Laney u′ Chart
The Laney u′ chart is an attribute chart in SPC Chart PRO that adjusts the classic u chart for overdispersion or underdispersion.
The classic u chart uses the Poisson model to estimate variation. When the observed variation is greater than the model predicts, its limits can be too narrow; when variation is lower, they can be too wide. The Laney method estimates the short-term variation of standardized values and adjusts the control-limit spread accordingly.
Select Laney u′ chart in the Chart Type dropdown.
When to Use It
Use a Laney u′ chart when the variation in your defect rate data differs from the Poisson prediction. Overdispersion is especially noticeable with large samples, where classic limits can become very narrow. Investigate the process and data before treating frequent violations as a dispersion issue.
When the adjustment factor is close to 1, the limits are close to those of the classic u chart. A factor above 1 widens them; a factor below 1 narrows them.
For proportion defective (binomial) data rather than defects per unit, the equivalent correction is the Laney p′ chart.
How It Works
A Laney u′ chart is set up exactly like the u chart it replaces — same data shape, same sample-size option. Each row is one sample; the value field holds the total defect count, and a sample-size column or fixed option gives the number of units inspected.
The difference is in the limits. The Laney method:
- Converts each sample's defect rate into a z-score — its distance from the center line in units of the Poisson standard error.
- Measures the short-term variation of those z-scores using their average moving range, giving a factor σz.
- Multiplies the classic control-limit spread by σz.
This adjustment accounts for variation that the classic u chart's model does not capture. It does not guarantee that every point from a stable process will fall inside the limits.
Setting It Up
In the SPC section, with the Laney u′ chart type selected:
| Option | Description |
|---|---|
| Sample size field | A numeric field giving the number of units inspected per sample. Hidden from the plot automatically. |
| Fixed sample size | A constant number inspected per sample, used when no field is selected. |
Rows with a missing or negative count, or a missing or non-positive sample size, are treated as gaps: they are not plotted or included in the estimates. Estimating σz requires at least one pair of consecutive valid, non-excluded samples and a positive model variance.
Methods and Formulas
The center line and the base spread are identical to the underlying u chart. The only addition is the σz factor multiplying the spread:
| Quantity | Formula |
|---|---|
| Plotted value | uᵢ = cᵢ ⁄ nᵢ |
| Center line | ū = Σcᵢ ⁄ Σnᵢ |
| Control limits | ū ± 3·σz·√( ū ⁄ nᵢ ), floored at 0 |
The σz factor. Each sample's defect rate is standardized against its within-sample sigma to give a z-score:
zᵢ = (uᵢ − ū) ⁄ √( ū ⁄ nᵢ )
The adjustment factor is:
σz = MR̄z ⁄ 1.128
Here, MR̄z is the average moving range of the z-scores, and 1.128 is the moving-range constant d₂(2). σz = 1 reproduces the classic u chart; σz > 1 widens the limits for overdispersion, and σz < 1 narrows them for underdispersion.
The limits step with the sample size just as on the classic u chart, and the representative scalar LCL/UCL use the average sample size. Run rules derive sigma from each point's upper limit. The sample-size configuration and chart type are saved with the dashboard, and the chart works with both a time and a numeric X-axis.
The Laney u′ chart is part of SPC Chart PRO. See requirements, licensing, and support.