The Normal Distribution
The normal distribution is the famous bell curve — the smooth,
symmetric hump that so many real-world measurements follow: heights, test scores, measurement
errors, and much more. It is the single most important distribution in statistics.
A normal distribution is pinned down by just two numbers: its mean
\mu and its
standard deviation
\sigma. We write
X \sim N(\mu, \sigma^2)
to say that X is normal with mean \mu and
variance \sigma^2 (so standard deviation
\sigma).
Centre and width
The two numbers play two very different roles:
-
\mu sets the centre — slide it and the whole bell
moves left or right without changing shape.
-
\sigma sets the width — a small
\sigma gives a tall, narrow bell; a large
\sigma a low, broad one.
The curve is perfectly symmetric about \mu, so for a
normal distribution the mean, median, and mode all coincide — they sit together
right at the peak.
The density formula
The height of the bell at each value x is given by its density:
f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).
You don't need to memorise this to use it — just notice that the value
x only ever appears as (x-\mu)^2, the squared
distance from the centre. That squared distance is why the curve falls away smoothly and
symmetrically on both sides of \mu. As always, the total area under the
curve is 1, so a narrower bell must rise taller to keep its area fixed.
(A deeper, measure-theoretic treatment of the same curve — standardisation, the Gaussian integral,
and more — lives at
the normal distribution in finance.)
Shape the bell
Slide \mu to move the curve, and
\sigma to make it wider or narrower. The faint curve
is the fixed standard normal N(0,1) for comparison. Watch how a narrower
bell grows taller — the area underneath stays 1.
- The normal N(\mu,\sigma^2) is the symmetric bell curve set by its mean \mu and standard deviation \sigma.
- \mu moves the centre; \sigma sets the width.
- It is symmetric about \mu, so mean = median = mode, all at the peak.
- Density: f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^2/(2\sigma^2)}; total area 1, so a narrower bell is taller.