Modules / Tools / Statistical process control chart

Statistical process control chart

Draw X-bar and R, X-bar and S, individuals and moving range, p, np, c and u charts from your data, with control limits and the run rules that flag a process out of control.

Quality statisticsFree to useHow it works Sources Where it helps

Start from a preset

Fills the inputs below with typical figures. Replace any of them with your own.

Chart and data

Chart type

Unit label (optional)

Shown on the chart axis.

Data

One subgroup per line, readings separated by commas, spaces or tabs. Every subgroup the same size, 2 or more. Best for subgroups up to about 10.

Limits and rules

Limits from

Points used for the limits (0 for all)

Use the first points as a baseline and judge the rest against those limits.

Points beyond the limits

Leave out only points whose cause is known and removed.

Run rules

Run rules apply to the main chart. The range, deviation and moving range charts use the limits only.

Width of the limits (sigma)

3 is the usual width.

X-bar chart: centre line: 10; Upper 10.158, lower 9.8417
R chart: centre line: 0.21725; Upper 0.49578, lower 0
Estimated sigma: 0.1055; average range / d2 (d2 = 2.0588 for subgroups of 4)
Points beyond the limits: 0; 20 points, 20 used for the limits
Points breaking any rule: 0; Western Electric rules 1 to 4, both charts

No point breaks the selected rules.

X-bar chart

Data (solid line, circles)
Centre line (dashed line, squares)
Upper limit (dotted line, triangles)
Lower limit (dash-dot line, diamonds)

Show the data as a table

X-bar chart
Point number	Data	Centre line	Upper limit	Lower limit
1	9.959	10	10.16	9.842
2	9.985	10	10.16	9.842
3	9.934	10	10.16	9.842
4	10.08	10	10.16	9.842
5	9.963	10	10.16	9.842
6	9.95	10	10.16	9.842
7	10.01	10	10.16	9.842
8	9.929	10	10.16	9.842
9	9.997	10	10.16	9.842
10	10.03	10	10.16	9.842
11	9.975	10	10.16	9.842
12	10.04	10	10.16	9.842
13	10.09	10	10.16	9.842
14	10.05	10	10.16	9.842
15	10.03	10	10.16	9.842
16	9.947	10	10.16	9.842
17	9.967	10	10.16	9.842
18	9.986	10	10.16	9.842
19	10.04	10	10.16	9.842
20	10.04	10	10.16	9.842

X-bar chart: every point and the rules it breaks
Point	Value	Lower limit	Centre	Upper limit	Rules broken
1	9.9585	9.8417	10	10.158	none
2	9.9848	9.8417	10	10.158	none
3	9.934	9.8417	10	10.158	none
4	10.076	9.8417	10	10.158	none
5	9.9627	9.8417	10	10.158	none
6	9.9503	9.8417	10	10.158	none
7	10.011	9.8417	10	10.158	none
8	9.9292	9.8417	10	10.158	none
9	9.9973	9.8417	10	10.158	none
10	10.029	9.8417	10	10.158	none
11	9.975	9.8417	10	10.158	none
12	10.044	9.8417	10	10.158	none
13	10.094	9.8417	10	10.158	none
14	10.047	9.8417	10	10.158	none
15	10.026	9.8417	10	10.158	none
16	9.9465	9.8417	10	10.158	none
17	9.9667	9.8417	10	10.158	none
18	9.9863	9.8417	10	10.158	none
19	10.043	9.8417	10	10.158	none
20	10.037	9.8417	10	10.158	none

R chart

Data (solid line, circles)
Centre line (dashed line, squares)
Upper limit (dotted line, triangles)
Lower limit (dash-dot line, diamonds)

Show the data as a table

R chart
Point number	Data	Centre line	Upper limit
1	0.095	0.2173	0.4958
2	0.077	0.2173	0.4958
3	0.305	0.2173	0.4958
4	0.193	0.2173	0.4958
5	0.235	0.2173	0.4958
6	0.296	0.2173	0.4958
7	0.366	0.2173	0.4958
8	0.297	0.2173	0.4958
9	0.208	0.2173	0.4958
10	0.189	0.2173	0.4958
11	0.309	0.2173	0.4958
12	0.231	0.2173	0.4958
13	0.181	0.2173	0.4958
14	0.207	0.2173	0.4958
15	0.114	0.2173	0.4958
16	0.217	0.2173	0.4958
17	0.243	0.2173	0.4958
18	0.3	0.2173	0.4958
19	0.197	0.2173	0.4958
20	0.085	0.2173	0.4958

R chart: every point and the rules it breaks
Point	Value	Centre	Upper limit	Rules broken
1	0.095	0.21725	0.49578	none
2	0.077	0.21725	0.49578	none
3	0.305	0.21725	0.49578	none
4	0.193	0.21725	0.49578	none
5	0.235	0.21725	0.49578	none
6	0.296	0.21725	0.49578	none
7	0.366	0.21725	0.49578	none
8	0.297	0.21725	0.49578	none
9	0.208	0.21725	0.49578	none
10	0.189	0.21725	0.49578	none
11	0.309	0.21725	0.49578	none
12	0.231	0.21725	0.49578	none
13	0.181	0.21725	0.49578	none
14	0.207	0.21725	0.49578	none
15	0.114	0.21725	0.49578	none
16	0.217	0.21725	0.49578	none
17	0.243	0.21725	0.49578	none
18	0.3	0.21725	0.49578	none
19	0.197	0.21725	0.49578	none
20	0.085	0.21725	0.49578	none

Centre lines and control limits
Item	Value
X-bar chart: centre line	10
X-bar chart: upper control limit	10.158
X-bar chart: lower control limit	9.8417
R chart: centre line	0.21725
R chart: upper control limit	0.49578
R chart: lower control limit	0

How it works

A control chart separates the ordinary, ever-present variation in a process from the signals that something has changed. This tool draws the Shewhart charts used in semiconductor lines: X-bar with R or S for subgroups, individuals with a moving range for single readings, and the p, np, c and u charts for counts. It computes the limits from your data (phase I) or from a standard you already have (phase II), and checks the run rules.

Choose the chart by the data. Subgroup readings (several parts measured together) go on an X-bar chart with R for small subgroups or S for larger ones. Single readings go on an individuals chart with a moving range. A count of defective units out of a sample uses p (fraction) or np (number); a count of defects found in an inspection unit uses c when the unit is always the same size and u when it varies.
Variables chart limits. Sigma is estimated from the average range, average standard deviation or average moving range with the constants d2, d3 and c4, which this tool computes exactly rather than reading from a table. Limits are three sigma (or the width you set) either side of the centre line; a negative lower limit on a range or standard deviation chart is set to zero.sigma = R-bar / d2, or s-bar / c4, or MR-bar / d2(2); X-bar limits = mean ± 3 sigma / sqrt(n); R limits = R-bar ± 3 d3 sigma; S limits = s-bar ± 3 sqrt(1 − c4^2) x s-bar / c4
Individuals chart limits. The moving range is the absolute difference between successive readings. Its average over d2 for ranges of 2 (1.128) estimates sigma. The individuals limits are the mean plus or minus three of those sigmas, which is 2.66 average moving ranges.limits = mean ± 3 MR-bar / 1.128; MR chart upper limit = MR-bar + 3 d3(2) MR-bar / d2(2) = 3.267 MR-bar
Attribute chart limits. Counts follow the binomial (defectives) or Poisson (defects) distribution, so the standard deviation comes from the centre line itself. With sample sizes that vary, p and u charts give each point its own limits. A lower limit below zero is shown as zero.p chart: p-bar ± 3 sqrt(p-bar (1 − p-bar) / n); np chart: n p-bar ± 3 sqrt(n p-bar (1 − p-bar)); c chart: c-bar ± 3 sqrt(c-bar); u chart: u-bar ± 3 sqrt(u-bar / n)
Phase I and phase II. In phase I the limits are estimated from data you believe represents the process; you can use only the first points as a baseline, and leave out points that are beyond the limits for a known cause. In phase II the limits come from a standard you already have (a known mean and sigma, or a known fraction or rate) and the new data are judged against them.
Run rules. The Western Electric rules flag one point beyond 3 sigma, two of three beyond 2 sigma on one side, four of five beyond 1 sigma on one side and eight in a row on one side of the centre line. The Nelson rules replace the last by nine in a row and add six steadily rising or falling, fourteen alternating, fifteen within 1 sigma and eight beyond 1 sigma on either side. The zones are drawn from the plotted point's own sigma, so they follow the limits when sample sizes vary. The range, standard deviation and moving range charts use only the limits rule.

Assumptions

Readings or counts are in time order and the points are independent. Autocorrelated data, such as a drifting bath, give many false alarms.
Subgroups for X-bar charts are rational: taken so that variation within a subgroup is common-cause only, and all subgroups are the same size.
Variables charts treat the subgroup means as normal (the individuals chart treats the readings as normal); attribute charts use the normal approximation to the binomial or Poisson, which needs about 5 expected events per sample.
Three sigma limits are the default. A different width changes the false alarm rate; the run rules still use 1 and 2 sigma zones.
At least 20 to 25 baseline points are used to estimate limits; fewer give limits that move as data arrive.

Limits and warnings

A point beyond the limits is evidence of a special cause, not proof; with 3 sigma limits about one in 370 points in a stable process is a false alarm, and more with the extra run rules.
Charts of rare events (very low p or c) with small samples need exact limits, which are not drawn here.
The tool does not detect small sustained shifts quickly; use a CUSUM or EWMA chart for that.
Subgroups of different sizes on X-bar charts are not supported; lines with an unusual size are left out and the tool says so.
The leave-out option does one pass only. Repeated clean-up until no point is beyond the limits hides real instability.

Sources

The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.

NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3, Univariate and multivariate control charts (opens in a new tab), National Institute of Standards and Technology, Online edition. Control limits for X-bar, R, S, p and c charts (sections 6.3.2.1, 6.3.3.1 and 6.3.3.2); the c chart wafer example used in the tests.
Statistical Quality Control Handbook, Western Electric Company, 1956. Named for the four zone rules (the Western Electric rules). Rules written in our own words.
The Shewhart Control Chart: Tests for Special Causes (opens in a new tab), L. S. Nelson, Journal of Quality Technology, volume 16, number 4, pages 238 to 239, October 1984. The eight tests for special causes (the Nelson rules).
Statistical Process Control (SPC) Reference Manual, second edition, Automotive Industry Action Group (AIAG), July 2005. Named for the phase I and phase II practice of chart use and the choice of chart by data type.

Where it helps

The lifecycle stages this tool serves: 10 stages in 7 lifecycles.

Method notev1Formula, ranges, sources and limits

Computes Shewhart control limits for X-bar and R or S, individuals and moving range, p, np, c and u charts from pasted data or a known standard, and flags points that break the Western Electric or Nelson rules.

Formula

sigma = R-bar / d2, or s-bar / c4, or MR-bar / d2(2); X-bar limits = mean ± 3 sigma / sqrt(n); R limits = R-bar ± 3 d3 sigma; S limits = s-bar ± 3 sqrt(1 − c4^2) x s-bar / c4; limits = mean ± 3 MR-bar / 1.128; MR chart upper limit = MR-bar + 3 d3(2) MR-bar / d2(2) = 3.267 MR-bar; p chart: p-bar ± 3 sqrt(p-bar (1 − p-bar) / n); np chart: n p-bar ± 3 sqrt(n p-bar (1 − p-bar)); c chart: c-bar ± 3 sqrt(c-bar); u chart: u-bar ± 3 sqrt(u-bar / n)

Inputs
Input	Unit	Range it holds for	Comes from
Chart type	choice	xbar-r, xbar-s, imr, p, np, c, u	Your figure
Limits from	choice	estimate, known	Your figure
Run rules	choice	limits, we, nelson	Your figure
Sample size, when counts are pasted without one	units	from 1	Your figure
Known centre line, for phase II	same unit as the chart	any	Your figure
Known sigma, for phase II variables charts	same unit as the readings	from 0	Your figure
Points used to estimate the limits, 0 for all	points	from 0	Your figure
Width of the limits	sigma	1 to 6	Your figure

Outputs: Centre line (chart units); Upper control limit (chart units); Lower control limit (chart units); Estimated sigma of the plotted statistic or process (chart units); Points beyond the control limits (points); Points that break any selected rule (points).

Assumptions

Readings or counts are in time order and the points are independent. Autocorrelated data, such as a drifting bath, give many false alarms.
Subgroups for X-bar charts are rational: taken so that variation within a subgroup is common-cause only, and all subgroups are the same size.
Variables charts treat the subgroup means as normal (the individuals chart treats the readings as normal); attribute charts use the normal approximation to the binomial or Poisson, which needs about 5 expected events per sample.
Three sigma limits are the default. A different width changes the false alarm rate; the run rules still use 1 and 2 sigma zones.
At least 20 to 25 baseline points are used to estimate limits; fewer give limits that move as data arrive.

Uncertainty: The limits are exact for the sigma estimated. With 20 to 25 baseline points the estimated limits themselves carry a sampling error of several percent, and the false alarm rate of the run rules depends on independent, normal data.

Known limits

A point beyond the limits is evidence of a special cause, not proof; with 3 sigma limits about one in 370 points in a stable process is a false alarm, and more with the extra run rules.
Charts of rare events (very low p or c) with small samples need exact limits, which are not drawn here.
The tool does not detect small sustained shifts quickly; use a CUSUM or EWMA chart for that.
Subgroups of different sizes on X-bar charts are not supported; lines with an unusual size are left out and the tool says so.
The leave-out option does one pass only. Repeated clean-up until no point is beyond the limits hides real instability.

Where the method comes from

Follows NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3, Univariate and multivariate control charts (opens in a new tab). The platform's version departs from it as follows.

The range and standard deviation constants are computed from their definitions rather than looked up Why: A computed constant works for any subgroup size, where a printed table stops at a small size.
Phase II charts, a baseline, a one-pass clean-up, and the Western Electric and Nelson run rules are added Why: These are how the charts are used on a line, and the handbook describes the rules in prose without a checker.

Worked example from the source: 25 wafers with 400 defects: the handbook finds c-bar = 16, upper limit 28 and lower limit 4. Source result: upper limit 28. This model, same inputs: 28.

Reference example, checked by test: NIST/SEMATECH e-Handbook, section 6.3.3.1, c chart example: 25 wafers with 400 defects in all give c-bar = 16, upper limit 28 and lower limit 4; wafer 24 (31 defects) is above the upper limit, and wafer 3 (28 defects) is on it. Centre line 16; Upper control limit 28; Lower control limit 4; Estimated sigma of the plotted statistic or process 4; Points beyond the control limits 1; Points that break any selected rule 1.

Versions

v1 (2026-10-25): First version, built from the tool idea.

All models in the register

Terms on this page

Control chart: A plot of a process measurement over time against its centre line and statistical control limits, used to spot when a process shifts or becomes unstable so it can be corrected before it produces ba...
Semiconductor: A material whose electrical conductivity sits between that of a conductor and an insulator, and can be controlled by doping, temperature, light or an applied electric field.
Warp: A group of 32 GPU threads that execute the same instruction in lockstep on an NVIDIA architecture.

The full glossary

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Statistical process control chart

How it works

Assumptions

Limits and warnings

Sources

Where it helps

More quality statistics tools

Formula

Assumptions

Known limits

Where the method comes from

Versions

Terms on this page