Control Limits Calculator — UCL and LCL from X̿, R̄, s̄, MR̄, p̄, c̄ or ū
Already have the averages? Get the centre line, UCL, LCL and the ±1σ and ±2σ zone lines for X̄-R, X̄-S, I-MR, p, np, c and u charts, with the A₂, D₃, D₄ or E₂ constant for your subgroup size printed beside them.
Loaded with Montgomery’s piston-ring example: X̿ = 74.001, R̄ = 0.023, n = 5.
X̄ chart
X̿ ± A₂·R̄
Zone lines for the run rules: ±1σ at 73.9966 / 74.0054, ±2σ at 73.9922 / 74.0098.
R chart
D₃·R̄ … D₄·R̄
Zone lines for the run rules: ±1σ at — / 0.0315, ±2σ at — / 0.0401. The lower limit was clamped at zero, so the zone tests below the centre line cannot be applied.
Constants computed from the normal distribution, not looked up — see every value to n = 100. To plot your data against these limits with the run rules applied, open the X̄-R chart tool; to check a series against limits you already have, use the run rules checker.
Everything on this page is calculated in your browser tab. Nothing you type is uploaded.
Control limits are not specification limits
Control limits come from the process: they are where three standard deviations of the plotted statistic fall on either side of its average, and they describe what the process does when nothing has changed. Specification limits come from the drawing and describe what the customer will accept. A process can be in control and out of specification, or the other way round, and the two sets of lines should never be drawn as if they meant the same thing.
The formulas, chart by chart
X̄-R: X̿ ± A₂·R̄ for the averages, D₃·R̄ to D₄·R̄ for the ranges. X̄-S: X̿ ± A₃·s̄ and B₃·s̄ to B₄·s̄. I-MR: X̄ ± 2.66·MR̄ for the individuals and 0 to 3.267·MR̄ for the moving ranges, where 2.66 is 3/d₂ for a span of two.
p: p̄ ± 3√(p̄(1 − p̄)/n). np: np̄ ± 3√(np̄(1 − p̄)). c: c̄ ± 3√c̄. u: ū ± 3√(ū/n). For p and u, n is the size of the sample being plotted, so with varying sample sizes every point has its own limits — the full p and u chart tools draw that staircase.
Whenever a lower limit works out below zero it is drawn at zero, and the calculator says so: a count cannot be negative, and the zone tests below the centre line stop meaning anything once the floor has been hit.
Where the constants come from
A₂, D₃, D₄, A₃, B₃, B₄ and E₂ are all functions of d₂, d₃ and c₄, which are properties of the normal distribution. This site computes them rather than copying a table, so any subgroup size from 2 to 100 works. The control chart constants page shows every value and the derivation.
Common questions
- How do you calculate the upper control limit?
- UCL = centre line + 3 × (standard deviation of the plotted statistic). For an X̄-R chart that is X̿ + A₂·R̄; for an individuals chart X̄ + 2.66·MR̄; for a c chart c̄ + 3√c̄.
- Why is my lower control limit zero?
- On range, standard deviation and attribute charts the formula can give a negative number for small subgroups or low defect rates. A range or a count cannot be negative, so the LCL is set to zero. D₃ is zero for subgroups of six or fewer for the same reason.
- How many subgroups do I need before calculating control limits?
- The usual guidance is 20 to 25 subgroups, or at least 20 to 30 individual values for an I-MR chart. Limits from fewer points are trial limits and should be recalculated once more data is in.
- Should control limits be recalculated?
- Recalculate when the process has been deliberately changed, or when trial limits were set on too little data. Do not recalculate simply because points went out of control: the limits are the baseline the signal is measured against.