Stat 312: Lecture 06
Quantile-quantile plots
Moo K. Chung
September 23, 2004
1. In order to compute 100(1 −α)% confidence
interval, it is required to find zα/2that satisfies
P(Z > zα/2) = α/2for given α. We will study
how to find zα/2and more. This lecture is based
on Chapter 4.6.
2. The p-th quantile point qfor random variable X
is the point such that
F(q) = P(X≤q) = p.
The textbook represent it in terms of percentile.
Note that p-th quantile = 100 ×p-th percentile.
So given p,
q=F−1(p).
For X∼N(0,1), it is easy to find the p-th qun-
tile using
> qnorm(1)
[1] Inf
> qnorm(0.5)
[1] 0
> qnorm(0)
[1] -Inf
> qnorm(0.5)
[1] 0
> qnorm(0.95)
[1] 1.644854
> qnorm(0.05)
[1] -1.644854
In order to find zα, we use command
qnorm(1−α).
3. Given nobservations x1,· · · , xn, we order them
from the smallest to the largest and we have
x(1),· · · , x(n). The i-th smallest observation is
defined as the (i−0.5)/n-th sample quantile
point or 100(i−0.5)/n sample percentile point.
> library(Devore6)
> data(xmp01.05)
20 40 60 80
10 20 30 40 50 60 70
q
sq
Figure 1: Plot of ordered data bingePct showing
sample sq-th quantile.
> attach(xmp01.05)
> sq <- sort(bingPct)
4. If bingePct really follows N(42,142), then
the sample quantiles should be resonably close
to the corresponding quantiles of the normal dis-
tribution. The corresponding quantile points for
bingePct can be computed using
> q=qnorm((1:140-0.5)/140,42,14)
We can check how closely the sample quantiles
corresponds to the normal distribution by plotting
the quantile-quantile plot (QQ-plot) of the sam-
ple quantiles vs. the corresponding quantiles of a
normal distribution (Figure 1).
> plot(q,sq)
5. If X∼N(µ, σ2), then
Z=X−µ
σ∼N(0,1).
