The normal distribution — common mistakes

Always draw a quick sketch of the bell curve and shade the area you are looking for. This helps you visualise the problem, decide whether the probability should be greater or less than 0.5, and correctly apply symmetry rules. It is a simple step that can prevent many common errors.

Continuity corrections are a common source of errors. Remember: for 'inclusive' inequalities ($le$ or $ge$), you expand the range (e.g., $X ge 10$ becomes $Y > 9.5$). For 'exclusive' inequalities ($<$ or $>$), you shrink the range (e.g., $X > 10$ becomes $Y > 10.5$). Visualising the discrete bars and the continuous curve can help.

This is a mathematical convention. Variance has useful additive properties that standard deviation does not. For example, the sum of two independent normal variables has a variance that is the sum of their variances. Using variance in the notation makes these theoretical properties clearer.

The standard normal table in the Cambridge formula book (MF19/MF20) is quite comprehensive. If a value is between two listed values, you can perform linear interpolation to get a more accurate probability. For values larger than those in the main table (e.g., z > 3.5), the probability $P(Z < z)$ is very close to 1.

You must use a continuity correction whenever you are using a continuous distribution (like the normal distribution) to approximate a discrete distribution (like the binomial or Poisson distributions). If the original problem involves a variable that can only take integer values, a continuity correction is required for the approximation.

Yes. A Poisson distribution $Po(\lambda)$ can be approximated by a normal distribution $N(\lambda, \lambda)$ provided that $\lambda$ is large (typically $\lambda > 15$). Just like with the binomial approximation, you must use a continuity correction.