Statistics for Engineers

Descriptive Statistics

Measures of Central Tendency

• Mean (μ or x̄): Average = Σx / n
• Median: Middle value (50th percentile)
• Mode: Most frequent value

Measures of Spread

• Range: max - min
• Variance: σ² = Σ(xᵢ - μ)² / N (population) or s² = Σ(xᵢ - x̄)² / (n-1) (sample)
• Standard Deviation: σ = √(variance)
• Coefficient of Variation: CV = σ/μ × 100%

Probability Distributions

Normal Distribution

• Bell-shaped, symmetric about mean
• Defined by μ and σ
• 68-95-99.7 Rule: 1σ, 2σ, 3σ from mean
• Standardize: z = (x - μ) / σ

Binomial Distribution

• n trials, p = probability of success
• P(X=k) = C(n,k) × pᵏ × (1-p)^(n-k)
• Mean = np, Variance = np(1-p)

Poisson Distribution

• Rare events in time/space
• P(X=k) = λᵏe⁻λ / k!
• Mean = Variance = λ

Confidence Intervals

For a population mean (large sample): CI = x̄ ± z_(α/2) × (σ/√n)

• 90% CI: z = 1.645
• 95% CI: z = 1.960
• 99% CI: z = 2.576

Linear Regression

Fit a straight line to data: ŷ = b₀ + b₁x

Least-squares estimates:

• b₁ = [Σ(xᵢ-x̄)(yᵢ-ȳ)] / Σ(xᵢ-x̄)²
• b₀ = ȳ - b₁x̄

Correlation coefficient r: -1 ≤ r ≤ 1

• |r| = 1: perfect linear relationship
• r = 0: no linear relationship

Engineering Applications

• Quality control (control charts, Six Sigma)
• Material testing (sample size, confidence in strength)
• Reliability analysis (failure rates)
• Hydrologic frequency analysis

Summary

Engineers use statistics to make decisions under uncertainty. Know mean, standard deviation, normal distribution, and regression — they appear on the FE exam and in practice.