pnorm() — Probability below a value

Tells you how much of the group is below a number you pick.

pnorm(q, mean = 0, sd = 1, lower.tail = TRUE)

Key idea

• pnorm gives the probability — the chance, from 0 to 1 — that a normal value lands below a number you pick. (e.g. pnorm(195, 180, 10) = 0.93)
• It is the area to the LEFT under the bell curve, so the answer is always between 0 and 1.
• At the mean — the centre of the curve — the area is exactly half, because the bell shape is symmetric. (e.g. pnorm(180, 180, 10) = 0.5)
• To get the probability ABOVE a value, subtract from 1. (e.g. 1 - 0.93 = 0.07)
• For a range, subtract two pnorm values. (e.g. pnorm(190) - pnorm(170))

Definition

$$F(q)=\Phi \left(\frac{q-\mu }{\sigma }\right)={\int }_{-\infty }^q\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(t-\mu {)}^2}{2{\sigma }^2}}dt$$

$q$

the value you ask about (the cut-off)

$\mu$

the mean — centre of the curve

$\sigma$

the standard deviation — the spread

$\Phi$

the standard normal cumulative function

$F(q)$

the probability of being at or below q

$$F(195)=\Phi \left(\frac{195-180}{10}\right)=\Phi (1.5)=0.9332$$

In plain words

Line up a crowd of kids from short to tall and pick a spot. pnorm counts what share of the kids are standing at or before that spot — the shaded part of the crowd, given as a fraction between 0 and 1.

Where you'd use it

• Exam scores are roughly Normal with mean 75 and sd 8. To find what fraction of students scored below 70, put 70 into pnorm — the shaded area to the left is about 27%, so roughly a quarter of the class fell short.
• A shop’s light bulbs last Normal(1000, 100) hours. The chance a bulb dies before 800 hours is pnorm(800, 1000, 100) ≈ 2%, so almost every bulb lasts longer than that.
• If adult heights are Normal(170, 7) cm, the share shorter than 175 cm is the area left of 175 — about 76%, meaning most people are under 175 cm.

Worked example

A courier’s delivery times follow Normal(μ = 28, σ = 5) minutes. What share of parcels arrive within 35 minutes? Standardize first — z = (35 − 28) / 5 = 1.4 — so the answer is the area left of that: pnorm(35, 28, 5) = Φ(1.4) ≈ 0.92, i.e. about 92% arrive on time. Try it yourself: type 35 into q and watch the shaded area.

Frequently asked questions

What does pnorm actually return?

The probability that a normal value is less than or equal to q — the area to the left under the bell curve.

How do I get the probability above a value?

Use 1 − pnorm(q, …) or pnorm(q, …, lower.tail = FALSE).

I only have a z-score — what now?

Call pnorm(z) with the defaults mean = 0, sd = 1.