Laney p chart

Laney p chart is an enhanced version of the traditional p chart that adjusts for overdispersion in the data. It accounts for between-subgroup variation that exceeds what would be expected from binomial distribution alone.

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 defective items in each lot.
• Sample Size: This field indicates the number of items inspected in each lot.
• 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 p-chart calculations are performed as follows:

Proportion defective (p̄)

p̄ = Total Defectives / Total Samples

Individual proportions and standardized residuals

For each sample:

pi = Defectives / Sample Size
σpi = √(p̄ × (1 - p̄) / Sample Size)
zi = (pi - p̄) / σpi

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 = p̄ + (Z-Score × σz × σpi)

Lower Control Limit (LCL):

LCL = p̄ - (Z-Score × σz × σpi)

Center Line (CL):

CL = p̄ (overall proportion)

Proportion for Each Sample:

Proportion = Defectives / 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 p-chart reduces to traditional p-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