np Control Chart Analysis

Background & Problem Context

Silicon wafers undergo a series of highly precise fabrication steps — including photolithography, etching, deposition, and chemical mechanical planarization (CMP) — before being diced into individual integrated circuits. At each stage, wafers are inspected for surface defects such as particles, scratches, and pattern misalignments. Even a single defect can render a chip non-functional, making defect monitoring critical to maintaining yield.

A fabrication facility (fab) produces wafers in lots. At the end of the photolithography stage, quality engineers inspect n = 200 wafers per lot and record the number of defective wafers in each lot. Data were collected over k = 50 consecutive lots during a period believed to represent a stable, in-control process.

Your goal is to construct an np control chart using the qcc package in R to confirm process stability and establish control limits for ongoing monitoring.

The Data

Use the following data representing the number of defective wafers out of 200 sampled wafers in each of the 50 lots:

defects<-c(8, 7, 11, 3, 6, 11, 4, 4, 5, 8, 8,  6,  7,  6, 10,  7,  9,  5,  6,  8, 8, 10,  4, 11,  4, 11,  6,  6,  9,  5, 4,  7,  5,  7,  8,  5, 10,  9,  7,  9, 4,  9,  4,  7, 11,  8, 10,  8,  7,  5)

Control Chart Formulas

\[ CL = n\bar{p} \]

where

\[\bar{p} = \frac{\text{Total number of defects}}{\text{Total wafers inspected}}\]

The upper and lower control limits are:

\[UCL = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}\]

\[LCL = n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}\]

Control Chart

library(qcc)
## Package 'qcc' version 2.7
## Type 'citation("qcc")' for citing this R package in publications.
defects <- c(8,7,11,3,6,11,4,4,5,8,
             8,6,7,6,10,7,9,5,6,8,
             8,10,4,11,4,11,6,6,9,5,
             4,7,5,7,8,5,10,9,7,9,
             4,9,4,7,11,8,10,8,7,5)
n = 200
np_chart <- qcc(defects,
                type = "np",
                sizes = n,
                title = "np Control Chart for Wafer Defects",
                xlab = "Lot Number",
                ylab = "Number of Defective Wafers")

pbar <- mean(defects) / n
npbar <- n * pbar

pbar
## [1] 0.0357
npbar
## [1] 7.14