Estabilidade e o gráfico XmR
Indice
Por que a estabilidade precisa de seu próprio estudo
Um estudo Gauge R e R, a bias study e a capability study all characterize a sistema de medição at one moment. They cannot detect mudança, porque mudança é only visible across time. Gauges wear, reference steards drift, fixtures loosen e environments shift, e none de that announces itself.
Um estudo de estabilidade responde a uma pergunta diferente: has anything mudançad since the last time we checked? You measure a stable peça de referência on a regular schedule e chart the results. O part é assumed not para mudança, so any sinal em the chart é attributed para the sistema de medição.
Por que um gráfico de indivíduos em vez de X-barra e R
An X-bar e 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 é no subgroup para média.
O individuals e amplitude móvel chart, usually written XmR or I-MR, é built para exactly that situation. It plots each leitura as its own point e estimates the short-term variação de the differences between consecutive leituras em vez de de within-subgroup spread.
Um amplitude móvel
With no subgroups available, variação é estimated de how much the process moves de one leitura para the next. Um amplitude móvel é the absolute difference between consecutive leituras.
Moving range
MRi = | xi − xi−1 |
There são always one fewer amplitude móvels than leituras, porque the first leitura has no predecessor. O média de these, MR-bar, é 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 em vez de being absorbed into wider limits.
Exemplo desenvolvido
Illustrative data. Five weekly checks de a 25 mm master, em 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 leitura x-bar = 25.0046 | MR̄ = 0.0035 | |
De onde vêm as constantes
O constants em control chart formulas são not arbitrary. They convert an média range into an estimate de the steard deviation, e they follow de the statistics de ranges drawn de a normal distribution.
Estimating sigma de the média amplitude móvel
σ̂ = MR̄ ÷ d2 = MR̄ ÷ 1.128
d2 = 1.128 é the value para a subgroup de size 2, que é what a amplitude móvel is: a comparison de two consecutive points.
O expected range de a sample de n values de a normal distribution é d2 multiplied by sigma. For n = 2 that expected value é 1.128 sigma, so dividing the média amplitude móvel by 1.128 recovers an estimate de sigma. Every other constant follows de that one.
| Constant | Value | Where it comes from |
|---|---|---|
| d2 | 1.128 | O expected range de two values drawn de a normal distribution, expressed em steard deviations. |
| Individuals limit factor | 2.66 | Three sigma expressed em units de MR-bar: 3 divided by 1.128 equals 2.66. This é why the individuals limits são three-sigma limits despite the formula never mentioning sigma. |
| D4 | 3.267 | O three-sigma upper limit para a range de two values, again em units de MR-bar. There é no lower limit, porque a amplitude móvel cannot be negative e the lower three-sigma bound falls below zero. |
Knowing the derivation matters em an audit. If someone asks why you multiplied by 2.66, the answer é that it é three sigma restated em terms de the média amplitude móvel, 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 amplitude móvel chart first. If the amplitude móvel chart é out de control, the estimate de sigma é unreliable, que makes the limits on the individuals chart meaningless. Fix the amplitude móvel sinal before interpreting anything above it.
Regras de detecção
Um single point outside the limits é the strongest sinal, but a sistema de medição that drifts slowly may stay inside the limits para a long time while clearly trending. O Western Electric regras add patterns that catch these cases. O four most commonly aplicada são below, using zones de one sigma each measured de the linha central.
| Rule | Pattern | Typical cause em a sistema de medição |
|---|---|---|
| 1 | One point beyond three sigma | Um 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 | Um 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 | Um modest sustained offset, often following maintenance or a mudança de operator or environment. |
| 4 | Eight consecutive points on the same side de the linha central | Classic gauge drift or wear. This é the regra that most often catches a estabilidade problem first. |
Falsos alarmes: o custo de mais regras
Every regra has a falso alarme rate. Applied para a process that has not mudançad at all, the three-sigma regra alone sinals on about 0.27 percent de points, roughly 1 em 370. That low rate é the reason three sigma was chosen em vez de a tighter bound.
Additional regras increase sensitivity para real drift, but they also add their own falso alarme rates, e those rates accumulate. Applying all four regras together raises the combined falso alarme rate para roughly 1 percent de points, que é about a fourfold increase over the three-sigma regra on its own.
O que isso significa na prática
Checking a master weekly com all four regras active, you should expect a false sinal roughly every two years de a perfectly stable gauge. Checking daily, expect one every few months. Neither é a reason para abeon the regras, but it é a reason para investigate a sinal em vez de immediately assuming the gauge é broken.
O practical compromise most organizations reach é para apply the three-sigma regra e the run-of-eight regra as steard, since between them they catch both sudden events e slow drift, e para add the zone regras only onde drift é a known risk.
Como executar bem um estudo de estabilidade
- Use a stable master that é representativa de the parts you measure. Checking estabilidade at 25 mm tells you little about a gauge used mostly at 200 mm.
- Measure at a intervalo fixo under condições normais. 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 leituras before treating the chart as a monitoring tool. Limits computed de five points são extremely unstable, as the worked example above would be.
- Record what happened when a sinal occurs, e what you did about it. Um chart com unexplained sinals e no annotations é weaker evidence em an audit than no chart at all.