Z-Scores
You scored 85 on a Maths test (mean 80, SD 5). Your friend scored 78 on English (mean 70, SD 8). Who performed better? Raw scores cannot answer this. The z-score converts any score into a universal measure — standard deviations from the mean — making fair comparison possible across any two distributions.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
You score 72 on Test A (class mean = 60, SD = 8). Your friend scores 85 on Test B (class mean = 75, SD = 10). Who performed better relative to their class?
Before calculating — write your gut feeling. We will revisit this at the end.
Z-score formula: $z = \dfrac{x - \bar{x}}{s}$ (sample) or $z = \dfrac{x - \mu}{\sigma}$ (population).
Convert back: $x = \bar{x} + z \cdot s$. Positive z means above the mean; negative z means below.
Use $|z| > 2$ as a threshold for an unusual value; $|z| > 3$ for very unusual.
Key facts
- The z-score formula
- Positive vs negative z-scores
- Standard normal properties (mean 0, SD 1)
- Approximate percentiles for z = ±1, ±2, ±3
Concepts
- Why z-scores enable fair comparison across different scales
- What the magnitude of a z-score tells you
- How standardisation works
Skills
- Calculate z-scores from raw data
- Compare performance across two distributions
- Convert a z-score back to a raw score
The z-score tells you how many standard deviations a value lies from the mean:
$$z = \frac{x - \bar{x}}{s}$$Example 1: IQ score of 115, mean = 100, SD = 15.
$$z = \frac{115 - 100}{15} = \frac{15}{15} = 1.0$$The score is exactly 1 standard deviation above the mean.
Example 2: Height of 160 cm, mean = 170 cm, SD = 8 cm.
$$z = \frac{160 - 170}{8} = \frac{-10}{8} = -1.25$$This height is 1.25 standard deviations below the mean. The negative sign indicates below-average; the magnitude tells you by how much in SD units.
What to write in your book
- $z = \dfrac{x - \bar{x}}{s}$. Positive = above mean; negative = below mean; zero = at the mean.
- To convert back: $x = \bar{x} + z \cdot s$.
- $|z| > 2$ is unusual; $|z| > 3$ is very unusual.
Quick check: A student scores 88 on a test where the mean is 80 and SD is 4. What is their z-score?
Z-scores allow fair comparison of performance on different tests, because they remove the effect of different means and spreads.
Example:
- Student A — Maths: 72, mean = 60, SD = 8 → $z = (72-60)/8 = 1.5$
- Student B — English: 85, mean = 75, SD = 10 → $z = (85-75)/10 = 1.0$
Student A performed better relative to their class ($z = 1.5 > 1.0$), even though their raw score is lower.
What to write in your book
- To compare across tests: calculate z-scores for each and compare them directly.
- Higher z-score = stronger relative performance regardless of the raw mark or scale.
- Converting back: $x = \bar{x} + z \cdot s$.
True or false: A student with a raw score of 90 always performed better than a student with a raw score of 75, regardless of the test.
| Z-score | Position | Approx. percentile |
|---|---|---|
| −3 | Far below average | 0.15th |
| −2 | Well below average | 2.5th |
| −1 | Below average | 16th |
| 0 | Average | 50th |
| +1 | Above average | 84th |
| +2 | Well above average | 97.5th |
| +3 | Far above average | 99.85th |
What to write in your book
- $z = 0$ → 50th percentile (exactly average). $z = +1$ → approx 84th percentile.
- $|z| > 2$: only about 5% of values fall this far from the mean (unusual).
- $|z| > 3$: only about 0.3% of values — very unusual.
Fill the gap: A test has mean = 65 and SD = 10. A student with $z = -1.5$ has a raw score of $x =$ .
Activities · two in-class tasks
Calculate z-scores for: (a) $x=85$, mean $=75$, SD $=10$; (b) $x=55$, mean $=70$, SD $=12$; (c) $x=120$, mean $=100$, SD $=15$. State whether each is unusual.
Three students sat different tests: Sarah — Maths: 78 (mean 70, SD 8); Tom — Science: 82 (mean 75, SD 6); Emma — History: 85 (mean 80, SD 12). Rank them by relative performance.
Match each z-score to its meaning:
Top 3 list: Name THREE things z-scores allow you to do that raw scores alone cannot.
Student A: $z = (72-60)/8 = 1.5$. Student B: $z = (85-75)/10 = 1.0$. Student A has the higher z-score and performed better relative to their class, even though their raw score (72) is lower than Student B's (85). This is exactly why z-scores exist — raw scores can be deeply misleading when scales differ.
SA 1. (a) Calculate z-scores for: (i) $x=92$, mean $=80$, SD $=8$; (ii) $x=65$, mean $=72$, SD $=6$; (iii) $x=110$, mean $=100$, SD $=15$. (b) Which value is most unusual? (c) Convert $z=-0.8$ back to a raw score with mean $=75$, SD $=10$. (3 marks)
SA 2. Three students sat different tests: Ana — Maths: 78 (mean 70, SD 8); Ben — Science: 85 (mean 80, SD 5); Carla — English: 82 (mean 72, SD 10). (a) Calculate the z-score for each student. (b) Rank them by relative performance. (c) The top 10% receive an award (approximately $z > 1.28$). Who qualifies? (3 marks)
Comprehensive answers (click to reveal)
SA 1: (i) $z=(92-80)/8=1.5$ (ii) $z=(65-72)/6=-1.17$ (iii) $z=(110-100)/15=0.67$. (b) $|z|=1.5$ is the largest — value (i) is most unusual. (c) $x=75+(-0.8\times10)=67$.
SA 2: Ana: $z=(78-70)/8=1.0$. Ben: $z=(85-80)/5=1.0$. Carla: $z=(82-72)/10=1.0$. (b) All have identical z-scores — equally strong relative to their class. (c) None qualify ($z=1.0 < 1.28$).
Five timed questions on z-scores and standardisation. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%).
⚔ Enter the arenaClimb platforms answering z-score questions. Pool: lesson 7.
Mark lesson as complete
Tick when you have finished the practice and review.