The normal distribution

What the normal distribution is

• A distribution describes how likely different values of a variable are.
• The normal distribution is a symmetric, bell-shaped curve that fits many natural measurements — heights, weights, exam marks.
• It is described by two numbers:
  • the mean $\mu$ (Greek "mu") — the centre of the bell, where it peaks.
  • the standard deviation $\sigma$ (Greek "sigma") — the spread; a larger $\sigma$ makes a wider, flatter bell.
• Key properties:
  • It is symmetric about the mean, so $P(X < \mu) = 0.5$ (half the data lies below the mean).
  • The total area under the curve is 1, and an area represents a probability.
  • About 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Standardising with the z-score

• To use the standard tables/calculator you convert any value to a z-score — the number of standard deviations the value is from the mean:

$z = \frac{x - \mu}{\sigma}$

• A positive $z$ is above the mean, a negative $z$ is below. $z = 0$ is exactly at the mean.
• Your calculator (or a standard normal table) turns a z-score into a probability (an area under the curve).

Finding a value from a probability (inverse)

• An inverse problem gives you a probability and asks for the value $x$.
• Work backwards: find the z-score matching that area first, then rearrange to $x = \mu + z\sigma$.