Reliable Performance Enterprises - Weibull Reliability Analysis Tool

Quick Start

1. Prepare your data: Excel file with columns: Tag, Failure/Suspension (f or s), Hours
2. Load data: Copy from Excel or upload file
3. Select ranking method: Choose Median Rank (recommended), Mean Rank, or Kaplan-Meier
4. Run analysis: Click "Run Weibull Analysis"
5. Review results: Check Summary, Charts, and Data Analysis tabs
6. Export: Save results to Excel for reporting

Data Format Requirements

Your data should have three columns:

• Tag/ID: Unique identifier for each item (numbers or text)
• Failure/Suspension: Use 'f' for failures, 's' for suspensions (censored data)
• Hours: Time to failure or suspension (positive numbers)

Three Ways to Load Data

1. Copy/Paste from Excel: Select cells including headers, copy (Cmd+C/Ctrl+C), paste into text area
2. Upload Excel File: Click "📁 Upload Excel File" button to browse and select .xlsx or .xls file
3. Load Sample Data: Click "Load Sample Data" to see an example dataset

Validation Process

After pasting data:

• Small datasets (<100 rows) process automatically
• Large datasets require clicking "Validate Data"
• Review the data preview to check for errors
• Fix any issues in Excel and re-paste if needed

Weibull Parameters

• β (Beta) - Shape Parameter:
  • β < 1: Decreasing failure rate (infant mortality, early failures)
  • β = 1: Constant failure rate (random failures, exponential distribution)
  • β > 1: Increasing failure rate (wear-out, aging)
• η (Eta) - Scale Parameter: Characteristic life, time at which 63.2% fail
• R - Correlation Coefficient: Measure of fit quality (R > 0.95 is excellent)

Reliability Metrics

• MTTF: Mean Time To Failure - average life expectancy
• B10 Life: Time at which 10% of units will fail
• B50 Life: Median life - time at which 50% will fail

Confidence Intervals

All parameters include 95% confidence intervals:

• Use lower bounds for conservative design decisions
• Use upper bounds for worst-case planning
• Narrower intervals indicate more confidence in estimates
• Larger sample sizes produce narrower intervals

Weibull Probability Plot

• Linearized plot: ln(Time) vs ln(-ln(1-F))
• Straight line indicates good Weibull fit
• Slope = β (shape parameter)
• Confidence bands show uncertainty in fit
• Points should fall mostly within confidence bands

Reliability Function

• Shows probability of survival over time
• Starts at 100% (R=1) and decreases
• Steep drop indicates rapid wear-out
• Gradual decline indicates slow aging

Hazard Function

• Instantaneous failure rate at time t
• Decreasing: β < 1 (infant mortality)
• Constant: β = 1 (random failures)
• Increasing: β > 1 (wear-out)

Multiple export options available in the Charts & Plots and Data Analysis tabs:

• Export Charts Data: Probability plot, reliability, and hazard function data points
• Export Complete Analysis: All input data and calculated parameters
• Export Analysis Data: Weibull parameters with confidence intervals
• Export Plot Data: Failure times and calculated probabilities

All exports are in Excel (.xlsx) format for easy integration into reports.

Common Issues

• Data won't load: Check format - must have headers and correct columns
• Poor fit (low R): Data may not follow Weibull distribution - check for outliers
• Wide confidence intervals: Small sample size - collect more failure data
• Export not working: Click "🧪 Test Export" to verify Excel library is loaded
• Analysis fails: Need minimum 5 data points and 3 failures

Important: This tool provides statistical analysis for reliability engineering. All results should be validated by qualified engineers before making critical decisions. The tool uses standard Weibull analysis methods but results depend on data quality and appropriateness of the Weibull distribution for your specific application.

This appendix documents the mathematical methods and formulas used in the Weibull Reliability Analysis Tool. All calculations follow standard reliability engineering practices and statistical methods.

Probability Density Function (PDF)

Cumulative Distribution Function (CDF)

Where F(t) is the unreliability or probability of failure by time t

Reliability Function

Where R(t) is the probability of survival beyond time t

Hazard Function

Where h(t) is the instantaneous failure rate at time t

Parameters

• β (beta): Shape parameter - determines failure pattern
• η (eta): Scale parameter - characteristic life (hours)
• t: Time (hours, cycles, etc.)

Ranking methods assign cumulative probability F(i) to each ordered failure time.

Median Rank (Bernard's Approximation)

where i = rank position (1, 2, 3, ...), n = total number of failures

• Approximation to the exact median rank
• Reduces bias in small samples
• Most commonly used in reliability engineering

Mean Rank

Provides unbiased estimate of the expected value of the rank

Kaplan-Meier Estimator

where:

• n_i = number at risk just before time t_i
• d_i = number of failures at time t_i
• ∏ = product over all failure times ≤ t

Non-parametric estimator that properly handles censored (suspended) data

Linearization

The Weibull CDF can be linearized by taking double logarithms:

This transforms to linear form: Y = β×X + intercept

• X = ln(t)
• Y = ln(-ln(1 - F))
• Slope = β
• Intercept = -β × ln(η)

Linear Regression

Least squares regression estimates β and η:

Correlation Coefficient

R measures goodness of fit (closer to ±1 is better)

Mean Time To Failure (MTTF)

where Γ is the gamma function

B_p Life (Percentile Life)

Examples:

• B10: Time at which 10% fail (p = 10)
• B50: Median life (p = 50)
• B90: Time at which 90% fail (p = 90)

Parameter Confidence Intervals (95%)

Based on asymptotic normality of maximum likelihood estimators:

Regression Confidence Bands

For the Weibull probability plot, confidence bands at each point X:

where:

• s = residual standard error
• X̄ = mean of ln(time) values
• S_xx = sum of squared deviations of X
• t_α/2 ≈ 1.96 for 95% confidence (large n)

Bands are narrowest at X̄ and widen at extremes, properly representing uncertainty.

The gamma function is required for MTTF calculation. Lanczos approximation is used:

where A_g(z) is a series expansion with coefficients for g = 7

Minimum Requirements

• Minimum 5 total data points
• Minimum 3 failures (non-censored observations)
• More data provides better estimates and narrower confidence intervals

Recommended Sample Sizes

• n ≥ 20: Acceptable for preliminary analysis
• n ≥ 30: Good for engineering decisions
• n ≥ 50: Excellent statistical confidence

Key Assumptions

• Data follows a two-parameter Weibull distribution
• Failures are independent
• Operating conditions are consistent
• Time-to-failure is the only variable of interest

Validation

• Check correlation coefficient (R > 0.90 preferred)
• Visual inspection of probability plot for linearity
• Data points should fall mostly within confidence bands
• Consider alternative distributions if fit is poor

• Abernethy, R.B. (2006). The New Weibull Handbook, 5th Edition
• ReliaSoft Corporation. Life Data Analysis Reference
• Nelson, W. (1982). Applied Life Data Analysis. Wiley
• Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, 2nd Edition
• Meeker, W.Q. & Escobar, L.A. (1998). Statistical Methods for Reliability Data. Wiley

Note: These methods represent standard practices in reliability engineering. Implementation details may vary slightly between software packages. For critical applications, verify results using multiple tools and consult with reliability engineers.