Laney u′ Chart
The Laney u′ chart is a SPC Chart PRO attribute chart that fixes a specific failure of the classic u chart: with large samples, the ordinary u limits become so tight that almost every point falls outside them, even for a perfectly stable process.
The cause is overdispersion. Classic u limits assume the Poisson distribution holds exactly, so the only variation between samples is sampling noise. Real processes rarely behave that way — there is usually extra, batch-to-batch variation the Poisson does not account for. With small samples this is masked by sampling noise, but as the inspected area grows the classic limits keep shrinking toward the center line while the real variation does not, and the chart drowns in false alarms. David Laney's u′ chart corrects this by measuring the actual short-term variation of the data and widening the limits to match.
Select Laney u′ chart in the Chart Type dropdown.
When to Use It
Reach for a Laney u′ chart when you have a u chart with large samples and the limits look implausibly tight — many or most points outside the control limits with no assignable cause. That is the signature of overdispersion.
With small samples, or when the process really is Poisson, the correction factor is close to 1 and the Laney u′ chart reproduces the classic u chart — so it is a safe default when in doubt, at the cost of slightly wider limits when they are not needed.
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 — how many within-sample sigma it sits from the center line.
- 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.
When the data are exactly Poisson, σz ≈ 1 and nothing changes. When the data are overdispersed, σz > 1 and the limits widen by exactly enough to absorb the extra variation — so a stable process stays inside its limits and only genuine special causes stand out.
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 count or a non-positive sample size are gaps: not plotted and not included in the estimates. A Laney u′ chart needs at least two valid samples to estimate σz.
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:
| 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 )
Then
σz = MR̄z ⁄ 1.128
the average moving range of the z-scores divided by the XmR constant d₂ = 1.128. σz = 1 reproduces the classic u chart; σz > 1 widens the limits to absorb overdispersion.
Because σz comes from a moving range, 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 test each point against its own per-point limits. 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. For requirements, licensing, and support, see the SPC Chart PRO documentation.