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X̄-s chart: paste your subgroups, get the chart

For subgroups big enough that the range wastes them. Limits from A₃, B₃ and B₄, sigma recovered with the c₄ correction, and the s chart read first because it decides whether the averages chart can be trusted.

Nothing you paste leaves this calculation. It is not stored, not logged and not sent anywhere else — the numbers are computed and returned, and that is all.

One subgroup per line, values separated by spaces, commas or tabs. Nine or more readings per line is where this chart earns its keep over X̄-R.

Why the standard deviation, and not the range

The range is a wonderfully cheap estimator of spread. It uses two numbers — the largest and the smallest — and on a subgroup of four or five that is most of the information there is. On a subgroup of twelve it is a waste: ten readings are collected, measured, recorded, and then ignored by the calculation.

The X̄-s chart replaces the range with the subgroup standard deviation, which uses every reading. The result is a more efficient estimate of variation, tighter and more stable control limits, and a chart that reacts to a change in spread sooner. The usual crossover is around n = 9 or 10 — below that the X̄-R chart loses little and is easier to compute by hand, which is why it dominated when charts were drawn on paper.

The formulas, and the constant nobody expects

Limits on the averages chart are X̿ ± A₃s̄, and on the s chart B₄s̄ and B₃s̄. Every constant is a lookup on subgroup size — see the control chart constants table.

The one that surprises people is c₄. The sample standard deviation is a biased estimator of the population standard deviation — it runs low, and it runs low by more the smaller the subgroup. c₄ is the correction: σ̂ = s̄ / c₄. At n = 5, c₄ is 0.94, so ignoring it understates sigma by about six per cent, which understates your control limits and manufactures false alarms. Any capability figure computed from a subgrouped chart depends on this correction having been applied.

Read the s chart first

Same discipline as the range chart. The averages chart's limits are computed from s̄, so an out-of-control s chart means those limits came from an unstable estimate of variation. Fix the spread before interpreting the centre.

Subgrouping still decides everything

A larger subgroup does not rescue a badly chosen one. The subgroup must still be a set of parts made under conditions as nearly identical as you can arrange, so that within-subgroup variation is the process's own noise. If you subgroup across four spindles, or across a shift change, that difference lands inside s̄, inflates the limits, and the chart stops being able to see the very thing you built it for.

These limits are computed from this data

Right for a one-off look, wrong for monitoring. A live chart freezes its limits to a baseline window when the process was behaving, so that later drift shows up as drift rather than quietly widening the limits until nothing can ever signal again.

Questions people ask about this

When should I use an X bar s chart instead of X bar R?

Once subgroups reach about nine or ten. The range uses only the largest and smallest reading, so on a subgroup of twelve it throws away ten measurements you paid to take. The standard deviation uses all of them, giving a tighter estimate of variation and a chart that reacts sooner to a change in spread.

What is the c4 constant for?

The sample standard deviation is a biased estimator of the population standard deviation — it runs low, and it runs lower the smaller the subgroup. c₄ corrects it: σ̂ = s̄ / c₄. At n = 5 the constant is 0.94, so ignoring it understates sigma by about six per cent, which understates the control limits and manufactures false alarms.

How are X bar s control limits calculated?

X̿ ± A₃s̄ on the averages chart, and B₄s̄ and B₃s̄ on the s chart, where s̄ is the mean of the subgroup standard deviations. Every constant is a lookup on subgroup size.

Do I use the population or sample standard deviation?

The sample standard deviation, with the n−1 denominator, computed per subgroup. The bias that leaves is what c₄ exists to remove.

Want this to keep itself up to date?

A calculator answers for the data you pasted. A control chart answers for the data your line produced this morning — limits frozen at a baseline you locked, rules evaluated on every new reading, an alert when one trips.

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More control charts tools

  • Control limit calculator — Give it subgroups or individual readings and it returns the control limits for the chart, the range or sigma chart beneath it, and every constant it used to get there.
  • Control chart generator — Paste a column of numbers, or rows of subgroups, and get a real control chart: limits from the data, Nelson rules 1–4 evaluated, out-of-control points marked.
  • Nelson & Western Electric rules checker — Every run rule evaluated on your data, each violation named in plain English with what it usually indicates — a shift, a trend, tool wear, two machines mixed.
  • X̄-R chart calculator — Subgroups in, X̄ and R charts out — limits from A₂, D₃ and D₄, run rules evaluated, and the range chart shown first because it decides whether the averages chart can be trusted.
  • I-MR chart calculator — For processes that give you one number at a time — a batch, an oven, a destructive test. Limits from the mean moving range, run rules evaluated, and an honest note about what an individuals chart cannot see.

Everything else

  • Cp / Cpk calculator — Paste measurements — or type a mean and a sigma — with your tolerance, and get Cp, Cpk, Pp, Ppk, the sigma level and the expected parts per million out of spec.
  • Cpk confidence interval & sample size — A Cpk of 1.33 from 30 pieces has a 95% interval of roughly 0.97 to 1.69. See the uncertainty in your own number, and how many parts would settle it.
  • Cpk ↔ PPM ↔ sigma level converter — What PPM is a Cpk of 1.33? What Cpk does 3.4 PPM need? Both conventions shown side by side, because the 1.5 sigma shift is why two sources disagree by a factor of ten.
  • Control chart constants — The whole table, n = 2 to 25, with the formula each constant belongs to. The same values the charts on this site are computed from.
  • Levey-Jennings chart with Westgard rules — Paste your QC results with the assigned mean and SD from the package insert, and get the chart with every sigma band drawn and all six Westgard rules evaluated — each one named, and labelled random or systematic error.
  • Bland-Altman plot — Paired readings from two methods, plotted as difference against average — with the bias, the limits of agreement, their confidence intervals, and a test for whether the disagreement depends on the magnitude.
  • Pareto chart generator — Categories and counts in, ranked bars and the cumulative line out — with the vital few named, an "Other" bucket that always sorts last, and an honest warning when the distribution is flat and there is no dominant cause to attack.
  • Box and whisker plot generator — One column per group, boxes side by side. Quartiles, the 1.5×IQR fences, whiskers that stop at real readings and outliers drawn individually — with the quartile method stated, because that is why your plot and Excel's disagree.
  • Histogram and normality test — A histogram with the fitted normal curve, both standard bin rules with a reasoned recommendation, and an Anderson-Darling test that refuses to tell you your data is normal — because no test can.
  • Gage R&R calculator (ANOVA) — Paste a crossed study — part, operator, reading — and get the full ANOVA: repeatability and reproducibility separated, the part-by-operator interaction tested rather than assumed away, %GRR, %Tolerance and ndc against AIAG's bands.