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: makie 0.22.10 | Julia 1.11.9
# Quality: 91/100 | Created: 2026-06-07
using CairoMakie
using Colors
using Random
using Statistics
Random.seed!(42)
# Theme tokens — Imprint palette
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const ELEVATED_BG = THEME == "light" ? colorant"#FFFDF6" : colorant"#242420"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const INK_MUTED = THEME == "light" ? colorant"#6B6A63" : colorant"#A8A79F"
const IMPRINT_PALETTE = [
colorant"#009E73", # 1 — brand green (failure data points)
colorant"#C475FD", # 2 — lavender
colorant"#4467A3", # 3 — blue (Weibull fit line)
colorant"#BD8233", # 4 — ochre (censored points)
colorant"#AE3030", # 5 — matte red
colorant"#2ABCCD", # 6 — cyan
colorant"#954477", # 7 — rose
colorant"#99B314", # 8 — lime
]
# Data: bearing fatigue-life qualification test (hours)
# True Weibull: shape β = 2.5, characteristic life η = 8500 h
n_total = 35
beta_true = 2.5
eta_true = 8500.0
all_lifetimes = eta_true .* (-log.(rand(n_total))).^(1.0 / beta_true)
censor_time = 9200.0 # inspection end time — survivors are right-censored here
observed_times = min.(all_lifetimes, censor_time)
censored_mask = all_lifetimes .> censor_time
# Sort all observations by observed time
order = sortperm(observed_times)
times_sorted = observed_times[order]
censored_sorted = censored_mask[order]
# Johnson's method: adjusted mean order number for each failure
# accounting for right-censored suspensions
adj_ranks, failure_times, cens_times = let
prev = 0.0
_ranks = Float64[]
_ftimes = Float64[]
_ctimes = Float64[]
for i in 1:n_total
if !censored_sorted[i]
prev += (n_total + 1 - prev) / (n_total - i + 2)
push!(_ranks, prev)
push!(_ftimes, times_sorted[i])
else
push!(_ctimes, times_sorted[i])
end
end
_ranks, _ftimes, _ctimes
end
# Bernard's median rank: F_i = (rank_i - 0.3) / (n + 0.4)
F_vals = clamp.((adj_ranks .- 0.3) ./ (n_total + 0.4), 1e-6, 1.0 - 1e-6)
# Weibull linearization: y = ln(−ln(1−F)), x = ln(t)
log_t_fail = log.(failure_times)
y_fail = log.(-log.(1.0 .- F_vals))
# OLS regression in linearized space → β̂, η̂
x_bar = mean(log_t_fail)
y_bar = mean(y_fail)
beta_hat = sum((log_t_fail .- x_bar) .* (y_fail .- y_bar)) /
sum((log_t_fail .- x_bar).^2)
intercept = y_bar - beta_hat * x_bar
eta_hat = exp(-intercept / beta_hat)
# B10 life: time at which 10% of units have failed
t_b10 = exp((log(-log(1.0 - 0.10)) - intercept) / beta_hat)
# Fitted line spanning the observed time range
t_lo = minimum(times_sorted) * 0.65
t_hi = maximum(times_sorted) * 1.40
t_fit = exp.(range(log(t_lo), log(t_hi); length = 200))
y_fit = beta_hat .* log.(t_fit) .+ intercept
# Custom y-axis ticks on the Weibull probability scale
prob_pct = [1.0, 5.0, 10.0, 20.0, 50.0, 63.2, 90.0, 99.0]
y_tick_vals = log.(-log.(1.0 .- prob_pct ./ 100.0))
y_tick_lbls = ["1%", "5%", "10%", "20%", "50%", "63.2%", "90%", "99%"]
# Grid color: INK at 15% opacity
grid_col = RGBAf(INK.r, INK.g, INK.b, 0.15f0)
# Figure
fig = Figure(
size = (1600, 900),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
ax = Axis(
fig[1, 1];
xscale = log10,
title = "probability-weibull · julia · makie · anyplot.ai",
titlesize = 20,
titlecolor = INK,
xlabel = "Time to Failure (hours)",
ylabel = "Cumulative Failure Probability",
xlabelsize = 14,
ylabelsize = 14,
xticklabelsize = 12,
yticklabelsize = 12,
xlabelcolor = INK,
ylabelcolor = INK,
xticklabelcolor = INK_SOFT,
yticklabelcolor = INK_SOFT,
xtickcolor = INK_SOFT,
ytickcolor = INK_SOFT,
backgroundcolor = PAGE_BG,
topspinevisible = false,
rightspinevisible = false,
leftspinecolor = INK_SOFT,
bottomspinecolor = INK_SOFT,
xgridcolor = grid_col,
ygridcolor = grid_col,
yticks = (y_tick_vals, y_tick_lbls),
)
# 63.2% reference line — time axis spans the full x range
y_632 = log(-log(1.0 - 0.632))
hlines!(ax, [y_632]; color = INK_MUTED, linewidth = 1.5, linestyle = :dash,
label = "63.2% (η)")
# Fitted Weibull line
lines!(ax, t_fit, y_fit; color = IMPRINT_PALETTE[3], linewidth = 2.5,
label = "Weibull fit")
# Failure observations — filled circles (first Imprint series, brand green)
scatter!(ax, failure_times, y_fail;
color = IMPRINT_PALETTE[1],
markersize = 12,
marker = :circle,
strokewidth = 0,
label = "Failure",
)
# Censored observations — hollow circles at their time, y interpolated from fit
if length(cens_times) > 0
y_cens = beta_hat .* log.(cens_times) .+ intercept
scatter!(ax, cens_times, y_cens;
color = :transparent,
strokecolor = IMPRINT_PALETTE[4],
strokewidth = 2.0,
markersize = 12,
marker = :circle,
label = "Censored",
)
end
# Parameter annotation (upper-left, away from dense data region)
ann_x = t_lo * 1.6
ann_y = y_tick_vals[end] - 0.10
ann_txt = "β = $(round(beta_hat, digits=2))\nη = $(round(Int, eta_hat)) h\nB10 = $(round(Int, t_b10)) h"
text!(ax, [ann_x], [ann_y];
text = [ann_txt],
fontsize = 13,
color = INK_SOFT,
align = (:left, :top),
)
# B10 life graphical marker — vertical dashed line from bottom of plot to the 10% level
y_b10 = log(-log(1.0 - 0.10))
lines!(ax, [t_b10, t_b10], [y_tick_vals[1] - 0.35, y_b10];
color = INK_MUTED,
linewidth = 1.5,
linestyle = :dash,
)
scatter!(ax, [t_b10], [y_b10];
color = INK_SOFT,
marker = :diamond,
markersize = 10,
strokewidth = 0,
)
text!(ax, [t_b10], [y_b10 + 0.08];
text = ["B10"],
fontsize = 12,
color = INK_SOFT,
align = (:center, :bottom),
)
# Legend
axislegend(ax;
position = :rb,
backgroundcolor = ELEVATED_BG,
framecolor = INK_SOFT,
labelcolor = INK,
labelsize = 11,
)
# Constrain y to the tick range with small padding
ylims!(ax, y_tick_vals[1] - 0.35, y_tick_vals[end] + 0.30)
save("plot-$(THEME).png", fig; px_per_unit = 2)
Runnable source as JSON, for any HTTP client: https://api.anyplot.ai/specs/probability-weibull/makie/code. Any spec id and library id listed in llms-full.txt fit the same URL shape; every URL below is complete and callable.
{
"spec_id": "probability-weibull",
"language": "julia",
"library": "makie",
"page": "https://anyplot.ai/probability-weibull/julia/makie",
"hub": "https://anyplot.ai/probability-weibull",
"code_json": "https://api.anyplot.ai/specs/probability-weibull/makie/code",
"spec_json": "https://api.anyplot.ai/specs/probability-weibull",
"render_light_png": "https://storage.googleapis.com/anyplot-images/plots/probability-weibull/julia/makie/plot-light.png",
"render_dark_png": "https://storage.googleapis.com/anyplot-images/plots/probability-weibull/julia/makie/plot-dark.png",
"quality_score": 91.0,
"license": "MIT",
"guide": "https://anyplot.ai/llms.txt"
}Part of Weibull Probability Plot for Reliability Analysis on anyplot.ai.