Key Idea: The normal distribution models data that clusters symmetrically around an average — heights, masses, exam marks. On Paper 2 you read probabilities straight off the GDC; on Paper 1 you use symmetry and the 68–95–99.7 rule.
🔔 The model: X ~ N(μ, σ²)
- the mean — the centre of the bell
- the variance (the second number) — take √ to get σ
- the standard deviation — sets the width
The bell is symmetric about μ, so the mean = median = mode. The total area is 1, and 0.5 lies on each side of the mean. Probabilities are areas under the curve.
| Change | Effect on the curve |
|---|---|
| Different mean μ | Shifts left/right — same shape and width. |
| Larger σ | Wider and flatter — more spread. |
| Smaller σ | Taller and narrower — more consistent data. |
💻 Finding P(a < X < b) — Paper 2
| You want | normalcdf(lower, upper, μ, σ) |
|---|---|
| P(a < X < b) | lower = a, upper = b |
| P(X < a) | lower = −1ᴇ99, upper = a |
| P(X > a) | lower = a, upper = 1ᴇ99 |
Without a calculator: P(X < μ) = 0.5, and about 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. The leftover splits into two equal tails — e.g. outside 1σ is 0.32, so each tail is 0.16.
✏️ IB-style worked examples
IB-style question — a probability and an expected number (Paper 2)
The masses of oranges are modelled by X ~ N(180, 15²) grams. An orange is rejected if it is lighter than 160 g. (a) Find P(X < 160). (b) In a crate of 500 oranges, find the expected number rejected.
Step by step:
The second number is the variance, so σ = √225 = 15.
P(X < 160): lower bound −1ᴇ99, upper 160.
Expected number = probability × total.
(a) P(X < 160) ≈ 0.0912. (b) About 46 oranges are expected to be rejected.
IB-style question — symmetry and comparing curves (Paper 1)
Two classes sit the same test. Both sets of marks are normal with mean 62, but class A has σ = 5 and class B has σ = 12. (a) Write down P(mark > 62) for class A. (b) Whose marks are more consistent, and what does the curve look like?
Step by step:
62 is the mean, and the curve is symmetric about it.
Smaller σ means less spread — taller and narrower.
(a) P(mark > 62) = 0.5 (no calculator needed). (b) Class A is more consistent; its curve is taller and narrower.
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Important: N(μ, σ²) gives the variance as the second number. normalcdf wants σ, so for N(180, 225) you must enter σ = √225 = 15, not 225. When the bracket already shows a square — N(180, 15²) — the σ is the 15.
Tap each card to reveal the answer.
In N(50, 8²), what is σ? σ = 8 — the number being squared is the standard deviation (variance = 64).
X ~ N(70, 6²). What is P(X < 70)? 0.5 — the mean splits the symmetric curve in half.
How do you enter P(X > 90) on the GDC? normalcdf(90, 1ᴇ99, μ, σ) — lower bound 90, upper bound 1ᴇ99.
About what % of data lies within 2σ of the mean? 95% — the 68–95–99.7 rule (1σ → 68%, 2σ → 95%, 3σ → 99.7%).
P one item meets a condition is 0.12. Expected number in 250? 30 — expected number = probability × total = 0.12 × 250.
Same mean, but σ goes from 4 to 10. What happens to the curve? It gets wider and flatter — a larger σ means more spread.
Exam tips
- N(μ, σ²): the second number is the variance — enter σ = √variance into the GDC.
- normalcdf needs a lower AND an upper bound; use −1ᴇ99 / 1ᴇ99 for a one-sided tail.
- Anything asked at the mean is 0.5 — by symmetry, no calculator needed.
- Expected number = (normalcdf probability) × total — show both, each earns a mark.
- Sketch the bell, mark μ, and shade the region to sanity-check your probability.