Mathematics Standard • Year 12 • Module 8 • Lesson 7

Z-Scores

Build fluency in the z-score formula z = (x − mean)/SD: calculate z-scores, convert back to raw scores, and interpret what the magnitude tells you.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the z-score formula:

z = ( __________ − __________ ) ÷ __________

Q1.2 Circle the correct sign of the z-score in each case:

A score above the mean has z ( + − 0 ). A score below the mean has z ( + − 0 ). A score equal to the mean has z ( + − 0 ).

Q1.3 Write the formula to convert a z-score back to a raw score x: x = ____________

Stuck? Revisit lesson § Key Ideas — z = (x − mean)/SD, and x = mean + z × SD.

2. Worked example — calculating a z-score

Follow each line of working. Every step has a reason on the right.

Problem. A Maths test has mean 70 and standard deviation 8. Find the z-score for a raw mark of 86.

Step 1 — Write the formula.

z = (x − mean) ÷ SD

Reason: the standard z-score definition.

Step 2 — Substitute the given values.

z = (86 − 70) ÷ 8

Reason: x = 86 (the raw score), mean = 70, SD = 8.

Step 3 — Simplify the numerator first.

z = 16 ÷ 8

Reason: order of operations — brackets before division.

Step 4 — Divide.

z = 2.0

Reason: 16 ÷ 8 = 2 exactly.

Conclusion. The mark of 86 is z = 2.0, i.e. 2 standard deviations above the class mean.

3. Faded example — fill in the missing steps

An IQ test has mean 100 and SD 15. A person scores 85. Fill in each blank line to find their z-score. 3 marks

Step 1 — Formula: z = (x − __________) ÷ __________

Step 2 — Substitute: z = (__________ − 100) ÷ __________

Step 3 — Numerator: z = __________ ÷ 15

Step 4 — Divide: z = ____________ (to 2 d.p.)

Conclusion. The IQ of 85 is z = ____________, i.e. ____________ standard deviations below the mean.

Stuck? Revisit lesson § Worked Example — Sarah/Tom/Emma z-score comparison.

4. Graduated practice — z-scores and raw scores

Show your working in the space below each part. Keep z-scores to 2 decimal places unless they are exact.

Foundation — single-step z-score calculations (4 questions)

Q	Problem	Answer
4.1 1	x = 85, mean = 75, SD = 10. Find z.
4.2 1	x = 55, mean = 70, SD = 12. Find z.
4.3 1	x = 120, mean = 100, SD = 15. Find z.
4.4 1	x = 70, mean = 70, SD = 10. Find z.

Standard — typical HSC difficulty (6 questions)

Show the formula line, then the substitution, then the answer with units (z is unitless).

4.5 A test has mean 60 and SD 8. Sarah scores 78. Calculate her z-score. 2 marks

4.6 Heights of Year 12 students have mean 170 cm and SD 8 cm. A student is 160 cm tall. Calculate their z-score. 2 marks

4.7 On a test with mean 65 and SD 10, convert z = −1.5 back to a raw score. 2 marks

4.8 On the same test (mean 65, SD 10), convert z = 2.2 back to a raw score. 2 marks

4.9 Ana scores 78 on a Maths test (mean 70, SD 8). Ben scores 85 on Science (mean 80, SD 5). Calculate both z-scores. Who performed better relative to their class? 3 marks

4.10 A student's z-score is −0.8 on a test with mean 75 and SD 10. What raw mark did they score? 2 marks

Extension — interpret magnitude or combine ideas (2 questions)

4.11 A test has mean 70 and SD 8. Mark calculates his z-score as 2.5. (i) Is this above or below the mean? (ii) Is z = 2.5 considered unusual? Refer to the |z| > 2 rule. (iii) Convert his z back to a raw mark. 3 marks

4.12 Three students sit different tests. Lina (Maths) z = 1.2, Owen (English) z = 1.6, Priya (Science) z = −0.4. (i) Rank them by relative performance. (ii) State which student performed below their class average. (iii) Explain in one sentence why raw marks would not have allowed you to rank them. 3 marks

Stuck on 4.12? The bigger the z, the further above the mean. Negative z means below the mean. Raw marks are on different scales so can't be compared directly.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (x = 85, mean = 75, SD = 10), I subtracted mean from x in the brackets (not x from mean), then divided by SD.

For 4.2 (x = 55), my answer is negative because 55 is below the mean of 70.

For 4.4 (x = 70, mean = 70), my answer is z = 0 because the score equals the mean.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Formula

z = ( x − mean ) ÷ SD.

Q1.2 — Sign of z

Above mean: +. Below mean: −. Equal to mean: 0.

Q1.3 — Convert back

x = mean + z × SD.

Q3 — Faded example (IQ = 85)

Step 1: z = (x − mean) ÷ SD.
Step 2: z = (85 − 100) ÷ 15.
Step 3: z = −15 ÷ 15.
Step 4: z = −1.00.
Conclusion: IQ of 85 is z = −1.00, i.e. 1 standard deviation below the mean.

Q4.1 — x = 85, mean = 75, SD = 10

z = (85 − 75) ÷ 10 = 10 ÷ 10 = 1.00.

Q4.2 — x = 55, mean = 70, SD = 12

z = (55 − 70) ÷ 12 = −15 ÷ 12 = −1.25.

Q4.3 — x = 120, mean = 100, SD = 15

z = (120 − 100) ÷ 15 = 20 ÷ 15 ≈ 1.33.

Q4.4 — x = 70, mean = 70, SD = 10

z = (70 − 70) ÷ 10 = 0 ÷ 10 = 0. Sarah is exactly at the mean.

Q4.5 — Sarah (78, mean 60, SD 8)

z = (78 − 60) ÷ 8 = 18 ÷ 8 = 2.25. (Magnitude > 2 — unusually high.)

Q4.6 — Student 160 cm (mean 170, SD 8)

z = (160 − 170) ÷ 8 = −10 ÷ 8 = −1.25.

Q4.7 — Convert z = −1.5 (mean 65, SD 10)

x = mean + z × SD = 65 + (−1.5)(10) = 65 − 15 = 50.

Q4.8 — Convert z = 2.2 (mean 65, SD 10)

x = 65 + (2.2)(10) = 65 + 22 = 87.

Q4.9 — Ana vs Ben

Ana: z = (78 − 70) ÷ 8 = 8 ÷ 8 = 1.00.
Ben: z = (85 − 80) ÷ 5 = 5 ÷ 5 = 1.00.
Both have z = 1.00, so they performed equally well relative to their respective classes.

Q4.10 — z = −0.8 (mean 75, SD 10)

x = 75 + (−0.8)(10) = 75 − 8 = 67.

Q4.11 — Mark, z = 2.5 (mean 70, SD 8)

(i) z is positive → above the mean.
(ii) |z| = 2.5 > 2, so Mark's mark is unusually high.
(iii) x = 70 + 2.5 × 8 = 70 + 20 = 90.

Q4.12 — Ranking by z

(i) Owen (1.6) > Lina (1.2) > Priya (−0.4).
(ii) Priya performed below her class average (her z is negative).
(iii) The three tests have different means and SDs, so raw marks are on different scales — z-scores remove the scale so the rankings are fair.