• Statistical methodology for improving process quality

• Focuses on reducing variability and defects

• Originated at Motorola

• Widely used in:

• Engineering

• Healthcare

• Finance

• Manufacturing

Core idea: better data → better decisions

A process operating at Six Sigma quality produces:

• 3.4 defects per million opportunities (DPMO)

• Extremely low variability

• Highly predictable outcomes

Achieved through:

• Normal distributions

• Process capability analysis

• Hypothesis testing

• Continuous improvement

Most Six Sigma models assume a normal process:

\[ X \sim \mathcal{N}(\mu, \sigma^2) \]

Where:

μ = process mean

σ = process standard deviation

Reducing σ is the key to improving quality.

A Six Sigma process satisfies:

\[ P(|X - \mu| > 6\sigma) \approx 0 \]

This implies nearly all observations fall within: \[ \mu \pm 6\sigma \]

Result: defects become extremely rare.

Defect rate is computed using the normal CDF:

\[ \text{DPMO} = (1 - \Phi(Z)) \times 10^6 \]

Where:

Z = sigma level

Φ = standard normal distribution

At \[ Z = 6 \]

\[ \text{DPMO} \approx 3.4 \]

Example: Simulated Manufacturing Process

We simulate a stable process with small variability.

set.seed(123)

process_data <- data.frame(
  value = rnorm(1000, mean = 50, sd = 2)
)

LSL <- 44
USL <- 56

Specifications: Target mean: 50 Lower specification limit: 44 Upper specification limit: 56

• Reduces waste and operational cost
• Improves reliability and consistency
• Demonstrates statistics in real-world decision making
• Widely recognized in industry

Six Sigma connects theory, data, and impact.

• Six Sigma is fundamentally statistical
• Reducing variation improves quality
• Normal distributions drive defect modeling
• R enables reproducible, professional analysis