中文（繁體）
中文（简体）
English
IB Maths: The Normal Distribution and z-Scores (Paper 1: basics)
IB Maths: The Normal Distribution and z-Scores (Paper 1: basics)
IB maths normal distribution paper 1, basics: from the standardising formula to table probabilities, with answers and short notes.
Direct answers Answers in order: 2, -3, the score equals the mean, z = 1.5 with 93.32 percent, 15.87 percent, 2.28 percent, 85 to 115, about 46 students, 74, and divide rather than multiply.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: IB Maths: The Normal Distribution and z-Scores |
| Level | IB |
02 Questions (10 items)
- FoundationA test has mean 60 and standard deviation 5. A student scores 70. What is the z-score?
- FoundationIn the same test, mean 60 and standard deviation 5, another student scores 45. What is the z-score?
- FoundationWhat does a z-score of 0 mean?
- IntermediateWith mean 50 and standard deviation 10, find the z-score of 65 and the proportion below it given that the table value at 1.5 is 0.9332.
- IntermediateGiven that the table value at -1 is 0.1587, what proportion lies more than one standard deviation below the mean?
- IntermediateGiven that the table value at 2 is 0.9772, what proportion lies more than two standard deviations above the mean?
- ChallengeWith mean 100 and standard deviation 15, between which two values do the middle 68 percent of scores lie?
- ChallengeIf 2.28 percent lie above two standard deviations, how many of 2 000 students is that?
- ChallengeA score has z = -1.5 where the mean is 80 and the standard deviation is 4. Find the score.
- Error redoA student writes z = (score – mean) times standard deviation. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 2 | The z-score is the score minus the mean divided by the standard deviation, so 10 over 5 gives 2. |
| 2 | -3 | 45 minus 60 is -15, and dividing by 5 gives -3, negative because the score is below the mean. |
| 3 | The score is exactly equal to the mean | A z-score of 0 means the numerator is zero, so the score equals the mean. |
| 4 | z = 1.5, about 93.32 percent are below 65 | 15 divided by 10 gives 1.5, and the standard normal table gives 0.9332 at that point. |
| 5 | 15.87 percent | The negative z is read straight from the table, and 0.1587 is 15.87 percent. |
| 6 | 2.28 percent | A greater-than question needs one minus the table value, so 1 minus 0.9772 gives 0.0228. |
| 7 | 85 to 115 | The 68 percent rule covers one standard deviation either side, so 100 plus or minus 15. |
| 8 | About 46 students | 2 000 times 0.0228 gives 45.6, which rounds to about 46 students. |
| 9 | 74 | Going backwards, multiply z by the standard deviation and add the mean: -1.5 times 4 is -6, and 80 minus 6 is 74. |
| 10 | The deviation should be divided by the standard deviation, not multiplied | Multiplying scales the units up a second time, while standardising divides so that the new standard deviation is 1. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Always subtract the mean and divide by the standard deviation first, then read the standard normal table; to go backwards, find z first and multiply by the standard deviation before adding the mean.Tutor Registration →
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.