CQE formula sheet (free, printable)
The core formulas, organized by topic and written in plain English. To get a copy, print this page or download the PDF. Check your exam's current rules on which references you can bring in.
Descriptive statistics
| Mean | x̄ = Σx / n |
| Sample std dev | s = √[ Σ(x − x̄)² / (n − 1) ] |
| Variance of a sum (independent) | σ²(X+Y) = σ²X + σ²Y |
| Standard error of the mean | σx̄ = σ / √n |
| z-score | z = (x − μ) / σ |
| Coefficient of variation | CV = s / x̄ |
Probability
| Addition rule | P(A or B) = P(A) + P(B) − P(A and B) |
| Multiplication rule (independent) | P(A and B) = P(A) × P(B) |
| Binomial | P(x) = C(n,x) pˣ (1−p)ⁿ⁻ˣ ; mean np, variance np(1−p) |
| Poisson | P(x) = e^(−λ) λˣ / x! ; mean = variance = λ |
| Combinations | C(n,r) = n! / [ r! (n−r)! ] |
SPC: control limits
| X̄–R chart | X̄: x̿ ± A₂R̄ R: D₃R̄ to D₄R̄ |
| σ estimate from ranges | σ̂ = R̄ / d₂ |
| Individuals–MR (n=2 moving range) | X: x̄ ± 2.66 MR̄ MR: UCL = 3.267 MR̄ |
| p chart | p̄ ± 3√[ p̄(1−p̄) / n ] |
| np chart | np̄ ± 3√[ np̄(1−p̄) ] |
| c chart | c̄ ± 3√c̄ |
| u chart | ū ± 3√(ū / n) |
| Rule | Negative lower limits are set to 0. |
Control chart constants
| n | A₂ | d₂ | D₃ | D₄ |
|---|
| 2 | 1.880 | 1.128 | 0.000 | 3.267 |
| 3 | 1.023 | 1.693 | 0.000 | 2.574 |
| 4 | 0.729 | 2.059 | 0.000 | 2.282 |
| 5 | 0.577 | 2.326 | 0.000 | 2.114 |
| 6 | 0.483 | 2.534 | 0.000 | 2.004 |
| 7 | 0.419 | 2.704 | 0.076 | 1.924 |
| 8 | 0.373 | 2.847 | 0.136 | 1.864 |
| 9 | 0.337 | 2.970 | 0.184 | 1.816 |
| 10 | 0.308 | 3.078 | 0.223 | 1.777 |
Process capability
| Cp / Pp | (USL − LSL) / 6σ (σ within for Cp, σ overall for Pp) |
| Cpk / Ppk | min(USL − x̄, x̄ − LSL) / 3σ |
| Cpm (Taguchi) | (USL − LSL) / 6√[ σ² + (x̄ − T)² ] |
| Defect rate from z | z = (limit − x̄) / σ → read the tail from the normal table |
| Rule | Use only for a stable (in-control) process. |
Measurement systems (MSA)
| Total GRR | σ²GRR = σ²repeatability + σ²reproducibility |
| Total variation | σ²total = σ²GRR + σ²part |
| %GRR (study variation) | 100 × σGRR / σtotal |
| %GRR (tolerance) | 100 × 6σGRR / (USL − LSL) |
| Number of distinct categories | ndc = 1.41 × σpart / σGRR (target ≥ 5) |
| Bias | bias = average measured − reference value |
Acceptance sampling
| Probability of acceptance (n, c) | Pa = Σ from x=0 to c of C(n,x) pˣ(1−p)ⁿ⁻ˣ |
| AOQ (rectifying inspection, approx.) | AOQ ≈ Pa × p |
| Producer's risk α | P(reject | lot at AQL) |
| Consumer's risk β | P(accept | lot at LTPD/RQL) |
Reliability
| Failure rate (constant) | λ = failures / total operating time |
| MTBF / MTTF | θ = 1 / λ |
| Exponential reliability | R(t) = e^(−λt) (R(θ) = 0.368) |
| Series system | Rsys = R₁ × R₂ × … × Rₙ |
| Parallel system | Rsys = 1 − (1−R₁)(1−R₂)…(1−Rₙ) |
| Availability | A = MTBF / (MTBF + MTTR) |
| Weibull | R(t) = e^[ −(t/η)^β ] β<1 infant, β≈1 random, β>1 wear-out |
Design of experiments
| Runs, full factorial | 2ᵏ (k factors, 2 levels) |
| Runs, fractional | 2^(k−p) |
| Main effect | avg(response at +) − avg(response at −) |
| Coded coefficient | effect / 2 |
| Resolution III / IV / V | main effects aliased with 2FI / 2FI aliased with each other / main effects and 2FI all clear |
Hypothesis tests and intervals
| 1-sample z | z = (x̄ − μ₀) / (σ/√n) |
| 1-sample t | t = (x̄ − μ₀) / (s/√n), df = n − 1 |
| 2-sample t (Welch) | t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) |
| Paired t | t = d̄ / (s_d/√n) |
| Chi-square, one variance | χ² = (n−1)s² / σ₀² |
| F, two variances | F = s₁² / s₂² (larger on top) |
| 1-proportion z | z = (p̂ − p₀) / √[ p₀(1−p₀)/n ] |
| Chi-square, counts | χ² = Σ (O − E)² / E |
| CI for a mean | x̄ ± t(α/2) × s/√n |
| Sample size for a mean | n = [ (z(α/2) + z(β)) σ / δ ]² |
| Errors | α = P(Type I) = reject true H₀; β = P(Type II); power = 1 − β |
Regression and correlation
| Slope | b = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)² |
| Intercept | a = ȳ − b x̄ |
| Coefficient of determination | R² = 1 − SSE/SST |
Quality costs and metrics
| DPMO | defects × 1,000,000 / (units × opportunities) |
| First-pass yield | good first time / units in |
| Rolled throughput yield | RTY = Y₁ × Y₂ × … × Yₙ |
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