CSSBB formula sheet (free, printable)
The formulas a Six Sigma Black Belt candidate uses most, grouped the way DMAIC uses them. Print this page or save it as a PDF from your browser's print dialog. Each section links a free calculator, so you can check your own numbers.
Check the current ASQ exam rules for which references you may bring. This sheet is our own summary, not an ASQ document.
1. Probability and distributions
| Addition rule | P(A or B) = P(A) + P(B) − P(A and B) |
| Multiplication rule | P(A and B) = P(A) × P(B|A); independent: P(A) × P(B) |
| Bayes | P(A|B) = P(B|A) × P(A) / P(B) |
| Combinations | C(n, x) = n! / (x!(n − x)!) |
| 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 |
| Hypergeometric | P(x) = C(D, x) C(N − D, n − x) / C(N, n) |
| Normal | z = (x − μ) / σ |
| Exponential | R(t) = e^(−λt); mean = 1/λ |
| Coefficient of variation | CV = s / x̄ |
2. Descriptive statistics and sampling
| Sample variance | s² = Σ(xᵢ − x̄)² / (n − 1) |
| Standard error of the mean | s / √n |
| CI for a mean (σ unknown) | x̄ ± t(α/2, n − 1) × s/√n |
| CI for a proportion | p̂ ± z(α/2) √(p̂(1 − p̂)/n) |
| CI for a variance | (n − 1)s² / χ²(α/2) ≤ σ² ≤ (n − 1)s² / χ²(1 − α/2) |
| Sample size, mean | n = (z(α/2) σ / E)² |
| Sample size, proportion | n = z(α/2)² p(1 − p) / E² |
| Sample size, test of a mean | n = ((z(α/2) + z(β)) σ / δ)² |
Calculators: sample size · t table · chi-square table · z table
3. Measurement system analysis
| Repeatability (EV) | R̿ × K₁ (K₁ = 1/d₂ for the number of trials) |
| Reproducibility (AV) | √((X̄diff × K₂)² − EV²/(n·r)) |
| Gage R&R | GRR = √(EV² + AV²) |
| Part variation | PV = Rp × K₃ |
| Total variation | TV = √(GRR² + PV²) |
| %GRR (study variation) | 100 × GRR / TV: <10% acceptable, 10–30% may be acceptable, >30% not |
| %Tolerance (P/T) | 100 × 6 × GRR / (USL − LSL) |
| Distinct categories | ndc = 1.41 × PV / GRR (want ≥ 5) |
| Variance form | σ²total = σ²part + σ²gage; σ²gage = σ²repeatability + σ²reproducibility |
Calculator: Gage R&R
4. Capability and performance
| Cp | (USL − LSL) / 6σ_within |
| Cpk | min(USL − μ, μ − LSL) / 3σ_within |
| Pp, Ppk | Same as Cp, Cpk with σ_overall (sample s) |
| Cpm | (USL − LSL) / (6√(σ² + (μ − T)²)), T = target |
| σ_within estimates | R̄/d₂ · S̄/c₄ · MR̄/1.128 |
| DPU | defects / units |
| DPMO | defects × 10⁶ / (units × opportunities per unit) |
| First pass yield (Poisson) | Y = e^(−DPU) |
| Rolled throughput yield | RTY = Y₁ × Y₂ × … × Yₙ |
| Normalized yield | RTY^(1/n) |
| Sigma level | z(DPMO) + 1.5 (long-term to short-term shift convention) |
| Sigma level (with 1.5σ shift) | DPMO |
|---|
| 3σ | 66,807 DPMO |
| 4σ | 6,210 DPMO |
| 5σ | 233 DPMO |
| 6σ | 3.4 DPMO |
Calculators: Cpk/Ppk from data · process capability · DPMO and sigma · RTY
5. Hypothesis tests and ANOVA
| 1-sample t | t = (x̄ − μ₀) / (s/√n), df = n − 1 |
| 2-sample t (pooled) | t = (x̄₁ − x̄₂) / (sₚ√(1/n₁ + 1/n₂)), sₚ² = ((n₁−1)s₁² + (n₂−1)s₂²)/(n₁+n₂−2) |
| 2-sample t (Welch) | t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) |
| Paired t | t = d̄ / (s_d/√n), df = n − 1 |
| 1 proportion | z = (p̂ − p₀) / √(p₀(1 − p₀)/n) |
| 2 proportions | z = (p̂₁ − p̂₂) / √(p̄(1 − p̄)(1/n₁ + 1/n₂)) |
| 1 variance | χ² = (n − 1)s² / σ₀², df = n − 1 |
| 2 variances | F = s₁² / s₂² (larger on top for a two-sided test) |
| Chi-square | χ² = Σ(O − E)²/E; independence df = (r − 1)(c − 1) |
| Errors and power | α = P(reject true H₀); β = P(miss a real effect); power = 1 − β |
| One-way ANOVA source | df | SS | MS | F |
|---|
| Between (factor) | k − 1 | SSB | MSB = SSB/(k − 1) | F = MSB/MSW |
| Within (error) | N − k | SSW | MSW = SSW/(N − k) | |
| Total | N − 1 | SST = SSB + SSW | | |
Calculators: t-test · F table · chi-square table
6. Correlation and regression
| Slope | b₁ = Sxy / Sxx = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² |
| Intercept | b₀ = ȳ − b₁x̄ |
| Correlation | r = Sxy / √(Sxx Syy) |
| R² | SSR / SST = 1 − SSE/SST (simple regression: r²) |
| Adjusted R² | 1 − (1 − R²)(n − 1)/(n − k − 1), k = predictors |
| Standard error of estimate | s = √(SSE / (n − 2)) for simple regression |
| Slope test | t = b₁ / SE(b₁), SE(b₁) = s / √Sxx, df = n − 2 |
| VIF (multicollinearity) | 1 / (1 − Rⱼ²); above 5 to 10 is a warning sign |
7. Design of experiments (2-level)
| Runs, full factorial | 2ᵏ |
| Runs, fractional factorial | 2ᵏ⁻ᵖ |
| Effect | avg(response at +) − avg(response at −) = contrast / (N/2) |
| Coded coefficient | effect / 2 |
| Sum of squares of an effect | SS = contrast² / N = N × effect² / 4 (N = total runs) |
| Resolution III / IV / V | III: main effects aliased with 2FIs · IV: mains clear, 2FIs aliased with each other · V: mains and 2FIs clear |
| Center points | Add to detect curvature without extra factorial runs |
Calculator: DOE effects
8. Control charts
| I-MR | x̄ ± 2.66 MR̄; MR UCL = 3.267 MR̄ |
| X̄-R | x̿ ± A₂R̄; R: D₃R̄ to D₄R̄ |
| X̄-S | x̿ ± A₃S̄; S: B₃S̄ to B₄S̄ |
| p | p̄ ± 3√(p̄(1 − p̄)/n) |
| np | np̄ ± 3√(np̄(1 − p̄)) |
| c | c̄ ± 3√c̄ |
| u | ū ± 3√(ū/n) |
| EWMA | zₜ = λxₜ + (1 − λ)zₜ₋₁; limits μ₀ ± Lσ√(λ/(2 − λ) × [1 − (1 − λ)²ᵗ]) |
| CUSUM (tabular) | C⁺ₜ = max(0, xₜ − (μ₀ + K) + C⁺ₜ₋₁); C⁻ₜ = max(0, (μ₀ − K) − xₜ + C⁻ₜ₋₁); K = kσ, signal when beyond H = hσ (often k = 0.5, h = 4 or 5) |
Calculators: control limits · constants n = 2–25 · chart picker
9. Lean and flow
| Takt time | available time / customer demand |
| OEE | availability × performance × quality |
| Little's law | WIP = throughput × lead time |
| Process cycle efficiency | value-added time / total lead time |
| Operators needed | total work content / takt time (round up) |
Calculators: takt time · OEE
10. Reliability
| MTBF | operating time / failures |
| Failure rate | λ = 1 / MTBF (constant-rate assumption) |
| Series system | R = R₁ × R₂ × … × Rₙ |
| Parallel system | R = 1 − (1 − R₁)(1 − R₂)…(1 − Rₙ) |
| Inherent availability | MTBF / (MTBF + MTTR) |
| MTBF lower bound (time-terminated) | 2T / χ²(CL, 2r + 2) |
Calculators: MTBF · system reliability
11. Project financials
| Future value | FV = PV(1 + i)ⁿ |
| Net present value | NPV = Σ CFₜ / (1 + i)ᵗ − initial investment |
| ROI | (benefit − cost) / cost |
| Simple payback | initial investment / annual net savings |
| IRR | the rate i at which NPV = 0 |
Guide: financial benefits and ROI
12. Risk
| RPN | Severity × Occurrence × Detection (each 1–10) |
| Action Priority (AIAG-VDA 2019) | Ranks S, then O, then D through a lookup table instead of multiplying; see the guide |
Guide: FMEA in DMAIC · RPN vs Action Priority
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