Formula Reference
All 20 calculators — formulas, notes, examples, benchmarks
Core Reliability Metrics
MTTR — Mean Time To Repair
Open Calculator →MTTR = Total Repair Time / Number of RepairsKey Notes
- • Repair time = detection to full restoration. Lower is better.
Example
(2h + 3h + 1.5h + 2.5h) ÷ 4 repairs = 2.25 hours
Benchmarks
MTBF — Mean Time Between Failures
Open Calculator →MTBF = Total Operating Time / Number of FailuresKey Notes
- • Operating time = time the system is running, excluding downtime.
Example
500 hours ÷ 10 failures = 50 hours
Benchmarks
MTTF — Mean Time To Failure
Open Calculator →MTTF = Total Operating Time / Number of Units FailedKey Notes
- • Used for non-repairable items. MTTF ≠ MTBF.
Example
10 bulbs × 1,000 h, 5 fail → MTTF = 10,000 ÷ 5 = 2,000 h
Benchmarks
Availability
Open Calculator →A = Uptime / (Uptime + Downtime) ≡ MTBF / (MTBF + MTTR)Key Notes
- • Expressed as % or decimal. High availability ≥ 99.9% (≤ 8.77 h downtime/year).
Example
MTBF=500h, MTTR=5h → A = 500 ÷ 505 = 99.01%
Benchmarks
OEE — Overall Equipment Effectiveness
Open Calculator →OEE = Availability × Performance × QualityKey Notes
- • Availability = Run Time / Planned Production Time
- • Performance = (Ideal Cycle Time × Total Count) / Run Time
- • Quality = Good Count / Total Count
Example
93.8% × 88.9% × 95.0% = 79.2%
Benchmarks
Downtime Cost
Open Calculator →Cost = (Lost Revenue + Labor Cost + Overhead + Penalty + Repair) × Downtime HoursKey Notes
- • Includes direct and indirect costs. Used for business case for maintenance investments.
Example
$5,000/h rate × 4 hours downtime = $20,000 total cost
Benchmarks
Spare Parts Optimization
Open Calculator →P(X ≤ k) = Σ [e⁻λ × λˣ / x!] (Poisson CDF)Key Notes
- • λ = expected demand per period. k = stock quantity. Targets service level (e.g., 95%).
Example
λ = 3 parts/yr, target 95% SL → stock 5 parts
Benchmarks
Advanced Reliability Analysis
Reliability R(t)
Open Calculator →R(t) = e^(−λt) = e^(−t / MTBF)Key Notes
- • λ = failure rate. t = mission time. Assumes constant failure rate (exponential distribution).
Example
MTBF=1000h, t=100h → R(100) = e^(−0.1) = 90.48%
System Reliability
Open Calculator →Series: Rs = R₁ × R₂ × … × RnParallel: Rs = 1 − (1−R₁)(1−R₂)…(1−Rn)k-of-n: Rs = Σ C(n,i) × Rⁱ × (1−R)^(n−i) for i ≥ kKey Notes
- • Parallel configurations improve availability through redundancy.
Example
2 components at 90% in parallel → Rs = 1−(0.1×0.1) = 99%
Failure Rate (λ)
Open Calculator →λ = Number of Failures / Total Operating TimeKey Notes
- • Units: failures per hour (fph), or × 10⁻⁶ (FIT). λ = 1 / MTBF.
Example
5 failures ÷ 10,000 hours = 0.0005 fph = 500 FIT
Weibull Analysis
Open Calculator →R(t) = e^−(t/η)^βKey Notes
- • η = characteristic life (scale parameter)
- • β = shape parameter: β<1 early failures, β=1 random, β>1 wear-out
- • MTTF = η × Γ(1 + 1/β)
Example
β=2, η=1000h → R(500) = e^−(0.5)² = e^−0.25 = 77.9%
Benchmarks
FMEA — Risk Priority Number
Open Calculator →RPN = Severity × Occurrence × DetectionKey Notes
- • Each factor rated 1–10. High RPN → prioritize corrective action.
Example
S=8, O=4, D=5 → RPN = 160 (High priority)
Benchmarks
Gage R&R
Open Calculator →%GRR = (σ_GRR / σ_Total) × 100%Where: σ²_Total = σ²_Part + σ²_GRRσ²_GRR = σ²_Repeatability + σ²_ReproducibilityKey Notes
- • %GRR < 10% → acceptable. 10–30% → marginal. > 30% → unacceptable.
Example
σ_GRR=0.05, σ_Total=0.20 → %GRR = 25% (marginal)
Benchmarks
Quality & Statistical Tools
Process Capability (Cp / Cpk)
Open Calculator →Cp = (USL − LSL) / (6σ)Cpk = min[ (USL − μ) / 3σ, (μ − LSL) / 3σ ]Key Notes
- • Cp measures potential; Cpk measures actual (accounts for centering).
Example
USL=205, LSL=195, μ=201, σ=1.2 → Cp=1.39, Cpk=1.11
Benchmarks
DPMO & Sigma Level
Open Calculator →DPMO = (Defects / (Units × Opportunities)) × 1,000,0006σ process: DPMO = 3.4Key Notes
- • DPU = Defects / Units. Yield = (1 − DPU) × 100%.
Example
12 defects, 1000 units, 3 opportunities → DPMO = 4,000 → ~4.1σ
Benchmarks
Sample Size
Open Calculator →n = (Z × σ / E)² or n = Z² × p(1−p) / E²Key Notes
- • Z = confidence z-score (95%→1.96). E = margin of error. σ = std dev.
Example
95% CI, σ=10, E=2 → n = (1.96×10/2)² = 96.04 → 97 samples
Control Chart (X̄-R)
Open Calculator →UCLₓ = X̄̄ + A₂R̄ LCLₓ = X̄̄ − A₂R̄UCL_R = D₄R̄ LCL_R = D₃R̄Key Notes
- • Constants for n=5: A₂=0.577, D₃=0, D₄=2.114. Points outside limits = special cause.
Example
X̄̄=50, R̄=0.06, n=5 → UCL = 50+0.577×0.06 = 50.035
Pareto Analysis
Open Calculator →Category % = (Category Value / Total) × 100Cumulative % = Σ previous % + Current %Key Notes
- • 80/20 Rule: ~20% of causes account for ~80% of problems.
Example
A=42%, B=27%, C=15% → Top 2 = 69% of issues
Correlation (Pearson r)
Open Calculator →r = Σ[(xᵢ−x̄)(yᵢ−ȳ)] / √[Σ(xᵢ−x̄)² × Σ(yᵢ−ȳ)²]Key Notes
- • r ranges −1 to +1. |r| > 0.7 = strong. |r| < 0.3 = weak.
Example
Temperature vs defect rate: r = −0.82 (strong negative correlation)
Benchmarks
Descriptive Statistics
Open Calculator →Mean (x̄) = Σx / nStd Dev (σ) = √[Σ(x−x̄)² / n]CV = (σ / x̄) × 100%Key Notes
- • CV = coefficient of variation. Describes spread relative to mean.
Example
Data: 10,12,11,13,14 → x̄=12, σ=1.41, CV=11.8%