Linear combination questions ask for the mean and variance of something like X − 3Y, or of a sum of several independent variables, and then a Normal probability. The reports show the same two weak points: the variance, and the choice of probability area. 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
- Not squaring the coefficient. The most common error in the reports. Using 3 instead of 3² in Var(X − 3Y), omitting to square 1.5 or m, not multiplying by 2², or giving 3.6 (0.4 × 9) rather than 1.44 (0.4² × 9).
- Subtracting variances. For X − Y, the variances are added, not subtracted. This appears in several reports.
- Wrong multiplier on a variance. Using 3.6² + 2 × 3.7² where 3.6² + 4 × 3.7² was needed. The reports describe several attempts at the variance in one question, with many wrong values, often by confusing methods.
- Standard deviation and variance. Mixing them up when combining, so that a standard deviation is used in place of a variance.
- Variance of a total. Calculating 140 × 20 + 80 × 50 for a variance when 140 × 20² + 80 × 50² was needed. Working per day when the question asked for per week.
- Units. Confusing dollars and cents, or errors in converting.
- The wrong probability area. Finding an area greater than 0.5 when it should be less than 0.5 (or the other way round). The reports say a diagram can be helpful, and that candidates should state which probability they are finding, such as P(< 25.5).
- An unrequired continuity correction. Using one when the variable is continuous and the Normal distribution applies directly.
- Using the wrong distribution. Using the distribution given in the question when a different distribution was needed, or omitting √15 when standardising, where one candidate went on to multiply the probability by 15 and obtained a probability greater than 1.
- Not forming the right combination. Weaker responses were often unable to make the correct approach and did not attempt to use X − 3Y.
- Squaring the standard deviations. Many candidates used 0.8 and 0.85 rather than 0.8² and 0.85² in the variance calculation.
- Wrongly signed z-value and wrong z-value (0.525 in one report).
- Reading carefully. The mean and variance were challenging in one question, with misinterpretation errors. For "write down", no calculation should be needed.
What examiners advise
- Write the combination, such as D = X₁ − 2X₂ + 3, then its mean and variance separately.
- Square the coefficients for the variance, and add the variances for independent variables (also when they are subtracted in the combination).
- State which probability you are finding, and sketch the Normal curve before reading the tables.
- Do not use a continuity correction for a continuous variable.
- Show working fully, and keep to at least 4 significant figures.
Habits that help
- Write the rule Var(aX + bY) = a²Var(X) + b²Var(Y) at the top of the page the first time you need it.
- Put every standard deviation into a variance before combining.
- Work out the mean first, then the variance, and keep them in separate lines.
Questions students ask
Why do I square the coefficient when finding a variance?
Because Var(aX) = a²Var(X). Not squaring the coefficient is the most common error in the reports, for example using 3 instead of 3² in Var(X − 3Y).
Do I subtract variances for X − Y?
No. For independent variables the variances are added, also when the variables are subtracted in the combination.
How do I handle standard deviations in a combination?
Square them to get variances before combining. Many candidates used 0.8 and 0.85 rather than 0.8² and 0.85².
Do I need a continuity correction when I use a Normal distribution here?
Not for a continuous variable. An unrequired continuity correction is a recorded error. State which probability you are finding and sketch the Normal curve first.
How do I set up a question about a total or a difference?
Write the combination first, such as D = X₁ − 2X₂ + 3, then its mean and variance on separate lines. For a total over several items, use 140 × 20² + 80 × 50² style working, not 140 × 20 + 80 × 50.
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.