Laney u chart

Laney u chart is an enhanced version of the traditional u chart that adjusts for overdispersion in the data. It accounts for between-subgroup variation that exceeds what would be expected from Poisson distribution alone when monitoring defects per unit.

Map the data fields

• Subgroup/Lot: This field represents the different lots or batches of items being inspected.
• Process variable: This field contains the number of defects found in each lot.
• Sample Size: This field indicates the number of units inspected in each lot (can vary between lots).
• UCL: Optional field for custom upper control limits.
• LCL: Optional field for custom lower control limits.
• Central Line: Optional field for custom center line values.

Calculation Methodology

The Laney u-chart calculations are performed as follows:

Average defects per unit (ū)

ū = Total Defects / Total Samples

Individual defect rates and standardized residuals

For each sample:

ui = Defects / Sample Size
σui = √(ū / Sample Size)
zi = (ui - ū) / σui

Between-subgroup standard deviation (σz)

z̄ = Average of all zi values
σz = √(Σ(z̄ - zi)² / (n - 1))

Adjusted Control Limits

For each data point:

Upper Control Limit (UCL):

UCL = ū + (Z-Score × σz × σui)

Lower Control Limit (LCL):

LCL = ū - (Z-Score × σz × σui)

Center Line (CL):

CL = ū (average defects per unit)

Defects per Unit for Each Sample:

Defects per Unit = Defects / Sample Size

Notes:

• If custom limits are provided, they override the calculated values
• The Z-Score determines the confidence level (typically 3 for 99.7% confidence)
• σz accounts for overdispersion - when σz > 1, data shows more variation than expected
• When σz ≤ 1, Laney u-chart reduces to traditional u-chart behavior
• Control limits vary for each sample based on both sample size and overdispersion factor
• Laney charts are particularly useful for high-volume processes with autocorrelated data
• Unlike traditional u-charts, Laney u-charts can handle processes with excessive variation