Packaging Statistics 101: AQL, Cpk, and Process Capability

If you have ever written or approved a packaging spec like “My box is 10.000 inches +/- 1/8 inch,” it feels clear. But it hides the real question: What does that tolerance mean when you are making 1,000,000 boxes?

How many will be out of spec? How many will jam your line? And can your supplier’s process even hold what you just put on the drawing?

This article is a practical, packaging-focused walkthrough of three ideas that every packaging engineer should understand:

• Cpk / process capability: can the process hold the tolerance?
• AQL acceptance sampling: what sample and decision rule should be used for a lot?
• Tolerance analysis / interference fit: what happens when product and package variation collide?

1. The “10 Inches +/- 1/8” Problem: What It Means at Scale

Consider a typical corrugated box dimension spec:

• Nominal = 10.000 in
• Tolerance = +/- 0.125 in
• LSL (Lower Spec Limit) = 9.875 in
• USL (Upper Spec Limit) = 10.125 in
• Total spec width = 0.250 in

That spec does not tell you:

• How tight the supplier’s process actually is.
• Whether the process is centered on the target.
• How many out-of-spec units you will see over time.

Two suppliers can both “target 10 inches” and have very different outcomes:

• Supplier A: Tight process, well-centered, almost no defects.
• Supplier B: Wider process, slightly off-center, thousands of defects per million.

This is exactly what process capability (Cpk) quantifies.

2. Cpk: “Can the Machine Actually Hold the Tolerance?”

Process capability compares specification limits with the spread and centering of an in-control process. Before interpreting Cpk, check process stability, measurement quality and whether a normal model is appropriate. A calculated index alone does not establish those conditions. NIST: What is Process Capability?

The formula:

Cpk = min(Cpu, Cpl)

where:
  Cpu = (USL - mean) / (3 * sigma)
  Cpl = (mean - LSL) / (3 * sigma)

Why “min()” matters: Centering matters. A process can have small variation but still produce defects if the mean drifts toward one spec limit. Cpk captures the worst side of the distribution relative to the nearest limit.

3. A Concrete Packaging Example: 1,000,000 Boxes

Using the spec from Section 1, assume independent observations from a stable, normally distributed process, with an adequate measurement system. For this illustrative calculation, take:

• Mean = 10.000 in (perfectly centered)
• Standard deviation (sigma) = 0.040 in

Computing Cpk:

Cpu = (10.125 - 10.000) / (3 * 0.040) = 0.125 / 0.120 = 1.04
Cpl = (10.000 - 9.875) / (3 * 0.040) = 0.125 / 0.120 = 1.04

Cpk = min(1.04, 1.04) = 1.04

Interpretation: The unrounded Cpk is 1.0417. Each specification limit is 0.125 / 0.040 = 3.125 standard deviations from the mean. Under the stated normal model, the tail probability is approximately 889 parts per million on each side, or 1,778 out-of-spec units per million in total. This is a model estimate, not a prediction from Cpk alone or a guarantee about the next lot. Do not round Cpk before calculating tail probabilities.

Now watch what happens if the process mean drifts by only 0.040 in (one sigma):

New mean = 10.040 in

Cpu = (10.125 - 10.040) / 0.120 = 0.085 / 0.120 = 0.71
Cpl = (10.040 - 9.875) / 0.120 = 0.165 / 0.120 = 1.38

Cpk = min(0.71, 1.38) = 0.71

The nearest limit is now only 0.085 / 0.040 = 2.125 standard deviations away. The normal model gives approximately 16,793 ppm above the upper limit and 19 ppm below the lower limit, or 16,812 ppm total. A one-sigma shift has changed the expected count substantially, even though the process spread stayed the same.

You can run these calculations directly using the Statistics Suite, which includes process capability analysis with Cpk, Cp, and sigma-level outputs.

4. What Cpk Number Should You Require for Packaging?

The meaning of a Cpk value is easiest to see in a centered normal-model example. In that special case the nearest limit is 3 × Cpk standard deviations away:

These are mathematical comparisons, not prescribed thresholds for medical, food or other regulated packaging.

Your “right answer” depends on four factors:

• Volume: Higher volume amplifies even small defect rates.
• Cost of a defect: A jammed line or a field return is far more expensive than a scrapped blank.
• Interaction: Does this dimension mate with another part? (See Section 7 on interference fit.)
• Measurement system quality: Your Cpk number is only as good as your gauge.

5. AQL in Packaging: “How Many Bad Boxes Before Rejecting the Lot?”

AQL means acceptance quality limit in ISO 2859-1. It indexes an attribute-sampling system for lot-by-lot inspection. The current ISO publication is ISO 2859-1:2026; specify the edition required by your customer or quality agreement. ISO 2859-1 publication and scope

AQL is not a guarantee that an accepted lot contains no more than that percentage of nonconforming units. A sampling plan has a probability of acceptance at each possible underlying defect rate. Sample size, acceptance number and the actual lot quality determine that behavior. NIST: lot acceptance sampling plans

Define defect categories before inspection:

• Critical: a defect whose consequence requires the highest level of control, such as a safety or sterile-barrier concern. An ordinary AQL table must not substitute for the applicable product-risk controls.
• Major: a defect that prevents intended use, such as a carton that will not erect or an opening that prevents assembly.
• Minor: a defect with a smaller functional consequence, such as an agreed cosmetic blemish.

Choose the sampling scheme and acceptance criteria with the quality owner. The category names do not establish universal AQL values. Specify whether the inspection counts nonconforming units or individual nonconformities; a package may have more than one defect.

PackCalc’s Statistics Suite includes a normal-inspection sampling-plan calculator and operating-characteristic analysis. Its implemented table is based on ANSI/ASQ Z1.4-2003 single sampling. It is not the complete ISO 2859-1:2026 system, including all switching rules and inspection states. Confirm the selected plan against the edition and scheme required for the actual inspection.

6. Where AQL and Cpk Get Mixed Up

Teams often confuse these two tools because both involve “defects” and “quality.” Here is the distinction:

• Cpk is about the process: Is it stable, centered, and tight enough to consistently produce conforming output?
• AQL sampling is about the inspection decision: Given a finite lot, how do we decide to accept or reject it?

These are complementary, not interchangeable:

1. Use Cpk to qualify the process and set specifications that the supplier can actually hold.
2. Use AQL sampling as an ongoing risk-based inspection check to catch lots that slip.

A passed incoming sample gives evidence about that lot under the chosen rule. It does not demonstrate that a drifting production process has been corrected. Investigate recurring failures through process and measurement data as well as lot inspection.

7. Interference Fit in Packaging: When Variation Makes the Product “Bigger Than the Box”

Product dimensions and packaging dimensions both vary. Even if both drawings look “fine” in isolation, overlap in the distributions creates insertion force spikes, jams, cracked trays, and field returns.

Worst-case stack analysis (conservative):

• Minimum clearance = minimum package opening minus maximum product size. Negative clearance indicates interference.
• This bounds the dimensional combination when the individual limits are valid; it does not by itself assess insertion forces or package deformation.

Statistical stack analysis (realistic):

• Treat both product and package dimensions as distributions.
• For independent dimensional errors, combine standard deviations using RSS (Root Sum of Squares). Include covariance when dimensions are correlated. A Monte Carlo simulation can represent other justified distributions and relationships.
• Compute the probability of interference at volume.

The clearance itself needs analysis. Two individually capable dimensions can still interfere if their nominal clearance is too small. For example, a package opening of 10.000 in and product size of 9.950 in have 0.050 in nominal clearance. With independent standard deviations of 0.020 in for both, clearance sigma is sqrt(0.020² + 0.020²) = 0.0283 in. Under normal assumptions, the nominal gap is only about 1.77 clearance sigmas. Review that fit directly rather than approving it from two Cpk values.

8. The Hidden Trap: Tolerances Without Capability Are Just Wishes

Teams set tolerances by:

• Copying old drawings.
• Eyeballing clearances.
• Picking “neat” fractions (1/8, 1/16).
• Over-trusting supplier claims.

Set functional limits from what the package must do, then assess whether the process can meet them. If capability is inadequate, improve the process, measurement or design; widening a tolerance is acceptable only when fit and function still meet the requirements.

The real workflow:

1. Stabilize the process: get it in statistical control (SPC charts).
2. Measure variation correctly: validate the gauge (R&R study), collect enough data.
3. Compute Cp/Cpk: quantify whether the process can hold the tolerance.
4. Set AQL sampling as a risk-based ongoing check, not a substitute for capability.

A) Glossary (short)

• Cpk (Process Capability Index): A statistical index that measures how well a process output fits within specification limits, accounting for both spread and centering. Cpk = min((USL - mean) / (3 × sigma), (mean - LSL) / (3 × sigma)).
• Cp (Process Capability): Similar to Cpk but assumes the process is perfectly centered. Cp = (USL - LSL) / (6 × sigma). Useful for understanding potential capability.
• AQL (Acceptance Quality Limit): A parameter used in acceptance sampling plans (ISO 2859-1, ANSI/ASQ Z1.4) to define the quality level that is the worst tolerable process average when a continuing series of lots is submitted for inspection.
• ISO 2859-1: The international standard for sampling procedures for inspection by attributes, indexed by AQL.
• ANSI/ASQ Z1.4: An acceptance-sampling standard used for inspection by attributes; identify the required edition and inspection scheme.
• Gauge R&R (Repeatability and Reproducibility): A study that quantifies how much of the observed measurement variation is due to the measurement system itself versus the actual part-to-part variation.
• SPC (Statistical Process Control): The use of control charts and statistical methods to monitor and control a process, ensuring it operates at its full potential.
• RSS (Root Sum of Squares): A way of combining independent standard deviations; correlations and the actual distributions must be considered.
• Interference Fit: A condition where the product dimension exceeds the package opening dimension, causing insertion difficulty or damage.
• LSL / USL: Lower Specification Limit and Upper Specification Limit, the boundaries of acceptable variation defined on a drawing or spec.

Sources

• NIST/SEMATECH: What is Process Capability?.
• NIST/SEMATECH: Lot Acceptance Sampling Plans and operating-characteristic curves.
• ISO 2859-1:2026: publication and scope.
• Worked examples are calculated from the stated hypothetical dimensions and normal-distribution assumptions.