deviation of the proportion is \( \sqrt{\bar{p}(1-\bar{p})/n} \). The capability index is:

\[
C_{pu} = \frac{USL - \bar{p}}{3 \sqrt{\frac{\bar{p}(1-\bar{p})}{n}}}
\]

For a Poisson process, the standard deviation of the rate is \( \sqrt{\bar{u}/n} \) (where n is the number of units per subgroup). The index is:

\[
C_{pu} = \frac{USL - \bar{u}}{3 \sqrt{\frac{\bar{u}}{n}}}
\]

Interpretation (common practice, consistent with ISO 22514 guidance):

• \( C_{pu} \ge 1.33 \) – process is capable (good).
• \( 1.0 \le C_{pu} < 1.33 \) – marginally capable (needs improvement).
• \( C_{pu} < 1.0 \) – not capable (process average is too close to, or above, the USL).

Note: The index is one-sided (upper only). If you have a lower limit as well, you would compute \( C_{pl} \) similarly and take the minimum, but this is rare for attribute data.

A Worked Illustrative Example

Example data (illustrative only):
A factory inspects 200 units per day for 20 days (n = 200, k = 20). Total defective units found = 80. The customer requires a maximum of 1.5% defective (USL = 0.015).

1. Estimate p:

\( \bar{p} = 80 / (200 \times 20) = 80 / 4000 = 0.020 \)

1. Compute the standard error:

\( \sqrt{0.020 \times 0.980 / 200} = \sqrt{0.000098} = 0.00990 \)

1. Calculate Cpu:

\( C_{pu} = (0.015 - 0.020) / (3 \times 0.00990) = (-0.005) / 0.0297 = -0.168 \)

Interpretation: The index is negative because the process average (2.0%) already exceeds the USL (1.5%). The process is not capable. You must reduce the defect rate — for example, by improving the upstream process — before capability can be achieved.

What Cpu would be acceptable? If you improved the process so that \( \bar{p} = 0.010 \), then:
\( C_{pu} = (0.015 - 0.010) / (3 \times \sqrt{0.010 \times 0.990 / 200}) = 0.005 / (3 \times 0.00704) = 0.005 / 0.0211 = 0.237 \). Still not capable. You would need a much lower defect rate or a larger sample size to detect capability reliably.

Common Pitfalls

• Calculating capability on an unstable process. ISO 22514 requires statistical control first. If your p-chart shows points beyond the control limits, your Cpu is meaningless.
• Using the wrong distribution. Use binomial for pass/fail (defective items) and Poisson for counts of nonconformities per unit. Mixing them up gives wrong standard errors.
• Ignoring sample size. The formula contains n in the denominator. A small n inflates the standard error and makes the index look worse than it is. Always report n.
• Treating Cpu like a normal Cpk. Attribute capability is one-sided and highly sensitive to small shifts in the average. Do not compare it directly to a continuous-data Cpk without context.
• Forgetting the upper limit only. If your customer gives a "defects per million" target, convert it to a proportion or rate first, then apply the correct formula.

Closing

Attribute capability (binomial/Poisson) gives you a rigorous, ISO 22514-aligned way to judge whether your defect rate is good enough for your customer's limit — provided you check stability first and choose the right model. To avoid manual calculation errors and get instant results, use the free 6SQ attribute capability calculator at:
https://www.6sq.com/tools/attr_cap/