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: plotly 6.8.0 | Python 3.13.13
Quality: 87/100 | Updated: 2026-06-16
"""
import os
import numpy as np
import plotly.graph_objects as go
from scipy import stats
THEME = os.getenv("ANYPLOT_THEME", "light")
# Theme-adaptive chrome
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
GRID = "rgba(26,26,23,0.15)" if THEME == "light" else "rgba(240,239,232,0.15)"
# Imprint palette
BRAND = "#009E73" # brand green — within-range points (first categorical series)
DEVIATION = "#AE3030" # matte red — semantic anchor for deviation / poor fit
NEUTRAL = INK # totals / baseline / reference line (theme-adaptive)
MUTED = INK_MUTED # confidence-band fill (theme-adaptive)
# Data: mixture of normal + exponential simulating process drift
np.random.seed(42)
observed = np.random.normal(loc=50, scale=10, size=200) + np.random.exponential(scale=2, size=200)
observed_sorted = np.sort(observed)
n = len(observed_sorted)
# Fit normal distribution to observed data
mu, sigma = stats.norm.fit(observed_sorted)
# Empirical CDF using plotting position formula i/(n+1)
empirical_cdf = np.arange(1, n + 1) / (n + 1)
# Theoretical CDF from fitted normal
theoretical_cdf = stats.norm.cdf(observed_sorted, loc=mu, scale=sigma)
# Deviation threshold for storytelling — confine flagged points to the genuine
# right tail (theoretical CDF > 0.8), where the added exponential mass makes the
# data depart from the fitted normal. This keeps the red diamonds in the tail
# region the annotation describes, rather than the distribution body.
deviation = np.abs(empirical_cdf - theoretical_cdf)
in_right_tail = theoretical_cdf > 0.8
threshold = np.percentile(deviation, 75)
is_deviant = in_right_tail & (deviation > threshold)
# 95% confidence band (Kolmogorov-Smirnov)
ks_band = 1.36 / np.sqrt(n)
diag = np.linspace(0, 1, 200)
# Plot
fig = go.Figure()
# Confidence band
fig.add_trace(
go.Scatter(
x=np.concatenate([diag, diag[::-1]]),
y=np.concatenate([diag + ks_band, (diag - ks_band)[::-1]]),
fill="toself",
fillcolor="rgba(107,106,99,0.12)" if THEME == "light" else "rgba(168,167,159,0.12)",
line={"width": 0},
name="95% KS confidence band",
hoverinfo="skip",
)
)
# Reference line (perfect fit = structural baseline → neutral)
fig.add_trace(
go.Scatter(
x=[0, 1],
y=[0, 1],
mode="lines",
line={"color": NEUTRAL, "width": 2.5, "dash": "dash"},
name="Perfect normal fit",
hoverinfo="skip",
)
)
# Data points: within expected range
fig.add_trace(
go.Scatter(
x=theoretical_cdf[~is_deviant],
y=empirical_cdf[~is_deviant],
mode="markers",
marker={"size": 6.5, "color": BRAND, "opacity": 0.6, "line": {"width": 0.6, "color": PAGE_BG}},
name="Within expected range",
customdata=np.column_stack([observed_sorted[~is_deviant], deviation[~is_deviant]]),
hovertemplate=(
"Theoretical: %{x:.3f}<br>"
"Empirical: %{y:.3f}<br>"
"Value: %{customdata[0]:.1f}<br>"
"Deviation: %{customdata[1]:.3f}"
"<extra></extra>"
),
)
)
# Data points: tail deviations
fig.add_trace(
go.Scatter(
x=theoretical_cdf[is_deviant],
y=empirical_cdf[is_deviant],
mode="markers",
marker={
"size": 11,
"color": DEVIATION,
"symbol": "diamond",
"opacity": 0.9,
"line": {"width": 0.8, "color": PAGE_BG},
},
name="Tail deviation",
customdata=np.column_stack([observed_sorted[is_deviant], deviation[is_deviant]]),
hovertemplate=(
"Theoretical: %{x:.3f}<br>"
"Empirical: %{y:.3f}<br>"
"Value: %{customdata[0]:.1f}<br>"
"Deviation: %{customdata[1]:.3f}"
"<extra></extra>"
),
)
)
# Annotation pointing to right-tail deviation
tail_idx = np.where(is_deviant & (theoretical_cdf > 0.8))[0]
if len(tail_idx) > 0:
ax_pt = theoretical_cdf[tail_idx[len(tail_idx) // 2]]
ay_pt = empirical_cdf[tail_idx[len(tail_idx) // 2]]
fig.add_annotation(
x=ax_pt,
y=ay_pt,
ax=-55,
ay=-45,
text="Heavier right tail<br>→ process drift detected",
showarrow=True,
arrowhead=2,
arrowsize=1.2,
arrowwidth=1.6,
arrowcolor=DEVIATION,
font={"size": 11, "color": INK_SOFT},
bgcolor=ELEVATED_BG,
bordercolor=DEVIATION,
borderwidth=1.2,
borderpad=5,
)
# Layout
fig.update_layout(
autosize=False,
width=600,
height=600,
margin={"l": 70, "r": 40, "t": 80, "b": 60},
title={
"text": "pp-basic · python · plotly · anyplot.ai",
"font": {"size": 16, "color": INK},
"x": 0.5,
"xanchor": "center",
"y": 0.97,
},
xaxis={
"title": {"text": "Theoretical CDF (Normal)", "font": {"size": 12, "color": INK}},
"tickfont": {"size": 10, "color": INK_SOFT},
"range": [-0.02, 1.02],
"showgrid": False,
"showline": True,
"linecolor": INK_SOFT,
"linewidth": 1.2,
"zeroline": False,
"ticks": "outside",
"tickcolor": INK_SOFT,
"dtick": 0.2,
},
yaxis={
"title": {"text": "Empirical CDF", "font": {"size": 12, "color": INK}},
"tickfont": {"size": 10, "color": INK_SOFT},
"range": [-0.02, 1.02],
"showgrid": False,
"showline": True,
"linecolor": INK_SOFT,
"linewidth": 1.2,
"zeroline": False,
"ticks": "outside",
"tickcolor": INK_SOFT,
"dtick": 0.2,
"scaleanchor": "x",
"scaleratio": 1,
},
legend={
"x": 0.02,
"y": 0.98,
"xanchor": "left",
"yanchor": "top",
"font": {"size": 10, "color": INK_SOFT},
"bgcolor": ELEVATED_BG,
"bordercolor": INK_SOFT,
"borderwidth": 1,
},
paper_bgcolor=PAGE_BG,
plot_bgcolor=PAGE_BG,
font={"color": INK},
)
# Save — square canvas: 600 × 600 × scale 4 = 2400 × 2400
fig.write_image(f"plot-{THEME}.png", width=600, height=600, scale=4)
fig.write_html(f"plot-{THEME}.html", include_plotlyjs="cdn")
Part of Probability-Probability (P-P) Plot on anyplot.ai.