Revise The normal distribution for IB Mathematics: Applications and Interpretation (4.7)

A random variable $X$ is normally distributed when its values follow a symmetric bell curve. The distribution is fixed by exactly two numbers: the mean $\mu$, which locates the centre, and the standard deviation $\sigma$, which sets the width. We record this as $X\sim N(\mu,\sigma^2)$. Read the notation carefully — the second entry is the variance $\sigma^2$, but every calculation uses the standard deviation $\sigma$, so if a question quotes a variance you must square-root it before touching the GDC.

Because $X$ is continuous, the probability of any single exact value is zero: probability is area, and a single point has no width. A useful consequence is that $P(X<a)$ and $P(X\le a)$ are identical — you never have to worry about whether an endpoint is included.

To find the probability that $X$ falls in an interval, you find the area under the bell over that interval. Doing this by hand needs advanced integration, so on Paper 2 you use the GDC's normal cumulative distribution function. You supply a lower bound, an upper bound, the mean and the standard deviation, and it returns the area between the bounds.

To find $P(a<X<b)$, use: \n normalCdf(lower: a, upper: b, μ: mean, σ: std dev)

Sometimes you are handed the probability — a percentage, a percentile, a 'top 10%' — and asked for the boundary value. This runs the previous idea backwards, so you use the inverse normal function. It takes an area to the LEFT of the unknown value, together with $\mu$ and $\sigma$, and returns the value $x$ for which $P(X<x)$ equals that area.

To find the value $x$ such that $P(X<x)=p$, use: \n invNorm(area: p, μ: mean, σ: std dev)

The standardised $z$-score rewrites a value as the number of standard deviations it lies from the mean. If $X\sim N(\mu,\sigma^2)$ then a value $x$ has $z$-score $z=\dfrac{x-\mu}{\sigma}$. A positive $z$ sits above the mean, a negative $z$ below, and $z=0$ is exactly at the mean. Standardising converts any normal variable into the standard normal $Z\sim N(0,1^2)$, which is what lets you compare values that come from different distributions.

Standardised score: $z = \dfrac{x-\mu}{\sigma}$, so that $X\sim N(\mu,\sigma^2)$ becomes $Z\sim N(0,1^2)$.

For example, a student scores 70 in a test with mean 60 and standard deviation 8, giving $z=\frac{70-60}{8}=1.25$; a friend scores 65 in a different test with mean 50 and standard deviation 10, giving $z=\frac{65-50}{10}=1.5$. Although the raw score of 70 is higher, the friend's result is the more exceptional relative to its own distribution, because its $z$-score is larger. On Paper 2 you can usually feed the original $\mu$ and $\sigma$ straight into the GDC, but the $z$-score is the tool for these comparison questions and for reasoning about how unusual a value is.

Every normal distribution shares the same proportions once you measure distance in standard deviations. This is the empirical rule, and it gives fast estimates and a way to check GDC answers without a calculator.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an earlier wrong value need not cost you the marks that depend on it, provided the later step is carried out correctly on your figure. The engine also ignores subsequent working (ISW) once a correct answer appears, and accepts any equivalent form and any correctly-rounded value. But that protection only exists if the method is on the page. Study how each mark below is earned by a specific line — for the normal distribution, that means quoting the GDC function you used before the number it produced.

The normal distribution is where the descriptive statistics of earlier in the course become a working model: the mean you learned to read off a GDC is now $\mu$, the centre of the bell, and the standard deviation is now $\sigma$, the dial that sets its width. Percentiles return as inverse-normal values, and the $z$-score you meet here is the same standardising idea that underpins confidence intervals and hypothesis tests in the wider statistics course. Master the two-dial picture — a centre and a spread — together with the discipline of sketching the region, quoting the GDC function and checking against the 68–95–99.7 rule, and every normal-distribution question on Paper 2 becomes a variation on a method you already know how to write.