Weibull Distribution Calculator

Calculate PDF, CDF, and descriptive statistics for the Weibull distribution. Model equipment failure times, extreme values, and reliability data.

Last updated: March 2026

Weibull Calculator

Shape (k)

Determines shape (k > 1: increasing failure rate; k = 1: constant rate)

Scale (λ)

Scaling parameter (stretch factor)

x (Value)

Point at which to evaluate PDF/CDF (x ≥ 0)

Point Estimates

PDF f(x)0.744783

CDF F(x)0.297811

Distribution Characteristics

Mean (μ)0.902745

Variance (σ²)0.375690

Median

0.783220

Mode

0.480750

Std Dev (σ)

0.612936

What is the Weibull Distribution?

The Weibull distribution is a continuous probability distribution widely used to model the lifetimes of objects and phenomena. It describes how long something lasts before it fails, and is central to reliability engineering, quality control, and failure analysis.

The Weibull distribution is defined by two parameters: shape (k) and scale (λ). The shape parameter controls the distribution's behavior:

k < 1: Decreasing failure rate (early failures). Used for infant mortality in products.
k = 1: Constant failure rate (memoryless, like exponential distribution). Random failures.
k > 1: Increasing failure rate (wear-out). Typical for aging equipment. Creates a mode (peak) but is not bell-shaped like normal distribution.
k ≈ 3.5: Approaches normal distribution shape.

The scale parameter λ stretches the distribution horizontally and represents the characteristic life — the time at which about 63.2% of items have failed.

How to Calculate Weibull Values

Formulas

PDF: f(x) = (k/λ) × (x/λ)^(k−1) × exp(−(x/λ)^k)

CDF: F(x) = 1 − exp(−(x/λ)^k)

Mean: μ = λ × Γ(1 + 1/k)

Variance: σ² = λ² × [Γ(1 + 2/k) − Γ(1 + 1/k)²]

Median: m = λ × (ln 2)^(1/k)

Mode: λ × ((k−1)/k)^(1/k) if k > 1; else 0

Step-by-Step Process

Step 1: Choose k (shape) and λ (scale) parameters for your data

Step 2: Specify x (the value at which to evaluate the distribution)

Step 3: Compute PDF: probability density at x (probability per unit)

Step 4: Compute CDF: cumulative probability up to x (percent of population below x)

Step 5: Use mean, variance, median to understand distribution center and spread

When to Use Weibull

Use Weibull distribution for reliability data, failure rates, material strength, and extreme value phenomena. Fitting k and λ to historical failure data predicts future equipment lifetimes and guides maintenance schedules.

Example Calculation

Equipment Maintenance: Projected Failure Time

Setup:

Industrial pump with shape k = 1.5, scale λ = 1000 hours. Query: probability of failure by 500 hours?

Step 1:

Given: k = 1.5, λ = 1000, x = 500

Step 2:

Calculate CDF at x = 500:

(x/λ)^k = (500/1000)^1.5 = 0.5^1.5 = 0.3536
CDF = 1 − exp(−0.3536) = 1 − 0.7016 = 0.2984

Step 3:

Interpret result:

About 29.84% probability of failure by 500 hours

Result:

Maintenance team should schedule preventive replacement before 500 hours to avoid costly downtime. A higher k-value (>1.5) would shift the failure curve right (later failures), reflecting more reliable equipment.

Frequently Asked Questions

What's the difference between shape and scale?

Shape (k) determines the distribution's form: <1 means early failures, =1 means constant rate, >1 means wear-out. Scale (λ) stretches the curve horizontally; larger λ shifts failures later in time.

How do I fit k and λ to real data?

Maximum likelihood estimation (MLE) or least-squares regression on log-transformed data. Many software packages (R, Python) automate this. Once fitted, your k and λ values model the specific equipment.

What's the relationship to exponential distribution?

Exponential is a special case: k=1. Weibull generalizes it. Exponential models purely random failures (no memory). Weibull with k>1 models realistic wear-out dynamics.

Why is F(λ) = 0.632?

At x = λ: F(λ) = 1 − exp(−1) ≈ 0.632. This is why λ is called the 'characteristic life' — about 63.2% of items fail by this time.

Can k or λ be negative?

No. Both must be positive. k>0 ensures PDF integrates to 1. λ>0 ensures the scale is meaningful. Negative values violate the distribution's mathematical definition.

What if x=0?

At x=0: CDF=0 (no failures yet), but PDF(0) depends on k. When k>1: PDF(0)=0 (smooth start). When k=1: PDF(0)=1/λ (exponential). When 0<k<1: PDF(0) diverges to infinity. Be cautious with k<1 near x=0.

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