Revise The normal distribution for IB Mathematics: Analysis and Approaches (4.6)

The normal distribution models a continuous random variable. Its probability density function creates a smooth, symmetric, bell-shaped curve (you never need the formula itself). The whole curve — its position and its width — is fixed by just two parameters: the mean $\mu$ and the standard deviation $\sigma$.

The mean $\mu$ sets the location of the centre: change $\mu$ and the whole curve slides left or right. The standard deviation $\sigma$ sets the spread: a small $\sigma$ clusters the data tightly, giving a tall, narrow curve, while a large $\sigma$ spreads it out into a short, wide curve. Two normal curves with the same $\mu$ but different $\sigma$ have the same centre but different widths.

We write $X \sim N(\mu, \sigma^2)$ to say that $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$.

A probability for a normal variable is the area under the curve over an interval. We never integrate this by hand — we use the normal cumulative distribution function (normal CDF) on the GDC, entering a lower bound, an upper bound, the mean $\mu$ and the standard deviation $\sigma$. For a lower tail $P(X<a)$ use a very small lower bound such as $-10^{99}$; for an upper tail $P(X>a)$ use a very large upper bound such as $10^{99}$. Always sketch and shade the region first — it stops you entering the wrong tail.

The standard normal distribution is the special case with mean $0$ and standard deviation $1$, written $Z \sim N(0, 1^2)$. Any normal variable $X\sim N(\mu,\sigma^2)$ can be converted to $Z$ by standardising — replacing the value $x$ by its z-score, the number of standard deviations it lies from the mean.

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

Often we know a probability and need the value of the variable that produces it — for example the score needed to be in the top $10\%$. This is the reverse process, done with the inverse normal function, which requires the area to the LEFT of the unknown value together with $\mu$ and $\sigma$. If a question gives an area to the right (a 'top' percentage), subtract from $1$ first.

For any normal distribution the areas at whole numbers of standard deviations from the mean are fixed. About $68\%$ of the data lies within $1\sigma$ of the mean, about $95\%$ within $2\sigma$, and about $99.7\%$ within $3\sigma$. Because the curve is symmetric, these split evenly across the two tails — for instance, roughly $2.5\%$ of the data lies beyond $2\sigma$ above the mean.

On Paper 2 the marks are analytic: each is tied to a specific line — a method mark (M) for the correct approach or an accuracy mark (A) for the correctly-rounded value, with an A-mark dependent on the M-mark it follows. Follow-through (FT) means a wrong number early on need not cost the marks after it, and 'ignore subsequent working' (ISW) means a correct answer is not un-awarded by clumsy extra steps. Equivalent correct methods — a direct GDC probability or a z-score route — earn the same marks. Study how each mark below is earned by a specific line.