Reference
Formula sheet
Every formula the course uses, with the module that explains it and the assumption that decides whether it applies.
Every formula the course uses, in one place, with the module that explains it. Nothing here is a substitute for that explanation: a formula applied without its assumptions is how a Cpk of 1.4 gets reported on a process that is not stable, not normal, and measured with a gauge that eats a third of the tolerance. Where a formula needs a constant, it is in the tables.
Descriptive statistics
Mean: x̄ = (Σ xi) / n
Sample standard deviation: s = √[ Σ(xi − x̄)² / (n − 1) ]
Population standard deviation: σ = √[ Σ(xi − μ)² / N ]
Range: R = xmax − xmin
Standard error of the mean: SE = s / √n The n − 1 is what makes s an unbiased estimator of the variance from a sample; use it unless you genuinely have the whole population. Module 1.
Sample standard deviation: s = √[ Σ(xi − x̄)² / (n − 1) ]
Population standard deviation: σ = √[ Σ(xi − μ)² / N ]
Range: R = xmax − xmin
Standard error of the mean: SE = s / √n The n − 1 is what makes s an unbiased estimator of the variance from a sample; use it unless you genuinely have the whole population. Module 1.
The normal distribution and Z
Z = (x − μ) / σ (or (x − x̄) / s from a sample)
Fraction beyond an upper limit: 1 − Φ(ZUSL), ZUSL = (USL − x̄) / s
Fraction below a lower limit: Φ(−ZLSL), ZLSL = (x̄ − LSL) / s The predicted fraction is only as good as the normal assumption behind it. Check the shape first. Modules 1, 7 and 8.
Fraction beyond an upper limit: 1 − Φ(ZUSL), ZUSL = (USL − x̄) / s
Fraction below a lower limit: Φ(−ZLSL), ZLSL = (x̄ − LSL) / s The predicted fraction is only as good as the normal assumption behind it. Check the shape first. Modules 1, 7 and 8.
Process capability
Within-subgroup sigma: σ̂within = R̄ / d₂ or
S̄ / c₄ or, on individuals, MR̄ / 1.128
Overall sigma: σ̂overall = s, the sample standard deviation of all the data
Cp = (USL − LSL) / (6σ̂) Cpu = (USL − x̄) / (3σ̂) Cpl = (x̄ − LSL) / (3σ̂) Cpk = min(Cpu, Cpl)
Cpm = (USL − LSL) / (6√[ σ̂² + (x̄ − T)² ]) with target T
Predicted PPM (normal): 106 × [ 1 − Φ(3Cpu) + Φ(−3Cpl) ]
Approximate 95 % interval on Cpk: Ĉpk [ 1 ± 1.96 √( 1/(9nĈpk²) + 1/(2(n−1)) ) ] Which sigma you used decides which letter you may print. Under the current AIAG & VDA and ISO 22514 convention the C letters require a process shown to be stable and the P letters do not, and both use the overall sigma; under the legacy 2005 convention C came from the within-subgroup sigma and P from the overall one. State the convention and the sigma next to every index. Modules 7 and 8.
Overall sigma: σ̂overall = s, the sample standard deviation of all the data
Cp = (USL − LSL) / (6σ̂) Cpu = (USL − x̄) / (3σ̂) Cpl = (x̄ − LSL) / (3σ̂) Cpk = min(Cpu, Cpl)
Cpm = (USL − LSL) / (6√[ σ̂² + (x̄ − T)² ]) with target T
Predicted PPM (normal): 106 × [ 1 − Φ(3Cpu) + Φ(−3Cpl) ]
Approximate 95 % interval on Cpk: Ĉpk [ 1 ± 1.96 √( 1/(9nĈpk²) + 1/(2(n−1)) ) ] Which sigma you used decides which letter you may print. Under the current AIAG & VDA and ISO 22514 convention the C letters require a process shown to be stable and the P letters do not, and both use the overall sigma; under the legacy 2005 convention C came from the within-subgroup sigma and P from the overall one. State the convention and the sigma next to every index. Modules 7 and 8.
Defects, yield and sigma level
DPMO = (defects / (units × opportunities per unit)) × 106
DPU = defects / units First-time yield of a step: Y = e−DPU (Poisson) or good/total
Rolled throughput yield: RTY = Y₁ × Y₂ × … × Yk Normalised yield = RTY1/k Total DPU = −ln(RTY)
Z (long term, no shift) = Φ−1(1 − DPMO / 106) Sigma level (Motorola convention) = Z + 1.5 The 1.5 is a convention, not a measurement. Say which of the two numbers you are quoting. Module 6.
DPU = defects / units First-time yield of a step: Y = e−DPU (Poisson) or good/total
Rolled throughput yield: RTY = Y₁ × Y₂ × … × Yk Normalised yield = RTY1/k Total DPU = −ln(RTY)
Z (long term, no shift) = Φ−1(1 − DPMO / 106) Sigma level (Motorola convention) = Z + 1.5 The 1.5 is a convention, not a measurement. Say which of the two numbers you are quoting. Module 6.
Measurement systems analysis
σ²total = σ²part + σ²repeatability + σ²reproducibility
σGRR = √( σ²repeatability + σ²reproducibility )
%Contribution = 100 × σ²GRR / σ²total %Study Variation = 100 × σGRR / σtotal
%Tolerance = 100 × 6σGRR / (USL − LSL) ndc = 1.41 × σpart / σGRR Variances add; standard deviations do not. %Contribution and %Study Variation therefore tell different-looking stories about the same gauge, and both are correct. Module 4.
σGRR = √( σ²repeatability + σ²reproducibility )
%Contribution = 100 × σ²GRR / σ²total %Study Variation = 100 × σGRR / σtotal
%Tolerance = 100 × 6σGRR / (USL − LSL) ndc = 1.41 × σpart / σGRR Variances add; standard deviations do not. %Contribution and %Study Variation therefore tell different-looking stories about the same gauge, and both are correct. Module 4.
Control charts
X̄-R: X̄̄ ± A₂R̄ | R chart: D₃R̄ to D₄R̄
X̄-S: X̄̄ ± A₃S̄ | S chart: B₃S̄ to B₄S̄
I-MR: x̄ ± 3 MR̄ / 1.128 | MR chart: 0 to 3.267 MR̄
p chart: p̄ ± 3 √[ p̄(1 − p̄) / ni ] np chart: np̄ ± 3 √[ np̄(1 − p̄) ]
c chart: c̄ ± 3√c̄ u chart: ū ± 3 √( ū / ni )
EWMA: zi = λxi + (1 − λ)zi−1, limits target ± Lσ̂√[λ/(2 − λ)] in the steady state
Tabular CUSUM: C⁺i = max(0, C⁺i−1 + (xi − target)/σ̂ − k), signal when C⁺ or C⁻ exceeds h Every limit is the centre line plus or minus three standard deviations of the statistic being plotted. Specification limits are never drawn on any of these charts. Modules 16 and 17.
X̄-S: X̄̄ ± A₃S̄ | S chart: B₃S̄ to B₄S̄
I-MR: x̄ ± 3 MR̄ / 1.128 | MR chart: 0 to 3.267 MR̄
p chart: p̄ ± 3 √[ p̄(1 − p̄) / ni ] np chart: np̄ ± 3 √[ np̄(1 − p̄) ]
c chart: c̄ ± 3√c̄ u chart: ū ± 3 √( ū / ni )
EWMA: zi = λxi + (1 − λ)zi−1, limits target ± Lσ̂√[λ/(2 − λ)] in the steady state
Tabular CUSUM: C⁺i = max(0, C⁺i−1 + (xi − target)/σ̂ − k), signal when C⁺ or C⁻ exceeds h Every limit is the centre line plus or minus three standard deviations of the statistic being plotted. Specification limits are never drawn on any of these charts. Modules 16 and 17.
Hypothesis tests and intervals
One sample: t = (x̄ − μ₀) / (s/√n), df = n − 1
Two samples, pooled: t = (x̄₁ − x̄₂) / (sp√(1/n₁ + 1/n₂)), sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)
Welch: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂), with the Welch–Satterthwaite df
Paired: t = d̄ / (sd/√n) on the n differences, df = n − 1
Confidence interval on a mean: x̄ ± tα/2, n−1 s/√n
Two-sample sample size: n per group = 2(zα/2 + zβ)² σ² / δ², then iterate with t
Effect size: Cohen's d = (x̄₁ − x̄₂) / sp
Chi-square on a contingency table: χ² = Σ (O − E)² / E, df = (r − 1)(c − 1) A p-value is the probability of data at least this extreme if the null hypothesis were true. It is not the probability that the hypothesis is true, and a large p-value is not evidence of no difference. Always report the effect size and its interval beside it. Module 11.
Two samples, pooled: t = (x̄₁ − x̄₂) / (sp√(1/n₁ + 1/n₂)), sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)
Welch: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂), with the Welch–Satterthwaite df
Paired: t = d̄ / (sd/√n) on the n differences, df = n − 1
Confidence interval on a mean: x̄ ± tα/2, n−1 s/√n
Two-sample sample size: n per group = 2(zα/2 + zβ)² σ² / δ², then iterate with t
Effect size: Cohen's d = (x̄₁ − x̄₂) / sp
Chi-square on a contingency table: χ² = Σ (O − E)² / E, df = (r − 1)(c − 1) A p-value is the probability of data at least this extreme if the null hypothesis were true. It is not the probability that the hypothesis is true, and a large p-value is not evidence of no difference. Always report the effect size and its interval beside it. Module 11.
Correlation and regression
r = Σ(x − x̄)(y − ȳ) / √[ Σ(x − x̄)² Σ(y − ȳ)² ]
Slope: b₁ = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² Intercept: b₀ = ȳ − b₁x̄
R² = SSregression / SStotal = 1 − SSresidual / SStotal
Adjusted R² = 1 − (1 − R²)(n − 1)/(n − k − 1)
Residual standard deviation: s = √( SSresidual / (n − k − 1) ) R² measures how much variation the model accounts for in the data you have, not whether the model is right. The residual plot decides that. Module 12.
Slope: b₁ = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² Intercept: b₀ = ȳ − b₁x̄
R² = SSregression / SStotal = 1 − SSresidual / SStotal
Adjusted R² = 1 − (1 − R²)(n − 1)/(n − k − 1)
Residual standard deviation: s = √( SSresidual / (n − k − 1) ) R² measures how much variation the model accounts for in the data you have, not whether the model is right. The residual plot decides that. Module 12.
Design of experiments
Main effect of A in a 2k: effect = ȳA high − ȳA low
Coefficient in the coded model: bA = effect / 2
Interaction AB: effect = ½[ (effect of A at B high) − (effect of A at B low) ]
Contrast form: effect = (Σ ±y) / (n2k−1) using the sign column of the design
Resolution of a fractional design: the length of the shortest word in the defining relation A fractional design buys runs with confounding, and you choose which confounding you can live with before you run it, not after. Module 13.
Coefficient in the coded model: bA = effect / 2
Interaction AB: effect = ½[ (effect of A at B high) − (effect of A at B low) ]
Contrast form: effect = (Σ ±y) / (n2k−1) using the sign column of the design
Resolution of a fractional design: the length of the shortest word in the defining relation A fractional design buys runs with confounding, and you choose which confounding you can live with before you run it, not after. Module 13.
Tolerances and loss
Worst case stack: Tassembly = Σ Ti
Root sum of squares: Tassembly = √( Σ Ti² )
Taguchi loss: L(x) = k(x − T)², k = A₀ / δ₀²
Average loss over a process: L̄ = k[ s² + (x̄ − T)² ] RSS predicts the spread of a stack correctly and says nothing about where it is centred; a small mean shift in each part is what turns a comfortable predicted capability into a marginal real one. Module 15.
Root sum of squares: Tassembly = √( Σ Ti² )
Taguchi loss: L(x) = k(x − T)², k = A₀ / δ₀²
Average loss over a process: L̄ = k[ s² + (x̄ − T)² ] RSS predicts the spread of a stack correctly and says nothing about where it is centred; a small mean shift in each part is what turns a comfortable predicted capability into a marginal real one. Module 15.