Reliable Performance Enterprises - Weibull Reliability Analysis Tool

Quick Start

1. Prepare your data: Excel file with columns: Tag, Failure/Suspension (f or s), and a run/exposure value (hours, cycles, days, miles, etc.)
2. Load data: Copy from Excel or upload file
3. Set the Data Unit: Use the "Data Unit" dropdown on this tab to tell the tool what that third column actually represents (see note below — the tool does not read this from your spreadsheet's column header)
4. Select ranking method: Choose Median Rank (recommended), Mean Rank, or Kaplan-Meier
5. Run analysis: Click "Run Weibull Analysis"
6. Review results: Check Summary, Charts, and Data Analysis tabs
7. Export: Save results to Excel, or generate a PDF report, for reporting

⚠️ Set the Data Unit — the tool does not infer it for you

The third data column can represent any run-based measure: hours, cycles, days, miles, starts, rounds, or anything else your organization tracks. The "Data Unit" dropdown on the Data Entry tab controls how that column is labeled everywhere in the tool — the Weibull math itself is unit-agnostic and works identically regardless of what the numbers represent.

Important: the tool does not read your spreadsheet's column header to determine the unit — it only uses header text to decide whether to skip the first row. If your spreadsheet's header says "Hours" but the values are actually cycles (or vice versa), the tool has no way to know that on its own. You must set the "Data Unit" dropdown yourself to match what the data actually represents. Getting this wrong won't affect β, η, MTTF, or any other calculated number — it only affects the label shown next to those numbers — but a mislabeled report can easily lead to a wrong real-world decision (e.g., reading a 172,357-cycle B10 life as 172,357 hours).

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)
• Run value: Time, cycles, or other exposure measure to failure/suspension (positive numbers). The column can be labeled anything in your spreadsheet ("Hours," "Cycles," a blank header, etc.) — the tool only checks whether row 1 looks like a header, it does not read that text as a unit. Set the "Data Unit" dropdown on the Data Entry tab to whatever this column actually represents; it drives the labels used throughout the results, charts, and reports (see callout above).

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 — the point at which 63.2% have failed, expressed in whatever Data Unit you selected (hours, cycles, days, etc.)
• R - Correlation Coefficient: Measure of fit quality (R > 0.95 is excellent)

Reliability Metrics

All three metrics below are expressed in your selected Data Unit (hours, cycles, days, etc.) — see the callout near the top of this tab.

• MTTF: Mean Time (or cycles/days/etc.) To Failure - average life expectancy
• B10 Life: Point at which 10% of units are expected to have failed
• B50 Life: Median life - point at which 50% are expected to have failed

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

The horizontal "Time" axis on the Reliability and Hazard charts (and the underlying η, MTTF, and B-life values) all use whichever Data Unit you selected — hours, cycles, days, etc.

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, plus an image of each chart embedded on its own sheet
• 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
• Results/report show the wrong unit (e.g., "hours" but your data is cycles): The tool doesn't read unit meaning from your spreadsheet's column header — set the "Data Unit" dropdown on the Data Entry tab and re-run the analysis (or, if results already exist, the labels update immediately when you change it)

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.

A note on units: every formula below is expressed in terms of a generic variable t — the "run" or exposure value from your data. The mathematics is completely unit-agnostic: t, η, MTTF, and B_p life are all in whatever unit your data uses (hours, cycles, days, miles, etc.). The tool does not infer this from your spreadsheet; it uses the "Data Unit" selector on the Data Entry tab purely to label results, charts, and reports correctly — it has no effect on any calculation.

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, in your selected Data Unit (hours, cycles, days, etc.)
• t: The run/exposure value from your data — hours, cycles, days, miles, or whatever unit you selected on the Data Entry tab

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. MTTF is in the same unit as η and t (whatever Data Unit you selected).

B_p Life (Percentile Life)

Also expressed in your selected Data Unit. Examples:

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