Normal distribution questions follow a pattern: standardise, find the probability from the tables, and sometimes work backwards from a probability to find a value, μ or σ. The reports show that the method is well known. The marks go missing in the choice of probability area, in the working and in the interpretation. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- The wrong probability area. Described as the most common error in several reports, and one report says a simple sketch of the Normal curve could have avoided it. Examples include linking 36 minutes with the wrong probability, and assuming that the area is symmetrical.
- Standardisation not shown. Partially evaluating or leaving out the standardisation formula, and not showing the numerical calculation for the probability area. Reports say that the formula with the values substituted must be seen.
- Working backwards from a probability.
- Equating the standardisation formula to a probability (0.5398 was the most common value seen), or using the tables the wrong way round.
- Using a positive z-value when the value must be below the mean, and then, on finding a negative standard deviation, ignoring the minus sign. One report says a final answer greater than the mean (47.1) should have shown the error.
- Using 0.8 as a probability and working with its associated z-value (0.842), or thinking P(Z > 0.8) is the same as P(Z < 0.2).
- Not knowing how to eliminate μ from the standardisation expression.
- Interpreting the percentage. Working out the minimum time for the wrong proportion of days (for example "less than 90 per cent"), and failing to interpret the success criteria, such as 1 − 0.95 where 1 − 0.94 was right.
- Two probabilities. Finding the sum when the product was required (or a difference when an earlier error made the sum greater than 1).
- Continuity correction. Using a continuity correction when none was needed (with a Normal distribution for a continuous variable), or not using it when a discrete variable was approximated.
- Premature rounding. Approximating the z-value, or the probability for one item, before using it again. Reports say to work to at least 4 significant figures until the end. In another session, students who did not use the tables truncated rather than rounded.
- Variance and standard deviation. Confusing them in the standardisation formula.
- Answering in context. Giving only the probability when the question asked for an expected number, rounding or truncating instead of interpreting the result in context, or not stating a single integer value as a conclusion when this is appropriate.
- Misreading values. 177 read as 117.
What examiners advise
- Sketch a Normal curve and shade the area you need, as the reports suggest, before using the tables.
- Show the standardisation formula with values substituted, then the probability.
- When working backwards, the sign of the z-value follows from the side of the mean. Check that the answer lies on the correct side of the mean.
- Use Φ carefully for two areas. One report records Φ(0.333) − (1 − Φ(1.333)) as the common method, but with confusion over signs.
- Do not round the z-value early.
- Give the final answer in the form the question requires: a 3 significant figure value for a non-exact answer, an expected number, or a whole number.
Habits that help
- Before you use the tables, write the probability you want in the form P(Z < z) or P(Z > z) and say in words what you are finding.
- When μ and σ are both unknown, write two standardisation equations and solve them as a pair.
- Check that your final probability is between 0 and 1.
Questions students ask
Why do I find the wrong probability in Normal distribution questions?
The wrong probability area is described as the most common error in several reports. Sketch a Normal curve and shade the area you need, then write the probability as P(Z < z) or P(Z > z) and say in words what you are finding.
Do I need to show the standardisation formula?
Yes, with the values substituted, followed by the probability. Partially evaluating or leaving it out lost marks.
How do I work backwards from a probability to find a value, μ or σ?
Use the tables in reverse and think about which side of the mean the value is on, because that gives the sign of z. A negative standard deviation or an answer on the wrong side of the mean signals an error. When μ and σ are both unknown, write two standardisation equations and solve them as a pair.
Should I round the z-value early?
No. Work to at least 4 significant figures until the end. Premature rounding of the z-value or an intermediate probability is a repeated problem.
When do I use a continuity correction?
When a discrete variable is approximated by a Normal distribution. Students used a continuity correction when none was needed for a continuous variable, or did not use it when a discrete one was approximated.
What if the question asks for an expected number?
Give the expected number, not just the probability, and state a single integer value when the situation requires one.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.