Twenty inspection units, an average of 5.5 defects each. The limit sits at 12.5356 and nothing exceeds it. The number worth reading is the exact tail: the true probability of a count above that limit is 0.004451, not the 0.00135 a three-sigma limit is supposed to give. At this mean the chart alarms about 3.3 times more often than it claims to.
20 inspections · 110 defects · count per unit
Centre 5.5000, limit 12.5356
Every inspection covers the same amount, so the limits are constant. 0 of 20 inspections exceed their limit. The exact probability of exceeding it under a stable process is 4.451e-3 — 3.3× the 0.00135 a three-sigma limit is supposed to give.
Centre
5.5000
110 defects over 20 units
Upper limit
12.5356
lower 0.0000
Out of control
0 of 20
none
True alarm rate
4.45e-3
3.3× the nominal 0.00135
That last figure is the exact Poisson tail above this chart’s own limit, summed rather than approximated. Three-sigma limits assume a symmetric distribution; a count distribution is skewed, and the gap is largest at low counts. At this mean the approximation is holding up reasonably well.
Every inspection
Each inspection with its count and limits
#
Defects
Limits
Status
1
5
0.0000 to 12.5356
—
2
3
0.0000 to 12.5356
—
3
7
0.0000 to 12.5356
—
4
4
0.0000 to 12.5356
—
5
6
0.0000 to 12.5356
—
6
8
0.0000 to 12.5356
—
7
2
0.0000 to 12.5356
—
8
5
0.0000 to 12.5356
—
9
4
0.0000 to 12.5356
—
10
9
0.0000 to 12.5356
—
11
6
0.0000 to 12.5356
—
12
3
0.0000 to 12.5356
—
13
7
0.0000 to 12.5356
—
14
5
0.0000 to 12.5356
—
15
4
0.0000 to 12.5356
—
16
11
0.0000 to 12.5356
—
17
6
0.0000 to 12.5356
—
18
3
0.0000 to 12.5356
—
19
5
0.0000 to 12.5356
—
20
7
0.0000 to 12.5356
—
A Poisson has its variance equal to its mean, so the limit is the centre plus three square roots of the centre. Nothing about the spread of the data enters it — which makes this the only chart here where the limits can be computed before any data is collected.
Poisson, not normal c and u charts Exact alarm rate shown Counts, not defective units
What this tool shows
A three-sigma limit is supposed to be a 0.00135 tail. On a count chart it is not. The exact probability of exceeding this chart’s own limit is 0.004451 at a mean of 5.5 and 0.01439 at a mean of 0.5 — ten times the nominal rate, at exactly the count levels where these charts are most used. That figure is computed live here by summing the Poisson tail, rather than assumed from the normal approximation the limit is built on.
c charts for defect counts on equal inspection units
u charts for defects per unit area, with limits that move as the area does
The exact Poisson tail above the chart’s own limit, computed rather than assumed
The distinction from a p chart, which counts defective units rather than defects
A rare-defect preset where the three-sigma limit alarms ten times too often
Why these limits can be drawn before any data is collected
Poisson based c and u charts Exact alarm rate Limits from the mean alone
The variance is the mean. If the data says otherwise, this is the wrong chart.
Updated 13 September 2026 · Works in any browser, no installation
A c chart tracks how many defects appear on each inspection unit, with limits from the Poisson. A unit can carry any number of defects, which is what separates it from a p chart, where each unit is simply good or bad. The Poisson has its variance equal to its mean, so the limits follow from the average count alone — no estimate of spread is needed, and none is made.
At a glance
Formula shown
c̄ = total defects / number of inspection units, and the limits are c̄ ± 3√c̄ with the lower one clamped at zero. For unequal inspection areas the same idea becomes a u chart: ū = total defects / total area, with limits ū ± 3√(ū/aᵢ) per point. Because a Poisson variance equals its mean, nothing about the observed spread enters those limits — which is both the convenience and the assumption.
Scenario support
Surface defects per panel or per square metre, errors per document or per thousand lines of code, complaints per week, infections per thousand patient-days, and any count of events over a consistent opportunity where a single unit can carry several.
Educational estimate
Planning support from the values you enter — not professional advice.
The three-sigma limit is not a 0.00135 tail
Three-sigma limits get their reputation from a normal distribution. A count distribution is skewed, and the limits inherit that.
At a mean of 5.5 the limit is 12.5356 and the exact probability of exceeding it is 0.004451. Not 0.00135 — 3.3 times that.
At a mean of 0.5 the limit is 2.6213 and the exact tail is 0.01439. Ten and a half times nominal, which is the regime most c charts actually operate in.
At a mean near 48 it falls to 0.002047, about 1.5 times nominal. The approximation improves as the count rises, and never quite arrives.
Every one of those is computed here by summing the Poisson exactly, for whatever data is in the box — so the chart states its own false-alarm rate instead of the one it is assumed to have.
Defects, not defective units
Choosing between this chart and a p chart is not a matter of preference. It decides which distribution sits under the limits.
A c chart counts defects. One unit can have five. The count has no upper bound, and the Poisson has none either.
A p chart counts defective units. Each unit contributes at most one, so the count is bounded by the sample size and the distribution is binomial.
Thirty scratches across 200 panels is a c chart. Ten scratched panels out of 200 is a p chart. Same inspection, two different questions.
The test is whether a count could exceed the number of units inspected. If it could, the binomial is the wrong model, and p-chart limits on that data would be far too narrow.
When the variance is not the mean
The whole chart rests on one assumption that nothing in the output checks, and real defect data frequently breaks it.
A Poisson has variance equal to its mean. Defects arriving independently at a constant rate produce exactly that.
Real defects cluster. A bad batch of material, a misaligned tool, a single bad shift — each produces several defects at once, and the variance exceeds the mean.
The symptom is a chart where most points are outside the limits with no pattern. The limits are too narrow because they assumed away the clustering.
Check it before charting: compute the variance of the counts and compare it against their mean. A ratio well above 1 means a Laney u′ chart or an individuals chart on the counts, both of which take the spread from the data rather than from a model.
When the inspection area varies
A c chart requires every inspection to cover the same amount of opportunity. When it does not, adding a second column changes the chart.
A u chart divides by the area and plots a rate. Defects per square metre, per thousand lines, per patient-day.
The limits then move with the area, for the same reason they move on a p chart with unequal samples: a rate from a small inspection is noisier.
On the shipped preset they run from 6.8920 on the largest inspection to 8.7723 on the smallest — a 27% spread, from areas differing by a factor of under three.
Charting raw counts when the areas differ is the error to avoid. A large inspection finds more defects because it looked at more, and the chart will report that as a special cause.
The missing lower limit
On most c charts the lower limit is zero, and that is a real loss rather than a formality.
c̄ − 3√c̄ stays above zero only when c̄ exceeds 9. Below that the limit clamps and the chart becomes one-sided.
So an improvement can never signal. A process that halves its defect rate produces no point outside the limits, because there is no lower limit to fall below.
Which matters when the chart is being used to confirm that a change worked. The run rules will catch a sustained improvement; the limits will not.
At a mean of 9 or more the lower limit exists and an unusually clean inspection becomes a signal — usually that the inspection changed rather than the process.
The only chart whose limits need no data
Every other chart here estimates a spread from the data. This one does not, and that has a practical consequence worth using.
A Poisson is a one-parameter distribution. Give it a mean and the variance follows, so the limits follow too.
Which means limits can be set from a target rather than from history: at a target of 4 defects per unit the limits are 0 and 10, before any inspection has happened.
That makes a c chart usable on a new process where there is no stable baseline to fit limits to — the problem that makes every other chart here need a phase-one study first.
The cost is that a wrong target gives wrong limits with no warning, and no amount of data will correct them, because the data never enters the calculation.
Reporting a count chart
Four items, and the first is what makes a count comparable at all.
Define the inspection unit precisely. Defects per what — per panel, per square metre, per shift? A count is meaningless without it.
Say whether it is a c chart or a u chart. They answer the same question on different axes and only one is valid when the areas differ.
Give the variance-to-mean ratio. It is the one number that says whether the Poisson assumption underneath the limits holds, and it takes one line to compute.
And say where the centre came from. Fitted from the data or set from a target — this is the one chart where both are defensible, and they are not the same.
Method. The centre is the total defect count over the number of inspection units, and the limits are three square roots either side of it — a Poisson variance equals its mean, so no estimate of spread is made and none is needed. The exact probability of exceeding the resulting limit is then computed by summing the Poisson tail term by term, rather than assumed to be the 0.00135 a three-sigma limit is nominally worth; that summation is checked against an independently written direct sum on 120 generated cases, and the suite asserts that the over-alarm shrinks as the mean rises and is still above nominal at a mean near 48. Unequal inspection sizes switch the chart to a rate with per-point limits, since a rate from a small inspection is noisier than one from a large one. Negative counts are dropped rather than charted, and the lower limit is clamped at zero with the clamping visible. That engine is verified on every change against 139 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
p Chartp and np charts with limits computed per point when the sample sizes differ, and the cost of flat limits measured on your own chart.
Poisson DistributionPoisson probabilities with a dispersion test against your own variance — because a Poisson forces variance to equal the mean, and real count data usually does not, which is exactly where the tail goes wrong.
Individuals Control ChartIndividuals and moving range charts with sigma from the average moving range, both sigma estimates shown, and all eight Nelson rules counted.
X-bar and R ChartX-bar with R or S charts, every Shewhart constant computed from its definition rather than looked up, and all eight Nelson rules.
DPMODefects per million opportunities with the sigma level computed both with and without the 1.5 shift, plus rolled throughput yield.
Incidence RateEvents per person-time with the exact Poisson interval, including at zero events: the exact two-sided upper limit is 3.688879 per unit of person-time, not the 3 the one-sided rule of three gives.
An educational tool. A c chart assumes defects arrive independently at a constant rate, so its variance equals its mean — real defects cluster, and clustered data produces limits that are far too narrow. Three-sigma limits also rest on a normal approximation that a skewed count distribution does not satisfy: at a mean of 0.5 the true alarm rate is ten times the nominal one, which this page computes rather than assumes.
Launched c and u charts for defect counts, with limits from the Poisson rather than from the observed spread.
Computed the EXACT probability of exceeding the chart own three-sigma limit by summing the Poisson tail: 0.01439 at a mean of 0.5, ten and a half times the nominal 0.00135.
Showed that over-alarm shrinking as the mean rises — 3.3x at 5.5 and 1.5x at 47.55 — and asserted the ordering in the suite.
Switched to a rate chart with moving limits whenever the inspection areas differ.
Documented the overdispersion failure and the missing lower limit below a mean of 9.
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