Home / Statistical Tools / Control Charts / Laney U' Chart / Math Details
Math Details¶
This page gives the exact formulas Quantum XL uses to build a Laney U' chart. It is a u chart whose standard deviation is scaled by the overdispersion factor σ_Z. Each equation lists what it computes and where it appears on the chart.
Notation¶
| Term | Description |
|---|---|
| \(n_i\) | size of subgroup \(i\) (its area of opportunity) |
| \(c_i\) | number of defects in subgroup \(i\) |
| \(\bar{u}\) | center line (pooled defects per unit) |
| \(\sigma_i\) | ordinary Poisson standard deviation at subgroup \(i\) |
| \(\sigma_Z\) | overdispersion factor |
| \(k\) | sigma multiplier (number of standard deviations), default \(3\) |
Center line and plotted statistic¶
The center line is the pooled defects per unit (or a supplied known value), and each point is the subgroup rate, exactly as on the u Chart.
Ordinary standard deviation¶
the Poisson standard deviation before the overdispersion correction.
Overdispersion-adjusted control limits¶
The Poisson standard deviation is multiplied by the overdispersion factor \(\sigma_Z\), estimated from the subgroup rates standardized by their ordinary sigma (here \(v_i = u_i\) and \(\text{CL} = \bar{u}\)):
Each subgroup is standardized by its own ordinary sigma:
where \(v_i\) is the plotted statistic at subgroup \(i\), \(\text{CL}\) is the center line, and \(\sigma_i\) is the ordinary binomial or Poisson standard deviation. If \(\sigma_i \le 0\) the value is set to \(z_i = 0\).
The factor is the average moving range of the \(z_i\) (a window of two, so \(MR_i = \lvert z_i - z_{i-1}\rvert\)) divided by \(d_2\):
The engine uses the full-precision table value rather than a rounded one, so a hand check from the printed digits will differ in the sixth significant figure.
Any moving-range window that spans a missing subgroup is dropped rather than bridged, and \(\overline{MR}_z\) is the plain mean of the surviving ranges. When \(\sigma_Z\) cannot be computed it is taken to be \(1\) (no correction).
The adjusted limits are:
When \(\sigma_Z = 1\) this reduces to the ordinary u chart. If Force Straight Limits is enabled, a single assumed size replaces \(n_i\) in \(\sigma_i\). Full details: Laney Overdispersion (Sigma Z).
Sigma zones¶
On the attribute charts and on Z-MR the one-sigma width is recovered from the limit spacing rather than recomputed, so the zones follow each point's limits exactly:
where \(\text{CL}\) is the center line and \(k\) the sigma multiplier. The zone boundaries are then placed at one and two of these units on each side of the center line:
| Zone | Region |
|---|---|
| Zone C | within \(\text{CL} \pm \sigma_1\) |
| Zone B | between \(\text{CL} \pm \sigma_1\) and \(\text{CL} \pm 2\sigma_1\) |
| Zone A | between \(\text{CL} \pm 2\sigma_1\) and the control limit |
Because \(\sigma_1\) comes from the upper half band and the lower boundaries mirror that same distance, a lower boundary can fall below a lower control limit that has been floored at \(0\), and can itself be negative. Z-MR always uses a multiplier of \(3\), so its zone lines are always at one and two standard deviations. If the center line or the upper limit is missing, or the multiplier is not positive, no zone boundaries are produced.
Full details, including the manual and disabled limit types: Control Limits and Zones.
Shared Math Details used here¶
This chart uses shared formulas defined once in Shared Math Details.
| Shared concept | Used here for | Reference |
|---|---|---|
| Laney overdispersion (σ_Z) | the standard-deviation correction | Laney Overdispersion (Sigma Z) |
| Control limits and zones | the UCL/LCL and zone construction | Control Limits and Zones |
| Out-of-control tests | flagging out-of-control points | Out-of-Control tests |
See Also¶
References¶
- Laney, D. B. (2002). Improved control charts for attributes. Quality Engineering, 14(4), 531-537.
- Montgomery, D. C. (2013). Introduction to Statistical Quality Control, 7th ed. Wiley.