A diagnostic plot comparing the cumulative distribution function (CDF) of observed data against a theoretical distribution by plotting empirical CDF values against theoretical CDF values. Unlike Q-Q plots which compare quantiles, P-P plots compare cumulative probabilities on both axes (0 to 1), making them more sensitive to deviations in the center of the distribution. Points falling along the 45-degree diagonal indicate a good fit.
# anyplot.ai
# pp-basic: Probability-Probability (P-P) Plot
# Library: makie 0.22.10 | Julia 1.11.9
# Quality: 87/100 | Created: 2026-06-09
using CairoMakie
using ColorSchemes
using Colors
using Random
using Statistics
Random.seed!(42)
# Theme tokens
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const ANYPLOT_SEQ = cgrad([colorant"#009E73", colorant"#4467A3"])
# Data — process measurements (coating thickness, micrometers) with a right-skewed tail
# 85% from a normal base distribution, 15% with a positive boost → slightly right-skewed
n = 200
base_measurements = 50.0 .+ 8.0 .* randn(n)
boost_flags = Float64.(rand(n) .< 0.15)
boost_amounts = 20.0 .+ 5.0 .* abs.(randn(n))
coating_thickness = base_measurements .+ boost_flags .* boost_amounts
# Standardise to zero mean, unit variance for P-P comparison against N(0,1)
μ_fit = mean(coating_thickness)
σ_fit = std(coating_thickness)
z_scores = (coating_thickness .- μ_fit) ./ σ_fit
# Sort and compute empirical CDF (Hazen plotting positions: (i − 0.5) / n)
sorted_z = sort(z_scores)
empirical_cdf = [(i - 0.5) / n for i in 1:n]
# Theoretical normal CDF — Abramowitz & Stegun (1964) rational approximation 26.2.17
# Maximum absolute error ≤ 7.5e-8; no external packages required.
abs_z = abs.(sorted_z)
t_coeff = 1.0 ./ (1.0 .+ 0.2316419 .* abs_z)
poly_val = t_coeff .* (0.319381530 .+ t_coeff .* (
-0.356563782 .+ t_coeff .* (
1.781477937 .+ t_coeff .* (
-1.821255978 .+ t_coeff .* 1.330274429))))
p_upper = 1.0 .- (1.0 / sqrt(2π)) .* exp.(-abs_z .^ 2 ./ 2.0) .* poly_val
theoretical_cdf = ifelse.(sorted_z .>= 0.0, p_upper, 1.0 .- p_upper)
# Deviation from reference diagonal — key insight: S-curve departure from normality
dev = abs.(empirical_cdf .- theoretical_cdf)
max_dev = maximum(dev)
# 95% KS confidence band half-width (D_α = 1.36 / √n)
delta = 1.36 / sqrt(n)
xs_vec = collect(range(0.0, 1.0, length=200))
# Figure — square canvas (P-P plots use equal 0–1 probability axes)
fig = Figure(
size = (1200, 1200),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
ax = Axis(
fig[1, 1];
title = "pp-basic · julia · makie · anyplot.ai",
titlesize = 28,
titlecolor = INK,
xlabel = "Theoretical Cumulative Probability",
ylabel = "Empirical Cumulative Probability",
xlabelsize = 20,
ylabelsize = 20,
xticklabelsize = 16,
yticklabelsize = 16,
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 = RGBAf(INK.r, INK.g, INK.b, 0.12),
ygridcolor = RGBAf(INK.r, INK.g, INK.b, 0.12),
xminorgridvisible = false,
yminorgridvisible = false,
aspect = AxisAspect(1),
limits = (0.0, 1.0, 0.0, 1.0),
)
# 95% KS confidence envelope — defines acceptance region around the diagonal
band!(ax, xs_vec, clamp.(xs_vec .- delta, 0.0, 1.0), clamp.(xs_vec .+ delta, 0.0, 1.0);
color = RGBAf(INK.r, INK.g, INK.b, 0.07),
)
# Reference diagonal — perfect distributional fit
lines!(ax, [0.0, 1.0], [0.0, 1.0];
color = INK_SOFT,
linewidth = 1.5,
linestyle = :dash,
)
# P-P scatter — points colored by absolute deviation from diagonal to highlight S-curve
scat = scatter!(ax, theoretical_cdf, empirical_cdf;
color = dev,
colormap = ANYPLOT_SEQ,
colorrange = (0.0, max_dev),
markersize = 11,
strokewidth = 0.5,
strokecolor = PAGE_BG,
)
# Colorbar showing deviation magnitude from reference diagonal
Colorbar(fig[1, 2], scat;
label = "Deviation from Reference",
labelcolor = INK,
ticklabelcolor = INK_SOFT,
tickcolor = INK_SOFT,
labelsize = 16,
ticklabelsize = 14,
)
# Save
save("plot-$(THEME).png", fig; px_per_unit = 2)
Runnable source as JSON, for any HTTP client: https://api.anyplot.ai/specs/pp-basic/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": "pp-basic",
"language": "julia",
"library": "makie",
"page": "https://anyplot.ai/pp-basic/julia/makie",
"hub": "https://anyplot.ai/pp-basic",
"code_json": "https://api.anyplot.ai/specs/pp-basic/makie/code",
"spec_json": "https://api.anyplot.ai/specs/pp-basic",
"render_light_png": "https://storage.googleapis.com/anyplot-images/plots/pp-basic/julia/makie/plot-light.png",
"render_dark_png": "https://storage.googleapis.com/anyplot-images/plots/pp-basic/julia/makie/plot-dark.png",
"quality_score": 87.0,
"license": "MIT",
"guide": "https://anyplot.ai/llms.txt"
}Part of Probability-Probability (P-P) Plot on anyplot.ai.