Study Guide
Bivariate Data: Correlation and Lines of Best Fit
An HSC/VCE-style study guide to analysing two numerical variables: describing scatterplots, interpreting Pearson's r and the coefficient…
- Difficulty
- Level 3 · Challenge
- Time
- 25 min
Study GuideYear 12Level 3 · ChallengeHSCVCE
Free Year 12 The Normal Distribution and z-scores
An HSC/VCE-style study guide to the bell curve: calculating z-scores, comparing results from different tests, using the 68–95–99.7 rule, reading probabilities from a standard normal table or calculator and working backwards from a percentage to a raw score. Includes a six-question self-check with answers.
Instant download · No sign-up · Free for personal, classroom and homeschool use
- Year
- Year 12
- Subject
- Mathematics
- Topic
- Statistics
- Difficulty
- Level 3 · Challenge
- Estimated time
- 25 minutes
- Curriculum
- Australian Curriculum
- Answers
- Not applicable
- Format
- PDF (A4) + print
Students will practise
- calculating z-scores with z = (x − μ) ÷ σ and converting back to raw scores
- comparing scores from different distributions using z-scores
- applying the 68–95–99.7 (empirical) rule
- finding probabilities and percentiles for a normal distribution using a table or calculator
Curriculum: Australian Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.
What's next?
Completed: The Normal Distribution and z-scores
More resources for this topic are coming soon.
How to use this study guide
- Read it together first, pausing at each worked example to try the step before reading the answer.
- Attempt the "Check yourself" questions at the end without looking back.
- Then practise with a worksheet from the pathway above and finish with the topic test.
Common questions
Who is this study guide for?
Year 12 students (typically ages 17–18) working on statistics. It is pitched at level 3 · challenge.
Are the answers included?
This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.
How long does it take?
About 25 minutes. Short, regular sessions work best: two or three a week beats one long one.
Do I need to sign up to download?
No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.
What should we do next?
Explore more Year 12 mathematics resources.
Preview
Year 12 · Mathematics · Statistics
The Normal Distribution and z-scores
Success Tutoring
Student name:
Date:
What you need to know
Many measurements — heights, test scores, measurement errors — follow a normal distribution: a symmetric bell-shaped curve centred on the mean μ, with spread measured by the standard deviation σ. The mean, median and mode are all equal, and the total area under the curve is 1 (100%). A z-score tells you how many standard deviations a value is from the mean: z = [frac x − μ/σ]. Positive z is above the mean, negative z is below, and z = 0 is exactly the mean.
Converting to z-scores lets you compare values from different distributions on the same scale, and lets you use one standard normal table (μ = 0, σ = 1) for every normal distribution. The table or calculator gives P(Z < z); use symmetry and the complement for other regions: P(Z > z) = 1 − P(Z < z), and P(Z < −z) = P(Z > z).
| Interval | Approximate percentage of data (empirical rule) | Useful halves |
|---|---|---|
| μ − σ to μ + σ (z from −1 to 1) | 68% | 34% each side of the mean |
| μ − 2σ to μ + 2σ (z from −2 to 2) | 95% | 47.5% each side; 2.5% in each tail |
| μ − 3σ to μ + 3σ (z from −3 to 3) | 99.7% | 49.85% each side; 0.15% in each tail |
| between 1σ and 2σ above the mean | 13.5% | 47.5% − 34% |
Test scores have μ = 65 and σ = 8. Find the z-scores for 77 and 53.
- z = = = 1.5 (1.5 standard deviations above the mean)
- z = = = −1.5 (1.5 standard deviations below the mean)
Which result is better: 82 in Maths (μ = 70, σ = 8) or 75 in English (μ = 62, σ = 10)?
- Maths: z = = 1.5
- English: z = = 1.3
- The Maths result is better relative to its class, because it is further above the mean in standard-deviation units.
Heights are normal with μ = 170 cm and σ = 6 cm. Use the empirical rule.
- Percentage between 164 cm and 182 cm: 164 is z = −1 and 182 is z = 2. Area = 34% + 47.5% = 81.5%
- Percentage above 188 cm: 188 is z = 3. Only 0.15% lies above z = 3, so 0.15%.
- Height with z = −2: x = μ + zσ = 170 + (−2)(6) = 158 cm. About 2.5% of people are shorter than this.
Using a standard normal table with μ = 65 and σ = 8
- P(X < 77) = P(Z < 1.5) = 0.9332, so about 93.3% of scores are below 77.
- P(X > 77) = 1 − 0.9332 = 0.0668
- P(57 < X < 77) = P(−1 < Z < 1.5) = P(Z < 1.5) − P(Z < −1) = 0.9332 − 0.1587 = 0.7745
Working backwards: what score is needed to be in the top 10% (μ = 65, σ = 8)?
- Top 10% means P(Z > z) = 0.10, so P(Z < z) = 0.90.
- From the table (or inverse normal on a calculator), z ≈ 1.28.
- x = μ + zσ = 65 + 1.28 × 8 = 75.24…, so a score of about 75.3 (in practice, 76 if scores are whole numbers).
Common mistake: dropping the negative sign on a z-score below the mean, or reading the table as if P(Z < −1.5) were 0.9332. For a negative z, use symmetry: P(Z < −1.5) = P(Z > 1.5) = 1 − 0.9332 = 0.0668.
Common mistake: using the empirical rule for values that are not whole numbers of standard deviations. 68–95–99.7 only applies at z = ±1, ±2, ±3. For z = 1.5 you must use a table or calculator.
Check yourself
- A distribution has μ = 50 and σ = 5. Find the z-score of 60.
- Using μ = 50 and σ = 5, find the z-score of 42.
- What percentage of a normal distribution lies within one standard deviation of the mean?
- Using μ = 50 and σ = 5, what percentage of values are below 40?
- Using μ = 50 and σ = 5, find the value with a z-score of 1.5.
- Alex scored 60 in a test with μ = 50 and σ = 5. Bea scored 85 in a test with μ = 70 and σ = 10. Who did better relative to their class?
Answers: 1. 2 2. −1.6 3. 68% 4. 40 is z = −2, so 2.5% 5. 50 + 1.5 × 5 = 57.5 6. Alex (z = 2) did better than Bea (z = 1.5)
Preview — the PDF contains the full resource.
More Year 12 statistics resources
Was this resource helpful?
About this resource
- Created by
- Success Tutoring
- Last reviewed
- 1 October 2026
- How it was made
- Written by our team
Free to print and share for personal, classroom and homeschool use. Please don't resell. Spotted an error? Let us know.