Mathematics Standard • Year 12 • Module 8 • Lesson 7
Z-Scores
Practise HSC Mathematics Standard 2-style writing on z-scores, multi-mark short answers and one structured extended response.
1. Short-answer questions
1.1 A Year 12 Maths test has a mean of 65 and a standard deviation of 12. Jamie scored 83.
(a) Calculate Jamie's z-score.
(b) Interpret the z-score in one sentence in context. 3 marks Band 3
1.2 A driving theory test has mean 72 and SD 9. A pass requires a z-score of at least −0.5.
(a) Calculate the minimum raw mark needed to pass.
(b) A candidate scores 64. Calculate their z-score and state whether they pass. 3 marks Band 3-4
1.3 Three students sit different HSC trial subjects. Their results are:
Ana, Maths: 78 (class mean 70, SD 8).
Ben, Science: 85 (class mean 80, SD 5).
Carla, English: 82 (class mean 72, SD 10).
(a) Calculate the z-score for each student.
(b) The top 10% of standardised performance approximately corresponds to z > 1.28. State which students (if any) qualify, and write one sentence justifying your decision. 4 marks Band 4
2. Extended response
2.1 A NSW university uses z-scores from HSC subjects to decide a scholarship. The cutoff is z ≥ 1.5 within each school.
School P (selective): mean HSC mark = 86, SD = 5. Applicant Pia scored 94.
School Q (comprehensive): mean HSC mark = 70, SD = 12. Applicant Quan scored 92.
(a) Calculate Pia's z-score and state whether she meets the cutoff.
(b) Calculate Quan's z-score and state whether he meets the cutoff.
(c) Quan has a lower raw mark than Pia but a higher z-score. In a clear paragraph (4–6 sentences) explain in plain English why this is mathematically reasonable, and state which student the university would award the scholarship to (with justification). 7 marks Band 5-6
Explicit marking criteria
Part (a), 2 marks
• 1 mark correct substitution and value for Pia's z.
• 1 mark explicit statement of whether she meets z ≥ 1.5.
Part (b), 2 marks
• 1 mark correct substitution and value for Quan's z.
• 1 mark explicit statement of whether he meets z ≥ 1.5.
Part (c), 3 marks
• 1 mark refers to the different means and SDs across the two schools.
• 1 mark explains that z measures standing within school, not absolute mark.
• 1 mark explicit, justified award decision naming the winning student.
Your response:
Stuck on (c)? Write: "Even though Quan's raw mark of 92 is lower than Pia's 94, his z-score is higher because at School Q the typical student scored only 70 with a wider spread (SD 12), while at School P the typical student already scored 86 with little spread (SD 5). Quan therefore stood out further from his peers than Pia did from hers, so the scholarship, based on z, should go to Quan."How did this worksheet feel?
What I'll revisit before next class:
1.1, Jamie's z-score (3 marks)
Sample response.
(a) z = (83 − 65) ÷ 12 = 18 ÷ 12 = 1.50.
(b) Jamie's mark is 1.5 standard deviations above the class mean, indicating clearly above-average performance.
Marking notes. 1 mark, correct substitution. 1 mark, correct value of z. 1 mark, interpretation referencing "standard deviations above the mean" in context. A bare "z = 1.5" with no interpretation scores 2/3.
1.2, Pass mark and candidate check (3 marks)
Sample response.
(a) x = mean + z × SD = 72 + (−0.5)(9) = 72 − 4.5 = 67.5. The minimum pass mark is 67.5.
(b) Candidate: z = (64 − 72) ÷ 9 = −8 ÷ 9 ≈ −0.89. Since −0.89 < −0.5, the candidate does not pass.
Marking notes. 1 mark, correct minimum pass mark. 1 mark, correct z calculation for the candidate. 1 mark, explicit pass/fail conclusion comparing −0.89 with −0.5.
1.3, Three students vs z = 1.28 cutoff (4 marks)
(a) Sample response. Ana: z = (78 − 70)/8 = 8/8 = 1.00. Ben: z = (85 − 80)/5 = 5/5 = 1.00. Carla: z = (82 − 72)/10 = 10/10 = 1.00.
(b) Sample response. All three students have z = 1.00, which is less than 1.28. None of the three students qualify for the top-10% threshold, because their standardised performances are all the same (one standard deviation above the mean) and the cutoff requires a slightly higher z.
Marking notes. (a) 1 mark per correct z (3 marks total, capped). (b) 1 mark, explicit decision naming all three and citing z = 1.00 < 1.28. Common error: declaring one student a winner because their raw mark is highest, z-scores are tied, so all three are equal.
2.1, Pia vs Quan scholarship (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Pia (School P).
zPia = (94 − 86) ÷ 5 = 8 ÷ 5 = 1.60. [1 mark, substitution and value.]
1.60 ≥ 1.5, so Pia meets the cutoff. [1 mark, explicit cutoff check.]
(b) Quan (School Q).
zQuan = (92 − 70) ÷ 12 = 22 ÷ 12 ≈ 1.83. [1 mark, substitution and value.]
1.83 ≥ 1.5, so Quan also meets the cutoff. [1 mark, explicit cutoff check.]
(c) Explanation and decision.
Although Pia's raw mark of 94 is higher than Quan's 92, School P had a much higher mean (86 vs 70) and a much smaller spread (SD 5 vs SD 12). This means an above-average student at School P sits much closer to the rest of the cohort than a similarly above-average student at School Q. [1 mark, refers to different means and SDs.] The z-score measures how many standard deviations each student is above their own school's mean, not their absolute mark, so it is a fair within-school comparison. [1 mark, explains the role of z as relative standing.]
Quan's z = 1.83 is higher than Pia's z = 1.60, so on the university's z-based criterion Quan should be awarded the scholarship, since he stood out further from his own peers than Pia did from hers. [1 mark, explicit award decision with justification.]
Total: 7/7.
Band descriptors for marker.
Band 3: Computes one z-score correctly but does not compare to the cutoff or makes one calculation error. ≈ 3 marks.
Band 4: Both z-scores correct, both cutoff checks correct, but no real explanation in (c), just states an award based on raw marks. ≈ 5 marks.
Band 5: Full numerical work plus an explanation that references either the means or the SDs but not both. Award decision present. ≈ 6 marks.
Band 6: Complete, correctly substituted z-scores; explicit cutoff checks; coherent explanation referencing both the different means and the different SDs across the two schools; explicit, justified award decision that names the winning student. 7/7.