KEVOS® MAKEOS · Quality
p-Chart Check (Fraction Defective, Control Limits, Samples Out of Control)
From the number of defective items in each sample of the same size, what is the average fraction defective, where are the control limits, and is any sample out of control?
Proportions (p) control chart from up to ten samples of equal size: the fraction defective in each sample, the average fraction defective, the upper and lower control limits from the binomial standard deviation (lower limit not below zero), which samples fall outside them, and the defectives per sample that would trigger a signal.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Items inspected in each sample | items | Number of items inspected in every sample (equal sample sizes). |
| Number of samples used | samples | How many of the samples below to use, in time order. |
| Defective items in each sample | ||
| Sample 1 defectives | items | Number of defective (nonconforming) items found in sample 1. |
| Sample 2 defectives | items | Number of defective (nonconforming) items found in sample 2. |
| Sample 3 defectives | items | Number of defective (nonconforming) items found in sample 3. |
| Sample 4 defectives | items | Number of defective (nonconforming) items found in sample 4. |
| Sample 5 defectives | items | Number of defective (nonconforming) items found in sample 5. |
| Sample 6 defectives | items | Number of defective (nonconforming) items found in sample 6. |
| Sample 7 defectives | items | Number of defective (nonconforming) items found in sample 7. |
| Sample 8 defectives | items | Number of defective (nonconforming) items found in sample 8. |
| Sample 9 defectives | items | Number of defective (nonconforming) items found in sample 9. |
| Sample 10 defectives | items | Number of defective (nonconforming) items found in sample 10. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Items inspected in each sample
- What it is
- The number of items inspected in every sample, the same for each sample on this chart.
- Units and format
- Enter a number in items: a count of items. The form accepts 2 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Larger samples narrow the control limits, so smaller shifts show.
- Get it right
- Samples of different sizes need limits worked out for each sample; this tool assumes they are all the same.
What this means: Number of samples used
- What it is
- How many of the samples listed below are used for the chart, taken from the first sample in time order.
- Units and format
- Enter a number in samples: samples. The form accepts 2 samples to 10 samples. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 samples as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Samples beyond this count are ignored.
- Get it right
- Use 20 to 25 samples for settled limits; ten gives a first look.
Defective items in each sample
What this means: Sample 1 defectives
- What it is
- The number of defective (nonconforming) items found in sample 1, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 1.
What this means: Sample 2 defectives
- What it is
- The number of defective (nonconforming) items found in sample 2, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 2.
What this means: Sample 3 defectives
- What it is
- The number of defective (nonconforming) items found in sample 3, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 3.
What this means: Sample 4 defectives
- What it is
- The number of defective (nonconforming) items found in sample 4, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 4.
What this means: Sample 5 defectives
- What it is
- The number of defective (nonconforming) items found in sample 5, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 5.
What this means: Sample 6 defectives
- What it is
- The number of defective (nonconforming) items found in sample 6, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 6.
What this means: Sample 7 defectives
- What it is
- The number of defective (nonconforming) items found in sample 7, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 7.
What this means: Sample 8 defectives
- What it is
- The number of defective (nonconforming) items found in sample 8, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 8.
What this means: Sample 9 defectives
- What it is
- The number of defective (nonconforming) items found in sample 9, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 9.
What this means: Sample 10 defectives
- What it is
- The number of defective (nonconforming) items found in sample 10, out of the items inspected in each sample.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 10000000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises that sample’s fraction defective and, slightly, the average and the limits.
- Get it right
- Count defective items, not defects: an item with three faults counts once in sample 10.
Method and formulas
- Fraction defective pᵢ = dᵢ ÷ n; average p̄ = Σ dᵢ ÷ (k n).
- Control limits = p̄ ± 3 √(p̄ (1 − p̄) ÷ n) (NIST e-Handbook 6.3.3.2); a negative lower limit is set to zero.
- A sample above the upper or below a positive lower limit signals a special cause; signal threshold = the defectives count just above n × UCL.
- Example: average 2.5 % defective with an upper limit of 5.8119 %; 0 sample(s) outside.
Limitations
- Assumes equal sample sizes and independent items; varying sample sizes need limits for each sample.
- Limits from ten samples are rough: recalculate when 20 to 25 samples are available.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Chart the reject rate of an inspection station
- Check whether a bad batch is a special cause or normal variation
- Set alarm levels for daily defect counts
Who it is for
- Quality engineers and technicians
- Production supervisors
- Continuous improvement staff
What you get
- Fraction defective of each sample
- Average fraction defective
- Upper and lower control limits
- Samples outside the limits and the signal threshold
Interpreting the result
- With a small average fraction defective, the lower limit is zero, so the chart cannot show an improvement until the sample size is increased.
References
- NIST/SEMATECH e-Handbook of Statistical Methods, 6.3.3.2 Proportions Control Charts — https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm
- NIST Dataplot Reference Manual, P CONTROL CHART (lower limit bounded at zero) — https://www.itl.nist.gov/div898/software/dataplot/refman1/ch2/pcontrol.pdf
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Learning the defect rate
In Accept or Reject by the Numbers
What it covers: Estimate and update a defect rate from audits: a p-chart with the KEVOS® tool, Bayesian updating with a beta prior, separating special causes, and correcting an imperfect reject rate.
Method explained here · KEVOS® original book
The random-mixture limit: counting particles
In Blending Bulk Solids by the Numbers
What it covers: Why even a perfect random blend varies from sample to sample, how the variance follows from counting particles, why the tails matter more than the average, and what changes the limit.
Method explained here · KEVOS® original book
Charts for counts and proportions
In Control Charts by the Numbers
What it covers: Defectives and defects as binomial and Poisson random variables: p and c chart limits, why they are asymmetric, limits that change with sample size, missing lower limits, and when to use a u chart.
Method explained here · KEVOS® original book
Finding surface cracks
In Eddy-Current Inspection by the Numbers
What it covers: Scanning ground journals for cracks: reference notches, signal and noise, the alarm threshold, scan pitch and coverage, how scan speed meets the instrument's filters, and charting rejects.
Method explained here · Library collection
Statistical process control on the shop floor: control charts, real signals and when to leave a process alone
What it covers: Control charts separate the normal noise of a process from real change. How to choose a chart, set limits, read the signals and stop well-meant adjustments that make variation worse.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the p-Chart Check (Fraction Defective, Control Limits, Samples Out of Control) calculator