Statistical Methods in Quality Management

Introduction to Statistical Methods

• Statistics: The science of collecting, organizing, analyzing, interpreting, and presenting data.
• Purpose: Helps managers to understand data, variety, and make informed decisions in quality management through various applications:
  • Product and market analysis
  • Product and process design
  • Process control
  • Testing and inspection
  • Improvement identification and verification
  • Reliability analysis

Basic Probability Concepts

• Experiment: A process with a defined outcome.
• Outcome: A result observed from an experiment.
• Sample Space: All possible outcomes from an experiment.
• Probability: Likelihood of an outcome occurring, where 0 ≤ P(x) ≤ 1 and total probability sums to 1.
• Event: A collection of outcomes from the sample space.
• Complement: Outcomes not included in an event (A).

Rules of Probability

1. P(event) = Sum of probabilities of its outcomes.
2. P(A') = 1 - P(A).
3. For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
4. For non-mutually exclusive A and B, P(A or B) = P(A) + P(B) - P(A and B).

Conditional Probability

• Definition: Probability of event A given event B has occurred.
• Formula: P(A|B) = P(A and B) / P(B).
• Multiplication Rule: P(A and B) = P(A|B) * P(B) = P(B|A) * P(A).
• If A and B are independent, then P(A|B) = P(A) and P(A and B) = P(A) * P(B).

Example: RoadRunner Inc.

• Production involves 2 stages; defect probabilities:
  • Stage 1: 15% defective
  • Stage 2: 10% defective
• Repairable Units:
  1. Defective in Stage 1 and Stage 2 → Completely defective
  2. Defective in Stage 1, Not in Stage 2 → Repairable I
  3. Not in Stage 1, Defective in Stage 2 → Repairable II
  4. Not defective in either → Completely good
• Probabilities Calculated:
  • Completely defective: 0.015
  • Repairable I: 0.135
  • Repairable II: 0.085
  • Completely good: 0.765
• Total Repairable Units Probability: P = 0.135 + 0.085 = 0.220

Probability Distributions

• Random Variable: Numerical representation of outcomes.
• Probability Distribution: Characterizes possible values of a random variable and their probabilities.
• Cumulative Distribution Function (CDF): P(X ≤ x).

Discrete Probability Distributions

• Definition: Random variable can take finite or countable values.
  • Binomial Distribution: Used to model number of defective items in a sample.
  • Formula: P(x) = nCx * p^x * (1 - p)^(n-x)
  • Expected Value: E[X] = np
  • Variance: Var[X] = np(1 - p)
  • Standard Deviation: SD[X] = √np(1 - p)
• Excel Function: BINOM.DIST(number_s, trials, probability_s, cumulative)

Example: SAT Partners

• Sample of 250 bills: 196 overdue, 54 paid.
  • Probability of paid = 54/250 = 0.216; overdue = 1 - paid.
• Binomial calculations:
  1. P(exactly 2 bills paid) = 0.000483298
  2. P(20 or fewer bills paid) = 0.999058866
  3. P(40 or more overdue) = 0.47202

Poisson Distribution

• Definition: Models number of events occurring in a fixed interval of time or space.
• Parameter: λ (lambda), the average number of events.
• Formula: P(x) = (λ^x * e^-λ) / x!
• Expected Value and Variance: E[X] = λ, Var[X] = λ.
• Excel Function: POISSON.DIST(x, mean, cumulative)

Continuous Probability Distributions

• Definition: Random variable defined over intervals of real numbers; infinite outcomes.
• Probability Density Function: Curve characterizing outcomes; integrals give probabilities.

Normal Distribution

• Characteristics: Bell-shaped curve; symmetric around the mean.
• Standard Normal Distribution: Mean = 0, Standard Deviation = 1.
• Cumulative Probability: Calculated with functions such as NORM.DIST().
• Area Under the Curve: Represents probabilities; total area equals 1.
• 68-95-99.7 Rule: % of values within 1, 2, and 3 standard deviations from the mean.
• Example (Tire Warranty): Mean = 75,000 miles, SD = 6,000 miles.
  • Calculate probabilities for replacements at lower and higher mileages.

Exponential Distribution

• Definition: Models time between random events; relates to the Poisson distribution.
• Formula: f(t) = λe^(-λt); CDF: F(t) = 1 - e^(-λt)
• Excel Function: EXPON.DIST(x, lambda, TRUE)

Conclusion

• Understanding these statistical methods is crucial for effective quality management and decision-making in various organizational processes.

• Practice with examples will strengthen comprehension of probability, distributions, and statistical inference, which are key for managing quality.