I-MR chart: one reading at a time

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.

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.

When there is only one reading

Plenty of processes give you no choice. A batch has one viscosity. An oven has one temperature at 06:00. A destructive test costs a part. Measurement is expensive, or slow, or the thing being measured only exists once. Subgrouping is not available, so the chart has to work at n = 1.

The problem is that with a single reading there is no within-subgroup variation to estimate sigma from. The I-MR chart's answer is the moving range: the absolute difference between each reading and the one before it. Consecutive readings stand in for a subgroup of two, and short-term variation is recovered from how much the process moves between one observation and the next.

The formulas

Sigma is estimated as MR̄ / d₂, and with n = 2 the constant d₂ is 1.128 — so the limits come out as X̄ ± 2.66 MR̄, where 2.66 is simply 3 / 1.128. The moving range chart's upper limit is 3.267 MR̄, and it has no lower limit, because with two readings you cannot produce a range small enough to be surprising.

What it costs you

An individuals chart is markedly less sensitive than a subgrouped one. Averages of five readings vary √5 less than the readings themselves, so an X̄ chart notices a small shift in the centre far sooner than an I chart ever will. If you can subgroup, subgroup — the X̄-R chart is a better instrument and it is not close. I-MR is what you use when the process will not give you subgroups, not what you use because it is simpler.

The assumption people forget

The moving range assumes consecutive readings are independent — that reading n carries no memory of reading n−1. Many processes violate this openly. A tank that heats and cools, a bath whose concentration drifts between top-ups, anything with thermal inertia: those readings are autocorrelated, and autocorrelation deflates the moving range. Deflated MR̄ gives tight limits, tight limits give constant false alarms, and the chart gets ignored within a fortnight, which is the real failure.

If your readings are strongly serially dependent, an individuals chart with limits from MR̄ is the wrong tool and no amount of care in plotting will fix it. Sample further apart, or chart the deviation from the expected trajectory rather than the raw value.

The individuals chart is also more exposed to non-normality than a subgrouped chart, which gets the central limit theorem working in its favour. Strongly skewed data — anything bounded at zero, like a concentricity or a flatness — will produce more upper-side signals than the nominal false-alarm rate implies.

These limits are computed from this data

Right for a one-off look, wrong for monitoring. A live chart freezes limits to a baseline window when the process was behaving, so later drift shows up as drift instead of dragging the limits along with it.

Questions people ask about this

When do I use an I-MR chart?

When the process gives you one measurement at a time and subgrouping is not available — one viscosity per batch, one temperature per hour, a destructive test that costs a part. It is the chart you use when you have no choice, not the one you use because it is simpler.

Where does 2.66 come from?

Sigma is estimated as MR̄ / d₂, and for a moving range of two consecutive readings d₂ is 1.128. Three sigma is therefore 3 × MR̄ / 1.128, and 3 / 1.128 is 2.66. The moving range chart uses 3.267 MR̄ as its upper limit and has no lower limit at all.

Is an I-MR chart less sensitive than X bar R?

Considerably. Averages of five readings vary √5 less than individual readings, so an X̄ chart detects a small shift in the centre far sooner than an individuals chart ever will. If your process allows subgrouping, subgroup.

What if my readings are autocorrelated?

Then the moving range is deflated, the limits come out too tight, and the chart produces constant false alarms until somebody stops looking at it. Anything with thermal or chemical inertia — a heated tank, a plating bath between top-ups — does this. Sample further apart, or chart the deviation from the expected trajectory rather than the raw value.

Does an I-MR chart need normal data?

It is more exposed to non-normality than a subgrouped chart, which has the central limit theorem working for it. Strongly skewed data — anything bounded at zero, like a flatness or a concentricity — will signal on the upper side more often than the nominal false-alarm rate suggests.

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.
X̄-s chart calculator — 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.

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.