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: ggplot2 3.5.1 | R 4.4.1
#' Quality: 90/100 | Created: 2026-06-07
library(ggplot2)
library(scales)
library(ragg)
set.seed(42)
# Theme tokens
THEME <- Sys.getenv("ANYPLOT_THEME", "light")
PAGE_BG <- if (THEME == "light") "#FAF8F1" else "#1A1A17"
ELEVATED_BG <- if (THEME == "light") "#FFFDF6" else "#242420"
INK <- if (THEME == "light") "#1A1A17" else "#F0EFE8"
INK_SOFT <- if (THEME == "light") "#4A4A44" else "#B8B7B0"
INK_MUTED <- if (THEME == "light") "#6B6A63" else "#A8A79F"
IMPRINT_PALETTE <- c(
"#009E73", # 1 — brand green (failures)
"#AE3030", # 5 — matte red (censored — semantic: suspended/lost)
"#4467A3" # 3 — blue (fitted line)
)
# Data: turbine blade fatigue-life data (hours to failure)
# Simulate Weibull(shape=2.3, scale=8000) with some right-censored observations
n_total <- 40
true_shape <- 2.3
true_scale <- 8000
# Generate failure times
all_times <- true_scale * (-log(runif(n_total)))^(1 / true_shape)
# Right-censor at 10000 hours
censor_time <- 10000
is_censored <- all_times > censor_time
time_to_failure <- pmin(all_times, censor_time)
# Sort by time for proper rank ordering
ord <- order(time_to_failure)
time_to_failure <- time_to_failure[ord]
is_censored <- is_censored[ord]
# Count actual failures for median rank calculation
n_fail <- sum(!is_censored)
failure_idx <- which(!is_censored)
# Median rank (Benard's approximation): (i - 0.3) / (n + 0.4)
# Only assign plotting positions to failures; censored are plotted separately
median_ranks <- (seq_along(failure_idx) - 0.3) / (n_total + 0.4)
# Weibull linearized y-axis: y = ln(-ln(1 - F))
weibull_y <- log(-log(1 - median_ranks))
# Data frame for failure points
df_fail <- data.frame(
time = time_to_failure[failure_idx],
wb_y = weibull_y,
type = "Failure"
)
# For censored points: use the last assigned median rank as a ceiling (right-censored)
# Place censored obs at a y that reflects "at least survived this long"
# — just plot them at the axis mid-level annotated as censored; no formal rank
censored_times <- time_to_failure[is_censored]
df_cens <- data.frame(
time = censored_times,
wb_y = rep(median(weibull_y), length(censored_times)),
type = "Censored (suspended)"
)
# Fit line via OLS on linearised data (log(t) ~ wb_y)
log_times <- log(df_fail$time)
fit <- lm(weibull_y ~ log_times, data = df_fail)
beta_hat <- coef(fit)[2] # slope = shape parameter
eta_hat <- exp(-coef(fit)[1] / coef(fit)[2]) # scale parameter (characteristic life)
# Fitted line over the data range
x_seq <- seq(log(min(df_fail$time) * 0.7), log(max(df_fail$time) * 1.3), length.out = 200)
y_fit_seq <- coef(fit)[1] + coef(fit)[2] * x_seq
df_line <- data.frame(time = exp(x_seq), wb_y = y_fit_seq)
# 63.2% reference line: wb_y at F=0.632 = ln(-ln(1-0.632)) ≈ 0
ref_y <- log(-log(1 - 0.632)) # ≈ -0.0006 ≈ 0
ref_63 <- data.frame(
x_start = min(df_line$time),
x_end = exp(-coef(fit)[1] / coef(fit)[2]), # = eta_hat
y_val = ref_y
)
# Title
plot_title <- "probability-weibull · r · ggplot2 · anyplot.ai"
title_len <- nchar(plot_title)
base_title_size <- 12
title_size <- max(8, round(base_title_size * 67 / title_len))
# Custom Weibull probability y-axis breaks and labels
# Convert F values to wb_y = ln(-ln(1-F))
f_breaks <- c(0.01, 0.05, 0.10, 0.20, 0.30, 0.50, 0.632, 0.80, 0.90, 0.95, 0.99)
wb_breaks <- log(-log(1 - f_breaks))
f_labels <- paste0(formatC(f_breaks * 100, format = "g", digits = 3), "%")
# Plot
p <- ggplot() +
# Fitted Weibull line
geom_line(
data = df_line,
aes(x = time, y = wb_y),
color = IMPRINT_PALETTE[3],
linewidth = 1.0,
linetype = "solid"
) +
# 63.2% horizontal reference line
geom_segment(
aes(x = ref_63$x_start, xend = ref_63$x_end,
y = ref_63$y_val, yend = ref_63$y_val),
color = INK_MUTED,
linewidth = 0.5,
linetype = "dashed"
) +
# Vertical drop to x-axis at eta
geom_segment(
aes(x = ref_63$x_end, xend = ref_63$x_end,
y = ref_63$y_val, yend = min(wb_breaks) - 0.3),
color = INK_MUTED,
linewidth = 0.5,
linetype = "dashed"
) +
# Failure points (filled)
geom_point(
data = df_fail,
aes(x = time, y = wb_y, shape = "Failure", color = "Failure"),
size = 2.8,
alpha = 0.85,
fill = IMPRINT_PALETTE[1]
) +
# Censored points (hollow)
geom_point(
data = df_cens,
aes(x = time, y = wb_y, shape = "Censored (suspended)", color = "Censored (suspended)"),
size = 2.8,
alpha = 0.85,
stroke = 1.2
) +
# Parameter annotation
annotate(
"label",
x = min(df_fail$time) * 1.1,
y = max(wb_breaks) - 0.2,
label = sprintf("β (shape) = %.2f\nη (scale) = %.0f h", beta_hat, eta_hat),
hjust = 0,
vjust = 1,
size = 3.5,
color = INK_SOFT,
fill = ELEVATED_BG,
label.size = 0.3
) +
# 63.2% label
annotate(
"text",
x = ref_63$x_start * 1.05,
y = ref_63$y_val + 0.08,
label = "63.2%",
hjust = 0,
size = 2.5,
color = INK_MUTED
) +
scale_x_log10(
labels = label_comma(),
name = "Time to Failure (hours)"
) +
scale_y_continuous(
breaks = wb_breaks,
labels = f_labels,
limits = c(min(wb_breaks) - 0.3, max(wb_breaks) + 0.1),
name = "Cumulative Failure Probability"
) +
scale_color_manual(
name = NULL,
values = c("Failure" = IMPRINT_PALETTE[1], "Censored (suspended)" = IMPRINT_PALETTE[2])
) +
scale_shape_manual(
name = NULL,
values = c("Failure" = 19, "Censored (suspended)" = 1)
) +
labs(title = plot_title) +
theme_minimal(base_size = 8) +
theme(
plot.background = element_rect(fill = PAGE_BG, color = PAGE_BG),
panel.background = element_rect(fill = PAGE_BG, color = NA),
panel.grid.major = element_line(color = INK_SOFT, linewidth = 0.15),
panel.grid.minor = element_blank(),
panel.border = element_rect(color = INK_SOFT, fill = NA, linewidth = 0.4),
axis.title = element_text(color = INK, size = 10),
axis.text = element_text(color = INK_SOFT, size = 8),
axis.text.x = element_text(color = INK_SOFT, size = 8),
plot.title = element_text(color = INK, size = title_size, face = "bold",
margin = margin(b = 8)),
legend.background = element_rect(fill = ELEVATED_BG, color = INK_SOFT, linewidth = 0.3),
legend.text = element_text(color = INK_SOFT, size = 8),
legend.title = element_text(color = INK, size = 9),
legend.position = "bottom",
legend.key = element_rect(fill = NA, color = NA),
plot.margin = margin(12, 16, 12, 12)
)
# Save
ggsave(
filename = sprintf("plot-%s.png", THEME),
plot = p,
device = ragg::agg_png,
width = 8,
height = 4.5,
units = "in",
dpi = 400
)
Runnable source as JSON, for any HTTP client: https://api.anyplot.ai/specs/probability-weibull/ggplot2/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": "r",
"library": "ggplot2",
"page": "https://anyplot.ai/probability-weibull/r/ggplot2",
"hub": "https://anyplot.ai/probability-weibull",
"code_json": "https://api.anyplot.ai/specs/probability-weibull/ggplot2/code",
"spec_json": "https://api.anyplot.ai/specs/probability-weibull",
"render_light_png": "https://storage.googleapis.com/anyplot-images/plots/probability-weibull/r/ggplot2/plot-light.png",
"render_dark_png": "https://storage.googleapis.com/anyplot-images/plots/probability-weibull/r/ggplot2/plot-dark.png",
"quality_score": 90.0,
"license": "MIT",
"guide": "https://anyplot.ai/llms.txt"
}Part of Weibull Probability Plot for Reliability Analysis on anyplot.ai.