Chart nonconforming units with the right denominator

Chart nonconforming units with the right denominator

For nonconforming units, every inspected unit contributes at most one failure to the count. A p chart displays the fraction nonconforming and can accommodate different sample sizes with different limits. An np chart displays the count and assumes a constant sample size in this basic form. A larger sample narrows the p-chart limits because it changes the expected sampling variation; it does not change the customer specification. Estimate the baseline from appropriate totals and retain each denominator. The example uses a supplied baseline proportion. Dependence, changing product mix and unsuitable binomial assumptions require investigation rather than blind application of these formulas.

Follow the method

1. p: sample size may vary
2. Fraction nonconforming
3. Sample in time order
4. Center: 0.1
5. LCL: 0.01 / 0.0363604
6. UCL: 0.19 / 0.16364
7. np: fixed sample size 100
8. Nonconforming units (count)
9. Center: 10
10. LCL: 1
11. UCL: 19

Match each unit fraction to its own sample-size limits

Fictional case: an inspection desk classifies each item once as conforming or nonconforming. A supplied historical fraction is .10. The core p example uses sample sizes 100,200,100,200 and counts 8,24,11,17; the separate np example always inspects 100 and records 8,12,11,9. Analyst Pia explains why limits must reflect the observation design.

Follow the method

1. p at n100
2. p at n200
3. np at fixed n100
4. Interpretation

Twice the count, the same fraction

Keep nonconforming-unit counts linked to sample size, calculate the appropriate fraction or fixed-size count limits, and interpret a changed denominator.

Fictional case: an inspection desk classifies each item once as conforming or nonconforming. A supplied historical fraction is .10. The core p example uses sample sizes 100,200,100,200 and counts 8,24,11,17; the separate np example always inspects 100 and records 8,12,11,9. Analyst Pia explains why limits must reflect the observation design.

Role: Quality analyst and inspection owner

Normal condition

Every sample records its actual inspected amount; one item contributes at most one nonconforming classification; model and baseline are suitable for the declared process.

The gap

A supervisor compares 24 failed items in 200 with 12 in 100 and declares the larger sample twice as bad. Both are 12%.

• Historical pbar=.10 is supplied, not estimated from these four samples.
• Dependence, overdispersion or changing risk can require another model; these are not acceptance-sampling limits.

1. Confirm what is counted

   Pia checks that an item with several faults is still classified once in this unit-count record. She retains fault-category details separately if needed.

   Why: A p or np chart describes affected units, not the number of faults within them. Mixing those numerators would break the interpretation.

   Evidence: Each count is bounded by its inspected sample size.

2. Calculate fractions and preserve n

   She divides each sample count by its actual n and retains n beside the plotted point. Sample 2 is 24/200=.12, not 24/100.

   Why: Different sample sizes can produce different counts with the same fraction. The denominator also determines the expected sampling variation.

   Evidence: The sequence is .08,.12,.11,.085 with its four actual sample sizes.

3. Apply the supplied p baseline

   Using .10 ±3√(.10×.90/n), n100 gives .01–.19; n200 gives approximately .03636–.16364. The center stays .10.

   Why: The larger sample has narrower fraction limits under this model. Reusing fixed limits for all denominators would discard relevant information.

   Evidence: Each point receives limits matching its own n; limits remain within possible fraction values.

4. Explain the fixed-size count view

   For the separate fixed-n100 illustration, np center is 10 and limits are 1–19 nonconforming units. Pia labels counts and does not mix this axis with fractions.

   Why: A count view is interpretable when the inspected amount is fixed. A p view can also be used with fixed n if it is clearly labelled.

   Evidence: The np sequence 8,12,11,9 uses n100 throughout.

5. Record a response rather than a verdict

   Pia documents any signal against the supplied baseline and checks measurement, sampling and process context. Product containment and release follow the applicable quality procedure.

   Why: A point within statistical limits is not proof that the defect level is acceptable. A signal identifies changed behavior worth responding to, not its cause.

   Evidence: The record separates chart evidence, product decisions and investigation ownership.

Inspection amount changes on the np record

A busy shift inspects 200 instead of 100 and enters 20 in the count chart.

Retain 20/200=.10 and handle the sampling-design change through the chart owner; do not compare 20 directly with the old fixed-n100 upper limit 19.

The change in denominator alone can explain the count increase. The original chart was designed for fixed n.

The observation is preserved with n200 and a documented display/design decision.

Run a practice session

Materials

• Sample-size/count cards
• Calculator
• Blank response record

1. Define affected units · 5 minutes

   Can one item contribute three here?

2. Compare n100 and n200 · 8 minutes

   Why do fraction limits narrow?

3. Evaluate 36 of 200 · 10 minutes

   What denominator and limit apply?

4. Debrief · 5 minutes

   Why does a chart not authorize product release?

Debrief

• Ask learners to explain why 12/100 and 24/200 are equal.
• Challenge the unsupported 36% calculation by tracing the inspected population.

Write n next to each numerator before choosing a plotting scale.

Transfer into the work

Owner: SPC and inspection owners

Record: Count, denominator, baseline and sample-specific response record

Review: At each sampling interval and whenever sampling changes

Evidence: One classification per unit, correct n and verified response

Resolve denominator/model changes before applying a fixed-n interpretation.