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Q-Q Plot

Q-Q Plot (Quantile-Quantile)

Definition

Core Statement

A Q-Q Plot (Quantile-Quantile Plot) is a graphical tool to compare the distribution of a sample against a theoretical distribution (usually the Normal Distribution). It plots sample quantiles against theoretical quantiles. If the data follows the theoretical distribution, points will fall on a straight diagonal line.

Purpose

  1. Assess normality of data or residuals.
  2. Diagnose deviations: skewness, heavy tails, outliers.
  3. Complement statistical tests (e.g., Shapiro-Wilk Test) with visual evidence.

When to Use

Use Q-Q Plot When...

  • Checking normality assumption for t-tests, ANOVA, or regression residuals.
  • Sample size is moderate to large (where histograms are also useful).
  • You want to understand how normality is violated (skew? tails?).

Theoretical Background

Construction

  1. Sort the data.
  2. Calculate sample quantiles (percentiles).
  3. Calculate theoretical quantiles from N(0,1).
  4. Plot pairs: (Theoretical Z, Sample Value).
  5. Add a 45-degree reference line.

Reading the Plot

Pattern Interpretation Example
Straight Line Data is Normal. Normal residuals.
S-Shape Light tails (Uniform). Limited variation.
Curve (Concave) Right Skew. Income data. Use Log Transformations.
Curve (Convex) Left Skew. Ceiling effects.
Tails Depart Upward Heavy right tail (Outliers). Check for extreme values.
Tails Depart Downward Heavy left tail. Negative outliers.

Assumptions

Q-Q plot is a diagnostic tool, not a test; no formal assumptions. However, interpretation assumes data is continuous.

Limitations

Pitfalls

  1. Subjectivity: "Straight enough" is a judgment call, especially for small n.
  2. Small samples: Random variation can look like deviation.
  3. Not a formal test: Use Shapiro-Wilk Test for statistical confirmation.

Python Implementation

import statsmodels.api as sm
import matplotlib.pyplot as plt

# Create Q-Q Plot
sm.qqplot(data, line='45', fit=True)
plt.title("Q-Q Plot")
plt.show()

# For regression residuals
sm.qqplot(model.resid, line='45', fit=True)
plt.title("Q-Q Plot of Residuals")
plt.show()

R Implementation

# Q-Q Plot for a vector
qqnorm(data, main = "Q-Q Plot")
qqline(data, col = "red", lwd = 2)

# For regression residuals
model <- lm(Y ~ X, data = df)
qqnorm(resid(model))
qqline(resid(model), col = "red")

# Or use built-in plot for lm objects
plot(model, 2)  # Standard plot #2 is Q-Q

Interpretation Guide

Visual Interpretation
Points on line Data is approximately normal.
Points curve up at right Right skew; consider log transform.
Points flatten at tails Heavy tails (platykurtic/leptokurtic).
Single point far off Outlier. Investigate.