A Weibull probability plot displays failure or lifetime data on Weibull probability paper (logarithmic x-axis for time/cycles, linearized Weibull CDF on y-axis) with a fitted straight line. It is the standard tool in reliability engineering for estimating Weibull distribution parameters (shape and scale), assessing whether data follow a Weibull distribution, and extrapolating failure probabilities. The slope of the fitted line gives the shape parameter (beta), while the characteristic life (eta) is read at the 63.2% failure probability crossing.
// anyplot.ai
// probability-weibull: Weibull Probability Plot for Reliability Analysis
// Library: highcharts 12.6.0 | JavaScript 22.23.2
// Quality: 90/100 | Created: 2026-08-24
const t = window.ANYPLOT_TOKENS;
// --- Data: turbine blade fatigue-life test, cycles to failure (or suspension) --
// Sorted ascending; a handful of units were pulled from test before failing
// (right-censored / "suspended"), which is common in reliability testing.
const observations = [
{ cycles: 38000, censored: false },
{ cycles: 45000, censored: false },
{ cycles: 52000, censored: false },
{ cycles: 55000, censored: true },
{ cycles: 58000, censored: false },
{ cycles: 63000, censored: false },
{ cycles: 67000, censored: false },
{ cycles: 71000, censored: false },
{ cycles: 76000, censored: false },
{ cycles: 80000, censored: false },
{ cycles: 85000, censored: false },
{ cycles: 90000, censored: false },
{ cycles: 96000, censored: false },
{ cycles: 98000, censored: true },
{ cycles: 103000, censored: false },
{ cycles: 111000, censored: false },
{ cycles: 120000, censored: false },
{ cycles: 125000, censored: true },
{ cycles: 132000, censored: false },
{ cycles: 148000, censored: false },
{ cycles: 160000, censored: true },
{ cycles: 170000, censored: false },
];
const n = observations.length;
// --- Median-rank regression with rank adjustment for suspensions -----------
// Johnson's rank-increment method: each suspension leaves the pool of
// "at-risk" units without resolving a rank, so later failures inherit a
// larger increment. Suspended units are drawn at the prevailing rank so the
// censoring pattern is visible, but only failures feed the line fit.
let previousAdjustedRank = 0;
const points = observations.map((obs, i) => {
const reverseRank = n - i; // remaining items at/after this position
if (obs.censored) {
const cumulativeProbability = (previousAdjustedRank - 0.3) / (n + 0.4);
return { ...obs, cumulativeProbability };
}
const increment = (n + 1 - previousAdjustedRank) / (1 + reverseRank);
previousAdjustedRank += increment;
const cumulativeProbability = (previousAdjustedRank - 0.3) / (n + 0.4);
return { ...obs, cumulativeProbability };
});
// Weibull linearization: y = ln(-ln(1 - F)), plotted against ln(time).
const weibullY = (f) => Math.log(-Math.log(1 - f));
const failurePoints = points
.filter((p) => !p.censored)
.map((p) => [p.cycles, weibullY(p.cumulativeProbability)]);
const suspensionPoints = points
.filter((p) => p.censored)
.map((p) => [p.cycles, weibullY(p.cumulativeProbability)]);
// --- Least-squares fit on failures only: y = beta * ln(t) + intercept ------
const logT = failurePoints.map((p) => Math.log(p[0]));
const yVals = failurePoints.map((p) => p[1]);
const meanLogT = logT.reduce((a, b) => a + b, 0) / logT.length;
const meanY = yVals.reduce((a, b) => a + b, 0) / yVals.length;
let covariance = 0;
let variance = 0;
for (let i = 0; i < logT.length; i++) {
covariance += (logT[i] - meanLogT) * (yVals[i] - meanY);
variance += (logT[i] - meanLogT) ** 2;
}
const beta = covariance / variance; // shape parameter
const intercept = meanY - beta * meanLogT;
const eta = Math.exp(-intercept / beta); // scale parameter (characteristic life)
const allCycles = observations.map((o) => o.cycles);
const tMin = Math.min(...allCycles) * 0.9;
const tMax = Math.max(...allCycles) * 1.1;
const fitLine = [
[tMin, beta * Math.log(tMin) + intercept],
[tMax, beta * Math.log(tMax) + intercept],
];
// --- Y-axis: linear space in the linearized value, labeled as probability --
const probabilityTicks = [1, 5, 10, 20, 30, 40, 50, 63.2, 70, 80, 90, 95, 99];
const weibullTicks = probabilityTicks.map((p) => ({
p,
y: weibullY(p / 100),
}));
// --- X-axis: explicit, well-spaced tick positions (log10 of the cycle count,
// since Highcharts' logarithmic axis expects tickPositions in its internal
// linear/log space). The default tick algorithm packs a label every 10k in
// the 100k-200k decade, crowding "180k"/"190k" together at the right edge;
// thinning out above 100k keeps every label legibly separated.
const xAxisCycleTicks = [40000, 60000, 80000, 100000, 150000, 200000];
const xAxisTickPositions = xAxisCycleTicks.map((v) => Math.log10(v));
// --- Chart -------------------------------------------------------------------
Highcharts.chart("container", {
chart: {
backgroundColor: "transparent",
animation: false,
style: { fontFamily: "inherit" },
},
credits: { enabled: false },
colors: t.palette,
title: {
text: "probability-weibull · javascript · highcharts · anyplot.ai",
style: { color: t.ink, fontSize: "22px", fontWeight: "600" },
},
subtitle: {
text: `β (shape) = ${beta.toFixed(2)} · η (scale) = ${Math.round(eta).toLocaleString()} cycles — characteristic life at 63.2% cumulative probability`,
style: { color: t.inkSoft, fontSize: "14px" },
},
xAxis: {
type: "logarithmic",
title: {
text: "Cycles to Failure (log scale)",
style: { color: t.inkSoft, fontSize: "16px" },
},
tickPositions: xAxisTickPositions,
startOnTick: false,
endOnTick: false,
lineColor: t.inkSoft,
tickColor: t.inkSoft,
gridLineColor: t.grid,
gridLineWidth: 1,
labels: { style: { color: t.inkSoft, fontSize: "14px" } },
},
yAxis: {
title: {
text: "Cumulative Failure Probability",
style: { color: t.inkSoft, fontSize: "16px" },
},
tickPositions: weibullTicks.map((wt) => wt.y),
gridLineColor: t.grid,
lineColor: t.inkSoft,
labels: {
style: { color: t.inkSoft, fontSize: "14px" },
formatter() {
const match = weibullTicks.find((wt) => Math.abs(wt.y - this.value) < 1e-6);
return match ? `${match.p}%` : "";
},
},
plotLines: [
{
value: weibullY(0.632),
color: t.inkSoft,
dashStyle: "ShortDash",
width: 1.5,
zIndex: 4,
label: {
text: "63.2% · η",
style: { color: t.inkSoft, fontSize: "13px" },
align: "left",
x: 6,
},
},
],
},
legend: {
itemStyle: { color: t.inkSoft, fontSize: "14px" },
itemHoverStyle: { color: t.ink },
},
tooltip: {
pointFormatter() {
const probability = (1 - Math.exp(-Math.exp(this.y))) * 100;
return `Cycles: ${this.x.toLocaleString()}<br/>Probability: ${probability.toFixed(1)}%`;
},
},
plotOptions: {
series: { animation: false },
scatter: { marker: { radius: 7, lineWidth: 2 } },
},
series: [
{
type: "scatter",
name: "Failures",
data: failurePoints,
marker: {
symbol: "circle",
fillColor: t.palette[0],
lineColor: t.palette[0],
lineWidth: 0,
},
color: t.palette[0],
},
{
type: "scatter",
name: "Suspensions (censored)",
data: suspensionPoints,
marker: {
symbol: "circle",
fillColor: t.pageBg,
lineColor: t.palette[0],
lineWidth: 2,
},
color: t.palette[0],
},
{
type: "line",
name: "Weibull fit",
data: fitLine,
color: t.ink,
lineWidth: 2.5,
dashStyle: "ShortDash",
marker: { enabled: false },
enableMouseTracking: false,
},
],
});
Part of Weibull Probability Plot for Reliability Analysis on anyplot.ai.