qnorm() — Value for a probability

Works backwards from pnorm — you give it a fraction, it gives you the number.

qnorm(p, mean = 0, sd = 1, lower.tail = TRUE)

Key idea

• qnorm is the reverse of pnorm: you give a probability — a chance from 0 to 1 — and get back a value. (e.g. qnorm(0.975) = 1.96)
• The input p is the area to the LEFT, so it must be between 0 and 1.
• It answers questions like the cut-off below which 95 percent of values fall. (e.g. qnorm(0.95) = 1.645)
• Feeding 0.5 returns the mean — the centre — since half the area sits on each side. (e.g. qnorm(0.5, 180, 10) = 180)
• It is the tool for finding percentiles — the value a given share falls below — and cut-off values.

Definition

$$q=F^{-1}(p)=\mu +\sigma {\Phi }^{-1}(p)$$

$p$

a probability between 0 and 1

$\mu$

the mean

$\sigma$

the standard deviation

${\Phi }^{-1}$

the inverse standard normal

$q$

the value with p of the area to its left

$$q=180+10{\Phi }^{-1}(0.975)=180+10(1.96)=199.6$$

In plain words

You know what share of the crowd you want behind the mark, and qnorm tells you where to put the mark. Give it 0.975 — 97.5% behind the line — and it tells you the number that sits there.

Where you'd use it

• Exam scores are Normal(75, 8) and you want the mark that puts a student in the top 10%. That is the 90th percentile, so qnorm(0.90, 75, 8) ≈ 85 — score about 85 to beat 90% of the class.
• Adult heights are Normal(170, 7) cm. To be taller than 90% of people you need the 90th percentile, qnorm(0.90, 170, 7) ≈ 179 cm.
• Parcel weights are Normal(500, 40) g. The cut-off for the heaviest quarter is the 75th percentile, qnorm(0.75, 500, 40) ≈ 527 g, so anything above about 527 g is in the top 25%.

Frequently asked questions

How is qnorm different from pnorm?

They’re inverses: pnorm turns a value into a probability; qnorm turns a probability into the value (a quantile).

Which p gives a 95% confidence interval?

qnorm(0.975) ≈ 1.96 — 0.025 is left in each tail.