Question 10
A student is trying to determine which subject they performed best in compared to other students. Results from recent tests in four subjects (A to D) are shown. Assume student results in each subject are normally distributed.
In which subject did the student perform best compared to other students?
| Class mean | Class standard deviation | Student’s result | |
|---|---|---|---|
| (A) | 62 | 22 | 77 |
| (B) | 55 | 25 | 74 |
| (C) | 61 | 15 | 70 |
| (D) | 73 | 20 | 82 |
Not quite. Subject A gives $z=\frac{77-62}{22}\approx0.68$. That is above average, but subject B’s $z=0.76$ is higher. In 2023, 5.9% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2023, 67.6% of Queensland students chose (B).
Not quite. Subject C has the smallest spread, and $\frac{70}{15}\approx4.67$ is the biggest result divided by the standard deviation. But a $z$-score measures from the mean: $z=\frac{70-61}{15}=0.6$. In 2023, 16.1% of Queensland students chose (C).
Not quite. 82 is the highest mark, but the class mean was highest too. It is only $z=\frac{82-73}{20}=0.45$ standard deviations above the mean, the lowest of the four. In 2023, 9.6% of Queensland students chose (D).
Subject has , less than subject B’s $z=0.76$.
Subject B has the biggest $z$-score, 0.76. Its result is further above the class mean, measured in standard deviations, than in any other subject.
Each class has its own bell curve, so raw marks can’t be compared directly. Switch to $z$-scores to put all four on one scale.
On the $z$-score scale all four classes share one curve. The result furthest to the right is the best compared to other students.
Each subject has a different mean and spread, so the raw marks can’t be compared. Turn each result into a $z$-score: the number of standard deviations it sits above the mean.
$$z=\frac{x-\mu}{\sigma}$$
Standardise each result.
$$\text{(A)}\quad z=\frac{77-62}{22}\approx0.68$$
$$\text{(B)}\quad z=\frac{74-55}{25}=0.76$$
$$\text{(C)}\quad z=\frac{70-61}{15}=0.6$$
$$\text{(D)}\quad z=\frac{82-73}{20}=0.45$$
The biggest $z$-score is 0.76, in subject B. Compared to the other students, that is the student’s best result, even though 82 in subject D is a higher mark.
$\displaystyle z_A=\frac{77-62}{22}\approx0.68$$\displaystyle z_B=\frac{74-55}{25}=0.76$
$\displaystyle z_C=\frac{70-61}{15}=0.6$$\displaystyle z_D=\frac{82-73}{20}=0.45$
Subject B has the largest $z$-score, so the student performed best there compared to other students.
How Queensland students answered in 2023
From the QCAA Mathematical Methods subject report for 2023. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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