np Control Chart for Wafer Defects

Background

Problem Context

A semiconductor fabrication facility inspects 200 wafers per lot after the photolithography stage. The number of defective wafers is recorded for 50 lots. An np control chart will be used to determine whether the process was operating in statistical control.

Data

Wafer Defect Counts

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
k <- length(defects)

np Chart Formulas

Control Limits

The fraction defective is

\[ \bar{p} = \frac{\sum d_i}{nk} \]

The center line is

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

The control limits are

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

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

Calculations

pbar <- sum(defects)/(n*k)
npbar <- n*pbar
sigma <- sqrt(n*pbar*(1-pbar))

UCL <- npbar + 3*sigma
LCL <- max(0, npbar - 3*sigma)

pbar

## [1] 0.0357

npbar

## [1] 7.14

UCL

## [1] 15.01184

LCL

## [1] 0

np Chart

qcc Output

library(qcc)

## Package 'qcc' version 2.7

## Type 'citation("qcc")' for citing this R package in publications.

chart <- qcc(defects,
type = "np",
sizes = rep(n, k))

Interpretation

All sample points fall within the control limits. This suggests the photolithography process was operating under statistical control during the observed period. The variation in defective wafers appears to be due to common causes rather than special causes.

Complete Code

library(qcc)

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
k <- length(defects)

pbar <- sum(defects)/(n*k)
npbar <- n*pbar
sigma <- sqrt(n*pbar*(1-pbar))

UCL <- npbar + 3*sigma
LCL <- max(0, npbar - 3*sigma)

chart <- qcc(defects,
type = "np",
sizes = rep(n, k))