Statistical_Distributions_in_Quality_Control.pptx

Statistical analysis

1. Statistical Distributions in Quality Control Normal, Binomial, and Poisson Distributions with Examples and Solutions

2. Normal Distribution - Example 1 Given: μ = 50, σ = 0.4, limits = [49.2, 50.8] Find: P(49.2 ≤ X ≤ 50.8) Z1 = -2, Z2 = 2 P(-2 ≤ Z ≤ 2) = 0.9545 → 95.45% within limits

3. Normal Distribution - Example 2 Given: μ = 1.02 kg, σ = 0.04, USL = 1.08 kg Find: P(X > 1.08) Z = 1.5, P(Z>1.5)=0.0668 → 6.68% exceed

4. Normal Distribution - Example 3 Process mean μ=10 mm, σ=0.1 mm Find specs for 99.73% conformance (±3σ) → LSL=9.7, USL=10.3

5. Normal Distribution - Example 4 Mean μ=100, σ=2 Control limits = μ ± 3σ P(|Z|>3)=0.0027 → 0.27% outside control limits

6. Normal Distribution - Example 5 Given: LSL=18, USL=22, μ=20, σ=0.5 Cp = (USL-LSL)/(6σ) = 1.33 → Capable process

7. Normal Distribution - Example 6 Given: μ=100, σ=5, LSL=85, USL=110 Z_L=-3, Z_U=2 P(defect)=P(Z<-3)+P(Z>2)=0.02415 → 2.4% defective

8. Binomial Distribution - Example 1 Given: n=5, p=0.1 Find: P(X=2) P(X=2)=C(5,2)(0.1)^2(0.9)^3=0.0729

9. Binomial Distribution - Example 2 Given: n=5, p=0.1 Find: P(X≤1) P(0)+P(1)=0.9185 → 91.85%

10. Binomial Distribution - Example 3 (p-chart) n=100, p=0.04 σ_p=√(p(1−p)/n)=0.0196 UCL=0.099, LCL=0 → Control limits for p-chart

11. Poisson Distribution - Example 1 λ=2 defects/unit Find: P(X=3) P(X=3)=e^-2 * 2^3/3! = 0.1353 → 13.53%

12. Poisson Distribution - Example 2 λ=2 defects/unit Find: P(X>3) 1−P(X≤3)=1−0.8571=0.1429 → 14.29%

13. Poisson Distribution - Example 3 (c-chart) Average c̄=4 defects/unit σ=√4=2 UCL=10, LCL=0 → Control limits for c-chart

14. Comparison: Binomial vs Poisson Binomial: number of defective items (n,p) Poisson: number of defects per item (λ) Binomial → p-chart, np-chart Poisson → c-chart, u-chart