Introduction

This report analyzes defect data from 20 lots of semiconductor wafers, where each lot contains 200 wafers. The data represents the number of defective wafers per lot. The goal is to establish control limits for an np-chart to monitor process stability. An np-chart is suitable here because the sample size (n=200 wafers per lot) is constant, and we’re tracking the number of non-conforming (defective) items.

The dataset is:
3, 7, 11, 3, 6, 11, 4, 4, 5, 8, 10, 8, 6, 7, 9, 8, 4, 6, 10, 9

We’ll calculate summary statistics, generate the np-chart with control limits, and interpret the results. All analyses use R with the qcc package for quality control charting.

Data and Summary Statistics

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

# Sample size
n <- 200

# Calculate statistics
p_bar <- mean(defects) / n
np_bar <- mean(defects)
ucl <- n * p_bar + 3 * sqrt(n * p_bar * (1 - p_bar))
lcl <- max(0, n * p_bar - 3 * sqrt(n * p_bar * (1 - p_bar)))

# Display results
cat("Average defects per lot (np_bar):", round(np_bar, 2), "\n")
## Average defects per lot (np_bar): 6.95
cat("Proportion defective (p_bar):", round(p_bar, 5), "\n")
## Proportion defective (p_bar): 0.03475
cat("UCL for np-chart:", round(ucl, 2), "\n")
## UCL for np-chart: 14.72
cat("LCL for np-chart:", round(lcl, 2), "\n")
## LCL for np-chart: 0

np-Chart Visualization

# Create np-chart using qcc
np_chart <- qcc(defects, type = "np", sizes = rep(n, length(defects)),
                title = "np-Chart for Defective Wafers",
                xlab = "Lot Number", 
                ylab = "Number of Defective Wafers")

# Add center line and limits (optional visual enhancement)
abline(h = np_bar, col = "blue", lty = 2, lwd = 1.5)
abline(h = ucl, col = "red", lty = 3, lwd = 1.5)
abline(h = lcl, col = "red", lty = 3, lwd = 1.5)

Interpretation

All 20 data points lie within the control limits, indicating the process is in statistical control. The highest defect count is 11 (lots 3 and 6), which is well below the Upper Control Limit. For ongoing monitoring, continue sampling lots of 200 wafers and plot against these limits.