Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A continuous random variable can take any value in an interval. Intervals have probabilities; a single exact value has probability 0.
- A normal curve is continuous, unimodal, symmetric and bell-shaped. It can model a data distribution or a random variable.
- It is fixed by μ (centre) and σ (spread). Smaller σ: taller and more concentrated. Larger σ: shorter and more spread out.
- The standard normal distribution is N(0, 1).
- Percentiles from a normal model let you compare positions within or between distributions.
Key relationships
| Idea | Rule |
|---|---|
| Standardise | z = (x − μ) / σ; back again: x = μ + zσ |
| Empirical rule | about 68%, 95%, 99.7% within 1, 2, 3 σ of μ |
| Strips (one side) | 34%, 13.5%, 2.35%, 0.15% |
| Interval probability | P(a < X < b) = normalcdf(a, b, μ, σ) |
| Lowest p% | P(X < xₐ) = p/100 |
| Highest p% | P(X > x_b) = p/100, so area to the left is 1 − p/100 |
| Middle p% | (1 − p/100)/2 in each tail |
| Most extreme p% | ½(p/100) in each tail |
Assumptions and conventions
- Use a normal model only when the variable is roughly unimodal and symmetric. Check the context: can the values go below 0?
- A z-table gives the area to the left of z, with z to 2 decimal places. Table and technology answers may differ slightly.
- Show four things: distribution and parameters, boundary, direction (sketch), answer.
Mistakes to avoid
- Dividing by σ² instead of σ in a z-score.
- Wrong tail: forgetting 1 − area for “more than” or “highest”.
- Putting the whole “most extreme p%” in one tail.
- Quoting the empirical rule as exact. Say “about”.
- A bare calculator command with no labels or context.
- Comparing raw values from different distributions instead of percentiles.
Quick self-check
X ~ N(40, 5).
- Use the empirical rule to estimate the percentage of values above 50. (50 is μ + 2σ, so about 2.5%)
- Find P(X < 33). (z = −1.4; 0.0808)
- Find the value that separates the top 20% from the rest. (area to the left 0.80, z = 0.8416, x = 44.21)
Next: practice questions.