Stability and the XmR Chart
Table of Contents
Why stability needs its own study
A Gauge R and R study, a bias study and a capability study all characterize a measurement system at one moment. They cannot detect change, because change is only visible across time. Gauges wear, reference standards drift, fixtures loosen and environments shift, and none of that announces itself.
A stability study answers a different question: has anything changed since the last time we checked? You measure a stable reference part on a regular schedule and chart the results. The part is assumed not to change, so any signal in the chart is attributed to the measurement system.
Why an individuals chart rather than X-bar and R
An X-bar and R chart requires rational subgroups, meaning several measurements taken together under essentially identical conditions. Stability monitoring usually produces one measurement per check, taken days or weeks apart. There is no subgroup to average.
The individuals and moving range chart, usually written XmR or I-MR, is built for exactly that situation. It plots each reading as its own point and estimates the short-term variation from the differences between consecutive readings instead of from within-subgroup spread.
The moving range
With no subgroups available, variation is estimated from how much the process moves from one reading to the next. The moving range is the absolute difference between consecutive readings.
Moving range
MRi = | xi − xi−1 |
There are always one fewer moving ranges than readings, because the first reading has no predecessor. The average of these, MR-bar, is the basis for every limit on both charts. Using consecutive differences deliberately keeps the estimate short-term: it captures the noise between adjacent checks, so that a slow drift shows up as points moving outside the limits rather than being absorbed into wider limits.
Worked example
Illustrative data. Five weekly checks of a 25 mm master, in millimetres.
| Check | Reading | Moving range |
|---|---|---|
| 1 | 25.004 | — |
| 2 | 25.007 | 0.003 |
| 3 | 25.002 | 0.005 |
| 4 | 25.006 | 0.004 |
| 5 | 25.004 | 0.002 |
| Average reading x-bar = 25.0046 | MR̄ = 0.0035 | |
Where the constants come from
The constants in control chart formulas are not arbitrary. They convert an average range into an estimate of the standard deviation, and they follow from the statistics of ranges drawn from a normal distribution.
Estimating sigma from the average moving range
σ̂ = MR̄ ÷ d2 = MR̄ ÷ 1.128
d2 = 1.128 is the value for a subgroup of size 2, which is what a moving range is: a comparison of two consecutive points.
The expected range of a sample of n values from a normal distribution is d2 multiplied by sigma. For n = 2 that expected value is 1.128 sigma, so dividing the average moving range by 1.128 recovers an estimate of sigma. Every other constant follows from that one.
| Constant | Value | Where it comes from |
|---|---|---|
| d2 | 1.128 | The expected range of two values drawn from a normal distribution, expressed in standard deviations. |
| Individuals limit factor | 2.66 | Three sigma expressed in units of MR-bar: 3 divided by 1.128 equals 2.66. This is why the individuals limits are three-sigma limits despite the formula never mentioning sigma. |
| D4 | 3.267 | The three-sigma upper limit for a range of two values, again in units of MR-bar. There is no lower limit, because a moving range cannot be negative and the lower three-sigma bound falls below zero. |
Knowing the derivation matters in an audit. If someone asks why you multiplied by 2.66, the answer is that it is three sigma restated in terms of the average moving range, not a number copied from a table.
Calculating the limits
Individuals chart
UCL / LCL = x̄ ± 2.66 × MR̄
Moving range chart
UCLMR = 3.267 × MR̄
Read the moving range chart first. If the moving range chart is out of control, the estimate of sigma is unreliable, which makes the limits on the individuals chart meaningless. Fix the moving range signal before interpreting anything above it.
Detection rules
A single point outside the limits is the strongest signal, but a measurement system that drifts slowly may stay inside the limits for a long time while clearly trending. The Western Electric rules add patterns that catch these cases. The four most commonly applied are below, using zones of one sigma each measured from the centre line.
| Rule | Pattern | Typical cause in a measurement system |
|---|---|---|
| 1 | One point beyond three sigma | A discrete event: the gauge was dropped, recalibrated, adjusted, or the wrong master was measured. |
| 2 | Two of three consecutive points beyond two sigma on the same side | A shift that has begun recently but has not yet produced an obvious outlier. |
| 3 | Four of five consecutive points beyond one sigma on the same side | A modest sustained offset, often following maintenance or a change of operator or environment. |
| 4 | Eight consecutive points on the same side of the centre line | Classic gauge drift or wear. This is the rule that most often catches a stability problem first. |
False alarms: the cost of more rules
Every rule has a false alarm rate. Applied to a process that has not changed at all, the three-sigma rule alone signals on about 0.27 percent of points, roughly 1 in 370. That low rate is the reason three sigma was chosen rather than a tighter bound.
Additional rules increase sensitivity to real drift, but they also add their own false alarm rates, and those rates accumulate. Applying all four rules together raises the combined false alarm rate to roughly 1 percent of points, which is about a fourfold increase over the three-sigma rule on its own.
What that means in practice
Checking a master weekly with all four rules active, you should expect a false signal roughly every two years from a perfectly stable gauge. Checking daily, expect one every few months. Neither is a reason to abandon the rules, but it is a reason to investigate a signal rather than immediately assuming the gauge is broken.
The practical compromise most organizations reach is to apply the three-sigma rule and the run-of-eight rule as standard, since between them they catch both sudden events and slow drift, and to add the zone rules only where drift is a known risk.
Running a stability study well
- Use a stable master that is representative of the parts you measure. Checking stability at 25 mm tells you little about a gauge used mostly at 200 mm.
- Measure at a fixed interval under normal conditions. Checks taken only when someone suspects a problem produce a biased chart that cannot detect anything.
- Establish limits from at least 20 to 25 initial readings before treating the chart as a monitoring tool. Limits computed from five points are extremely unstable, as the worked example above would be.
- Record what happened when a signal occurs, and what you did about it. A chart with unexplained signals and no annotations is weaker evidence in an audit than no chart at all.