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Weibull Distribution Calculator

Free Weibull distribution calculator. Compute PDF, CDF, reliability, hazard rate, MTTF, B-life. Reli

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

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STATISTICSDistributions

Weibull — Reliability: PDF, CDF, Hazard Rate, MTTF, B-life

Ball bearings, light bulbs, turbine blades. Infant mortality (k<1), constant rate (k=1), wear-out (k>1).

Exponential →Rayleigh →

Real-World Scenarios — Click to Load

Inputs

PDF with P(a≤X≤b) Shaded

weibull_results.sh
CALCULATED
$ weibull_dist --k=2 --lambda=100 --x=50
Mean (MTTF)
88.6227
Median
83.2555
Mode
70.7107
Variance
2146.0184
P(X≤50)
22.1199%
P(20≤X≤80)
43.3497%
Reliability R(50)
77.8801%
Hazard h(50)
0.010000
B10
32.4593
B50 (Median)
83.2555
B10
32.4593
Share:
Weibull Distribution Weibull(k=2, λ=100)
P(X≤50) = 22.12%
P(20≤X≤80) = 43.35%
MTTF = 88.623B10 = 32.459Reliability R(50) = 77.88%
numbervibe.com/calculators/statistics/weibull-distribution-calculator

Hazard Rate by Shape k

Calculation Breakdown

COMPUTATION
Mean (MTTF)
88.6227
\text{lambda} Γ(1+1/k)
Median
83.2555
\text{lambda} (ln2)^(1/k)
Mode
70.7107
\text{lambda} ((k-1)/k)^(1/k) ext{when} k>1
MOMENTS
Variance
2146.0184
\text{lambda} ^{2}[Γ(1+2/k)-Γ(1+1/k)^{2}]
P(X≤x)
22.1199%
1 - \text{exp}(-(x/\text{lambda} )^k)
P(a≤X≤b)
43.3497%
\text{exp}(-(a/\text{lambda} )^k) - \text{exp}(-(b/\text{lambda} )^k)
RELIABILITY
Reliability R(x)
77.8801%
\text{exp}(-(x/\text{lambda} )^k)
Hazard h(x)
0.010000
(k/\text{lambda} )(x/\text{lambda} )^(k-1)
B-LIFE
B10
32.4593
ext{Time} ext{when} 10% ext{failed}

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • The Weibull distribution Weibull(k, λ) models failure times with flexible hazard: k<1 decreasing, k=1 constant (exponential), k>1 increasing
  • • PDF: f(x) = (k/λ)(x/λ)^(k-1) × exp(-(x/λ)^k) for x ≥ 0; CDF: F(x) = 1 - exp(-(x/λ)^k)
  • • Reliability R(x) = exp(-(x/λ)^k); Hazard h(x) = (k/λ)(x/λ)^(k-1)
  • • Mean = λΓ(1+1/k), Median = λ(ln2)^(1/k), Mode = λ((k-1)/k)^(1/k) when k>1
  • • B-life B_p = λ(-ln(1-p/100))^(1/k); MTTF = Mean

Did You Know?

🔧Ball bearings often follow Weibull with k≈2–3 — wear-out failure modeSource: Reliability engineering
💡Light bulbs with k<1 have decreasing failure rate (infant mortality)Source: Consumer products
🔋Battery cycle life is commonly modeled with Weibull k≈2–4Source: Battery research
✈️Turbine blade and aircraft component lifetimes use Weibull for fatigueSource: Aerospace
⚡k=1 reduces to exponential — constant failure rate, memorylessSource: Wolfram MathWorld
🌊k=2 gives Rayleigh — magnitude of 2D Gaussian vectorSource: Statistics
📐k≈3.6 approximates a normal distribution shapeSource: NIST

How It Works

1. Shape Parameter k

k controls hazard behavior: k<1 infant mortality (decreasing), k=1 constant (exponential), k>1 wear-out (increasing).

2. Scale Parameter λ

λ is the characteristic life — 63.2% of units fail by time λ (since F(λ)=1-e^(-1)≈0.632).

3. Reliability and Hazard

R(x)=1-F(x)=exp(-(x/λ)^k). Hazard h(x)=(k/λ)(x/λ)^(k-1) — power-law in x.

4. B-life

B_p is the time by which p% have failed. B10 = time when 10% failed; B50 = median. B_p = λ(-ln(1-p/100))^(1/k).

5. MTTF

Mean Time To Failure = Mean = λΓ(1+1/k). Requires the Gamma function.

Expert Tips

Weibull vs Exponential

Use Weibull when hazard varies with time. Exponential (k=1) assumes constant failure rate — often too simplistic.

Bathtub Curve

Real systems often show infant mortality (k<1), useful life (k=1), wear-out (k>1). Weibull fits each phase.

Parameter Estimation

Fit via MLE or linear regression on ln(-ln(R)) vs ln(x). Weibull paper gives straight line.

Special Cases

k=1: Exponential. k=2: Rayleigh. k≈3.6: approximately Normal. Know these for quick checks.

Why Use This Calculator vs Other Tools?

FeatureThis CalculatorExcelRSciPy
PDF + CDF + Reliability + Hazard✅⚠️ Multiple functions⚠️ Multiple functions✅
B-life percentiles✅❌⚠️ Manual⚠️ Manual
7 real-world presets✅❌❌❌
Hazard rate & bathtub chart✅❌❌⚠️ Requires plot
Educational content✅❌❌❌
AI analysis✅❌❌❌

Frequently Asked Questions

When should I use Weibull vs Exponential?

Use Weibull when failure rate changes with time (infant mortality, wear-out). Exponential assumes constant rate — only when k=1.

What does the shape parameter k mean?

k<1: decreasing hazard (infant mortality). k=1: constant (exponential). k>1: increasing hazard (wear-out).

What is B10 life?

Time by which 10% of units have failed. B10 = λ(-ln(0.9))^(1/k). Critical in reliability specs.

What is MTTF?

Mean Time To Failure = λΓ(1+1/k). Average lifetime. Same as mean of the distribution.

How do I interpret the scale λ?

Characteristic life. 63.2% fail by time λ. Larger λ = longer typical life.

What is the bathtub curve?

Many systems: high early failures (infant), constant middle (useful life), rising late (wear-out). Weibull fits each phase with different k.

Weibull vs Rayleigh?

Rayleigh is Weibull with k=2. Used for magnitude of 2D Gaussian (wind speed, signal envelope).

How do I fit Weibull to data?

MLE or Weibull paper: plot ln(-ln(R̂)) vs ln(x). Slope ≈ k, intercept relates to λ.

Weibull Distribution by the Numbers

λΓ(1+1/k)
Mean (MTTF)
λ(ln2)^(1/k)
Median
k=1,2,3.6
Exp, Rayleigh, ~Normal
B_p
B-life

Disclaimer: This calculator uses exact closed-form formulas for the Weibull distribution. Results are mathematically exact. For critical applications (reliability engineering, warranty analysis), verify assumptions and fit parameters from real data. This tool is for educational and professional reference purposes.

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