R Graphics · QQ Plot

Assessing Distributional Fit with QQ Plots in R

Diagnose normality, heavy tails, skewness, group differences, alternative distributions, and simulated envelopes.

6 examplesReproducible R code

Before you begin

Choose the variation that serves the analysis

Quantile–quantile plots compare ordered sample values with theoretical or observed quantiles. Shape matters more than isolated points: systematic curvature identifies departures from the reference distribution.

Examples: 06
Levels: Beginner + Intermediate

Example 01Base RBeginner

Create a Normal QQ Plot

A normal QQ plot compares sample quantiles with theoretical normal quantiles.

Points near the reference line indicate an approximately normal shape, subject to ordinary sampling variation.

RRun this code

x <- rnorm(150)
qqnorm(x, pch = 19, col = "#2563eb")
qqline(x, col = "#dc2626", lwd = 2)

Example 02Base RBeginner

Recognize Heavy Tails

Heavy-tailed data depart from the line in opposite directions at both extremes.

The S-shaped pattern indicates more extreme observations than the normal reference predicts.

RRun this code

x <- rt(180, df = 3)
qqnorm(x, pch = 19, col = adjustcolor("#2563eb",.6))
qqline(x, col = "#dc2626", lwd = 2)

Example 03Base RBeginner

Identify Skewness

Skewed distributions create systematic curvature rather than random scatter around the line.

A long upper tail typically bends the largest sample quantiles above the normal reference.

RRun this code

x <- rexp(180)
qqnorm(x, pch = 19, col = "#2563eb")
qqline(x, col = "#dc2626", lwd = 2)

Example 04Base RIntermediate

Compare Two Samples

A two-sample QQ plot compares the quantiles of two observed distributions.

qqplot() reveals whether samples differ mainly in location, scale, or overall shape.

RRun this code

a <- rnorm(150, 0, 1); b <- rnorm(150, 1, 1.4)
qqplot(a, b, pch = 19, col = "#2563eb",
       xlab = "Sample A quantiles", ylab = "Sample B quantiles")
abline(0, 1, col = "#dc2626", lwd = 2)

Example 05Base RIntermediate

Use an Exponential Reference

QQ plots can assess distributions other than the normal.

Supply theoretical quantiles from the chosen distribution and compare them with sorted sample values.

RRun this code

x <- rexp(150, rate = .5)
p <- ppoints(length(x))
plot(qexp(p, rate = .5), sort(x), pch = 19, col = "#2563eb",
     xlab = "Theoretical exponential quantiles", ylab = "Observed quantiles")
abline(0, 1, col = "#dc2626", lwd = 2)

Example 06Base RIntermediate

Add a Simulation Envelope

A simulation envelope shows the range of QQ patterns expected from repeated reference samples.

The band helps distinguish meaningful departures from ordinary sample-to-sample variation.

RRun this code

set.seed(42); x <- rnorm(100); p <- ppoints(100); theo <- qnorm(p)
sims <- replicate(200, sort(rnorm(100)))
plot(theo, sort(x), pch = 19, col = "#2563eb")
lines(theo, apply(sims,1,quantile,.025), col = "#94a3b8")
lines(theo, apply(sims,1,quantile,.975), col = "#94a3b8")