Year 13 · Mathematics · Statistics
The normal distribution and z-scores
Recognise normally distributed data, use the 68–95–99.7 rule to estimate proportions, and use z-scores to compare values from different distributions.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.
These lessons were written around Australian Curriculum strands and matched to New Zealand year levels by age (NZ Year 1 is the first year of school). They are not yet mapped to The New Zealand Curriculum, so check them against your own learning programme.
Learning goals
Learners will
- Describe the shape and key features of a normal distribution.
- Use the 68–95–99.7 rule to estimate percentages of data.
- Calculate and interpret z-scores.
Success looks like
- I can sketch a normal curve and mark the mean and one, two and three standard deviations either side.
- I can estimate the percentage of data between or beyond given values.
- I can calculate a z-score and use it to compare results from different tests.
The big idea
Many measurements, such as adult heights or the masses of packaged food, form a normal distribution when graphed. The graph is a symmetric bell shape with its peak at the mean. Half the data lies below the mean and half above.
The standard deviation (σ) describes the spread. A small standard deviation gives a tall, narrow curve; a large one gives a low, wide curve.
The 68–95–99.7 rule gives approximate percentages for any normal distribution:
- About 68% of values lie within 1 standard deviation of the mean.
- About 95% lie within 2 standard deviations, and about 99.7% within 3.
Because the curve is symmetric, these percentages split evenly either side of the mean. For example, about 34% of values lie between the mean and 1 standard deviation above it, and about 47.5% lie between the mean and 2 standard deviations above it. About 2.5% lie more than 2 standard deviations above the mean.
A z-score tells you how many standard deviations a value is from the mean: z = (x − μ) / σ. A positive z-score is above the mean, a negative one is below, and z = 0 is exactly at the mean. Rearranging gives x = μ + zσ, which finds the value for a given z-score.
z-scores let us compare results from different distributions. A mark with a higher z-score is better relative to the group, even if the raw mark is lower.
Worked examples
Calculating a z-score
Heights in a group are normally distributed with mean 170 cm and standard deviation 8 cm. Find the z-score for a height of 186 cm.
- z = (x − μ) / σ = (186 − 170) / 8.
- z = 16 / 8 = 2.
Answer: z = 2 (two standard deviations above the mean)
Using the 68–95–99.7 rule
For the same heights (mean 170 cm, standard deviation 8 cm), estimate the percentage of people taller than 186 cm.
- 186 cm has z = 2.
- About 95% of heights lie between z = −2 and z = 2, so about 5% lie outside this range.
- By symmetry, half of that 5% is above z = 2.
Answer: About 2.5%
Comparing results
Sam scored 75 in a Maths test (mean 60, standard deviation 10) and 78 in an English test (mean 70, standard deviation 4). In which test did Sam do better relative to the group?
- Maths: z = (75 − 60) / 10 = 1.5.
- English: z = (78 − 70) / 4 = 2.
- The English z-score is higher.
Answer: English (z = 2 compared with z = 1.5)
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
In a normal distribution, approximately what percentage of values lie within 1 standard deviation of the mean?
68% — By the 68–95–99.7 rule, about 68% of values lie within 1 standard deviation of the mean.
Scores have mean 70 and standard deviation 5. What is the z-score of a score of 85?
3 — z = (85 − 70) / 5 = 15 / 5 = 3.
Data are normally distributed with mean 50 and standard deviation 10. What percentage of values are above 50?
50% — The normal curve is symmetric about the mean, so half of the values (50%) are above it.
Data are normally distributed with mean 100 and standard deviation 15. Approximately what percentage of values are above 130?
2.5% — 130 is 2 standard deviations above the mean. About 95% lie within 2 standard deviations, leaving 5% outside, half of which (2.5%) is above.
What does a negative z-score tell you?
The value is below the mean — A negative z-score means x − μ is negative, so the value is below the mean.
Data have mean 40 and standard deviation 6. Find the z-score of 31.
−1.5 — z = (31 − 40) / 6 = −9 / 6 = −1.5.
Data have mean 20 and standard deviation 3. Which value has a z-score of 2?
26 — x = μ + zσ = 20 + 2 × 3 = 26.
Data are normally distributed with mean 70 and standard deviation 5. Using the 68–95–99.7 rule, approximately what percentage of values lie between 65 and 80? Give the number only.
81.5% — 65 is 1 standard deviation below the mean (about 34% between 65 and 70) and 80 is 2 above (about 47.5% between 70 and 80). Total: 34 + 47.5 = 81.5%.
Watch out for
- Thinking the rule gives exact percentages. The 68–95–99.7 values are approximations for data that are close to normal.
- Subtracting in the wrong order, such as μ − x instead of x − μ, which flips the sign of the z-score.
- Assuming all data are normally distributed. Skewed data, such as house prices or incomes, should not be analysed this way.
- Comparing raw scores from different tests directly. Different means and spreads make raw scores misleading; z-scores account for this.
Tips for parents
- Ask your learner to sketch a bell curve and mark the mean and standard deviations for every question. Most errors disappear with a sketch.
- Discuss real examples, such as how manufacturers set package weights so that very few packets are underweight.
- Ask your learner to explain what a z-score means in a sentence, such as "this height is two standard deviations above average".
Go further
- Measure the hand spans of as many family members and friends as possible, draw a histogram, and decide whether the data look approximately normal.
- Find the mean and standard deviation of a data set using a calculator's statistics mode, then check what percentage of values lie within one standard deviation.
- Research how standardised scores are used to compare results across different groups, and discuss the advantages and limitations.