dnorm() — Height of the curve

Tells you how tall the bell-shaped hill is at one spot.

dnorm(x, mean = 0, sd = 1)

Key idea

• dnorm gives the HEIGHT of the bell curve at a point — called the density — not a probability.
• The height is largest at the mean — the centre — and shrinks as you move away.
• You read density off the up-and-down axis, but these heights are never themselves probabilities.
• The whole area under the curve adds up to 1, even though heights are not areas.
• The point of highest density is always the mean of the distribution. (e.g. peak at x = 180)

Continuous probability distributions — the big idea

1. A continuous variable can take ANY value in a range — height, weight, time, temperature — so, unlike a die, you cannot list its outcomes one by one.
2. Because there are infinitely many possible values, the chance of any single EXACT value (like a height of exactly 180.0000… cm) is 0.
3. So probability is measured as the AREA under the curve over a RANGE — e.g. the chance a height is between 170 and 180 cm. (That area is what pnorm computes.)
4. The curve dnorm draws is a probability DENSITY: its height shows how crowded values are at each point, not a probability — which is why a density can even be above 1.
5. The total area under any continuous distribution is exactly 1, because the value must land somewhere.
6. The normal (bell) curve is the most famous continuous distribution, but the same area-not-height rule holds for all of them (uniform, exponential, and so on).

Definition

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$

$x$

the point on the axis

$\mu$

the mean (centre)

$\sigma$

the standard deviation (spread)

$f(x)$

the curve height (density) at x

$$f(185)=\frac{1}{10\sqrt{2\pi }}{e}^{-\frac{(185-180{)}^2}{2\cdot 10^2}}=0.0352$$

In plain words

How tall the bell-shaped hill is right over the spot x. Taller means values land near there more often. But the height by itself is a density, not a chance — it just shows how crowded values are at that point.

Where you'd use it

• Say test scores are Normal(75, 8) and you want to draw the bell curve. Feeding many x values into dnorm gives the curve’s height at each one, which traces the familiar hill — tallest right at 75 and tailing off toward 50 and 100.
• Heights are Normal(170, 7) cm. To see how common values near 185 cm are, dnorm(185, 170, 7) ≈ 0.006; that height is small, telling you 185 cm sits far out on the thin part of the curve.
• Compare two machines whose part sizes are Normal with sd 2 and sd 5. The first has a taller, narrower dnorm peak and the second a shorter, wider one, so the height shows you at a glance which machine is more consistent.

Frequently asked questions

Is dnorm a probability?

No — it’s the height of the curve (density) at a point. Probabilities are areas (use pnorm), and any exact value has probability 0.

Why is my dnorm value bigger than 1?

Density isn’t capped at 1; with a small SD the curve is tall and narrow. Only the total area equals 1.