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 calculatingwrite 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.
z-score formula: z = (x − μ) / σ. A z-score measures how many standard deviations a value x is from the mean. z > 0 means above average; z < 0 means below average; z = 0 means exactly at the mean.
Pause, copy z = (x − μ) / σ with definitions, the sign interpretation (z > 0: above mean; z < 0: below mean; z = 0: at the mean), and note the z-score gives the exact number of standard deviations from the mean into your book.
Quick check: A student scores 88 on a test where the mean is 80 and SD is 4. What is their z-score?
The z-score formula z = (x − μ) / σ converts any raw value to the number of standard deviations it is from the mean. A positive z means above the mean; negative means below. Raw marks in different subjects (e.g., 75 in Maths and 70 in English) cannot be compared directly because the distributions have different means and spreads, z-scores remove this problem by expressing both results on the same standardised scale.
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.
z-scores allow comparison across different distributions. If a student scores z = 1.5 in maths and z = 2.0 in English, their English performance is relatively better, even if their raw English mark was lower.
Pause, copy the cross-distribution comparison rule: to compare performance across different subjects or datasets, compare z-scores (not raw marks), the higher z-score indicates better relative performance regardless of the raw numbers into your book.
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 |
z-scores standardise results across different distributions: a student with z = 2.0 in English performed better relative to their cohort than a student with z = 1.5 in Maths, regardless of raw marks. The magnitude of z also signals how unusual a result is: |z| < 2 is typical (within 95% of the distribution), |z| > 2 is unusual (outer 5%), and |z| > 3 is very rare (outer 0.3%).
z-score magnitude: |z| < 2 is typical (within 95% of data); |z| > 2 is unusual; |z| > 3 is very rare (outside 99.7%). Use these thresholds to classify individual values as typical, unusual, or exceptional in context.
Pause, copy the three z-score magnitude thresholds and their interpretations: |z| < 2 → typical (within 95% of data); |z| > 2 → unusual (outer 5%); |z| > 3 → very rare (outer 0.3%) into your book.
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.
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