Probability Fundamentals
P(A or B) = P(A) + P(B) − P(A and B)
P(A and B) = P(A) × P(B|A)
P(A and B) = P(A) × P(B) [if independent]
P(Ā) = 1 − P(A) [complement]
Combinations & permutations
C(n,r) = n! / [r!(n−r)!] [order doesn't matter]
P(n,r) = n! / (n−r)! [order matters]
Key Distributions
| Distribution | Use | Mean / Variance |
|---|
| Normal | Continuous; symmetric; many natural phenomena | μ / σ² |
| Binomial | n trials, p prob of success; count of successes | np / np(1−p) |
| Poisson | Rare events per unit; defects per area, failures/time | λ / λ |
| Exponential | Time between Poisson events; constant failure rate | 1/λ / 1/λ² |
| Weibull | Flexible reliability model; β<1 (infant), β=1 (random), β>1 (wear-out) | β, η shape/scale |
| Hypergeometric | Sampling without replacement; finite population | nK/N / n(K/N)(1−K/N)(N−n)/(N−1) |
| Chi-square | Variance tests; goodness of fit; contingency tables | df / 2·df |
| t-distribution | Means testing with unknown σ; small n | 0 / df/(df−2) for df > 2 |
| F-distribution | Ratio of variances; ANOVA | — |
Normal Distribution & Z
z = (X − μ) / σ
z = (x̄ − μ) / (σ/√n) [for sample means]
Key z values
| z | One-tail | Two-tail coverage |
|---|
| 1.645 | 5% | 90% |
| 1.960 | 2.5% | 95% |
| 2.576 | 0.5% | 99% |
| 3.000 | 0.13% | 99.73% |
Six Sigma relationship
DPMO = P(defect) × 1,000,000
- 6σ process with 1.5σ shift: 3.4 DPMO
- Sigma level = (USL − μ)/σ (short-term, no shift)
Descriptive Statistics
x̄ = Σxᵢ/n [mean]
s² = Σ(xᵢ−x̄)² / (n−1) [sample variance]
s = √[Σ(xᵢ−x̄)²/(n−1)] [sample std dev]
CV = (s/x̄) × 100% [coefficient of variation]
Central tendency comparison
- Mean: sensitive to outliers; use for symmetric distributions
- Median: resistant to outliers; use for skewed data
- Mode: most frequent; useful for categorical data
Shape
- Skewness: asymmetry (right/positive skew → tail to right)
- Kurtosis: tail heaviness vs. normal; excess kurtosis: 0 = normal, + = leptokurtic, − = platykurtic
Regression & Correlation
ŷ = b₀ + b₁x [simple linear regression]
b₁ = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)²
r = Sxy / (Sx × Sy) [Pearson correlation; −1 to +1]
R² = r² [simple linear regression only; proportion of variation explained]
- r = +1: perfect positive linear relationship
- r = 0: no linear relationship (may still be nonlinear)
- Correlation ≠ causation
- Residual plot should show random scatter (no patterns)
ANOVA (One-way)
F = MS_between / MS_within = (SS_between/df_between) / (SS_within/df_within)
| Source | df | SS | MS |
|---|
| Between groups | k−1 | SS_B | SS_B/(k−1) |
| Within groups | N−k | SS_W | SS_W/(N−k) |
| Total | N−1 | SS_T | — |
- k = number of groups; N = total observations
- Reject H₀ (all means equal) if F > F_critical
- Post-hoc tests (Tukey, Bonferroni) identify which pairs differ
Design of Experiments (DOE)
Key terms
| Term | Definition |
|---|
| Factor | Controllable variable changed during the experiment |
| Level | Setting of a factor (high/low, −1/+1 in coded units) |
| Response | Measured output variable |
| Main effect | Average change in response when one factor changes from low to high |
| Interaction | Effect of one factor depends on level of another |
| Full factorial | All combinations; 2ᵏ runs for k factors at 2 levels |
| Fractional factorial | Subset of full factorial; 2^(k−p) runs; confounds some effects |
| Resolution III | Main effects confounded with 2-factor interactions |
| Resolution IV | Main effects clear; some 2FIs confounded with other 2FIs |
| Resolution V | Main effects and all 2FIs estimable; 3FIs confounded with 2FIs |