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Year 12 Mathematics · The normal distribution and z-scores

Name: ______________________

Date: ____________

1. In a normal distribution, approximately what percentage of values lie within 1 standard deviation of the mean?
- ☐ (a) 50%
- ☐ (b) 68%
- ☐ (c) 95%
- ☐ (d) 99.7%
2. Scores have mean 70 and standard deviation 5. What is the z-score of a score of 85?
- ☐ (a) 3
- ☐ (b) 15
- ☐ (c) −3
- ☐ (d) 1.5
3. Data are normally distributed with mean 50 and standard deviation 10. What percentage of values are above 50?
- ☐ (a) 68%
- ☐ (b) 34%
- ☐ (c) 50%
- ☐ (d) 95%
4. Data are normally distributed with mean 100 and standard deviation 15. Approximately what percentage of values are above 130?
- ☐ (a) 5%
- ☐ (b) 16%
- ☐ (c) 47.5%
- ☐ (d) 2.5%
5. What does a negative z-score tell you?
- ☐ (a) The value is above the mean
- ☐ (b) The value is below the mean
- ☐ (c) The value was recorded incorrectly
- ☐ (d) The standard deviation is negative
6. Data have mean 40 and standard deviation 6. Find the z-score of 31.
7. Data have mean 20 and standard deviation 3. Which value has a z-score of 2?
8. 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.

Answer sheet

1. 68% — By the 68–95–99.7 rule, about 68% of values lie within 1 standard deviation of the mean.
2. 3 — z = (85 − 70) / 5 = 15 / 5 = 3.
3. 50% — The normal curve is symmetric about the mean, so half of the values (50%) are above it.
4. 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.
5. The value is below the mean — A negative z-score means x − μ is negative, so the value is below the mean.
6. −1.5 — z = (31 − 40) / 6 = −9 / 6 = −1.5.
7. 26 — x = μ + zσ = 20 + 2 × 3 = 26.
8. 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%.