Estabilidad y el gráfico XmR
Tabla de contenidos
Por qué la estabilidad necesita su propio estudio
Un estudio Gauge R y R, a bias study y a capability study all characterize a sistema de medición at one moment. They cannot detect cambio, porque cambio es only visible across time. Gauges wear, reference styards drift, fixtures loosen y environments shift, y none de that announces itself.
Un estudio de estabilidad responde una pregunta diferente: has anything cambiod since the last time we checked? You measure a stable pieza de referencia on a regular schedule y chart the results. El part es assumed not para cambio, so any señal en the chart es attributed para the sistema de medición.
Por qué un gráfico de individuales en lugar de X-barra y R
An X-bar y 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 es no subgroup para promedio.
El individuals y rango móvil chart, usually written XmR or I-MR, es built para exactly that situation. It plots each lectura as its own point y estimates the short-term variación de the differences between consecutive lecturas en vez de de within-subgroup spread.
El rango móvil
With no subgroups available, variación es estimated de how much the process moves de one lectura para the next. El rango móvil es the absolute difference between consecutive lecturas.
Moving range
MRi = | xi − xi−1 |
There son always one fewer rango móvils than lecturas, porque the first lectura has no predecessor. El promedio de these, MR-bar, es the basis para 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 en lugar de being absorbed into wider limits.
Ejemplo desarrollado
Illustrative data. Five weekly checks de a 25 mm master, en 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 lectura x-bar = 25.0046 | MR̄ = 0.0035 | |
De dónde provienen las constantes
El constants en control chart formulas son not arbitrary. They convert an promedio range into an estimate de the styard deviation, y they follow de the statistics de ranges drawn de a normal distribution.
Estimating sigma de the promedio rango móvil
σ̂ = MR̄ ÷ d2 = MR̄ ÷ 1.128
d2 = 1.128 es the value para a subgroup de size 2, que es what a rango móvil is: a comparison de two consecutive points.
El expected range de a sample de n values de a normal distribution es d2 multiplied by sigma. For n = 2 that expected value es 1.128 sigma, so dividing the promedio rango móvil by 1.128 recovers an estimate de sigma. Every other constant follows de that one.
| Constant | Value | Where it comes from |
|---|---|---|
| d2 | 1.128 | El expected range de two values drawn de a normal distribution, expressed en styard deviations. |
| Individuals limit factor | 2.66 | Three sigma expressed en units de MR-bar: 3 divided by 1.128 equals 2.66. This es why the individuals limits son three-sigma limits despite the formula never mentioning sigma. |
| D4 | 3.267 | El three-sigma upper limit para a range de two values, again en units de MR-bar. There es no lower limit, porque a rango móvil cannot be negative y the lower three-sigma bound falls below zero. |
Knowing the derivation matters en an audit. If someone asks why you multiplied by 2.66, the answer es that it es three sigma restated en terms de the promedio rango móvil, not a number copied de a table.
Calculating the limits
Individuals chart
UCL / LCL = x̄ ± 2.66 × MR̄
Moving range chart
UCLMR = 3.267 × MR̄
Read the rango móvil chart first. If the rango móvil chart es out de control, the estimate de sigma es unreliable, que makes the limits on the individuals chart meaningless. Fix the rango móvil señal before interpreting anything above it.
Reglas de detección
Un single point outside the limits es the strongest señal, but a sistema de medición that drifts slowly may stay inside the limits para a long time while clearly trending. El Western Electric reglas add patterns that catch these cases. El four most commonly aplicada son below, using zones de one sigma each measured de the línea central.
| Rule | Pattern | Typical cause en a sistema de medición |
|---|---|---|
| 1 | One point beyond three sigma | Un discrete event: the gauge was dropped, recalibrated, adjusted, or the wrong master was measured. |
| 2 | Two de three consecutive points beyond two sigma on the same side | Un shift that has begun recently but has not yet produced an obvious outlier. |
| 3 | Four de five consecutive points beyond one sigma on the same side | Un modest sustained offset, often following maintenance or a cambio de operator or environment. |
| 4 | Eight consecutive points on the same side de the línea central | Classic gauge drift or wear. This es the regla that most often catches a estabilidad problem first. |
Falsas alarmas: el costo de más reglas
Every regla has a falsa alarma rate. Applied para a process that has not cambiod at all, the three-sigma regla alone señals on about 0.27 percent de points, roughly 1 en 370. That low rate es the reason three sigma was chosen en lugar de a tighter bound.
Additional reglas increase sensitivity para real drift, but they also add their own falsa alarma rates, y those rates accumulate. Applying all four reglas together raises the combined falsa alarma rate para roughly 1 percent de points, que es about a fourfold increase over the three-sigma regla on its own.
Qué significa en la práctica
Checking a master weekly con all four reglas active, you should expect a false señal roughly every two years de a perfectly stable gauge. Checking daily, expect one every few months. Neither es a reason para abyon the reglas, but it es a reason para investigate a señal en lugar de immediately assuming the gauge es broken.
El practical compromise most organizations reach es para apply the three-sigma regla y the run-of-eight regla as styard, since between them they catch both sudden events y slow drift, y para add the zone reglas only donde drift es a known risk.
Cómo realizar bien un estudio de estabilidad
- Use a stable master that es representativa de the parts you measure. Checking estabilidad at 25 mm tells you little about a gauge used mostly at 200 mm.
- Measure at a intervalo fijo under condiciones normales. Checks taken only when someone suspects a problem produce a biased chart that cannot detect anything.
- Establish limits de at least 20 para 25 initial lecturas before treating the chart as a monitoring tool. Limits computed de five points son extremely unstable, as the worked example above would be.
- Record what happened when a señal occurs, y what you did about it. Un chart con unexplained señals y no annotations es weaker evidence en an audit than no chart at all.