The big idea: Many measurements follow a bell-shaped pattern: symmetric, with values clustered near center and fewer at extremes.
This is the normal distribution.
The bell-shaped normal curve, symmetric about the mean; the shaded area under it is the probability.
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Examples: heights, test scores, measurement errors.
Not all data is normal (income is skewed), but natural phenomena often are.
Notation: X ~ N(μ, σ²) where μ is mean and σ² is variance.
Sometimes the question gives σ (SD) instead — always check.
Why this matters
Key characteristics
- Most data clusters near mean.
- Fewer values far from mean.
- Curve is symmetric left-right.
- Mean = median = mode.
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The 68-95-99.7 rule
One rule for all normal distributions: No matter what μ and σ are, the percentage of data within fixed distances from the mean is always the same.
The golden percentages
- 68% within 1 SD: between μ - σ and μ + σ
- 95% within 2 SDs: between μ - 2σ and μ + 2σ
- 99.7% within 3 SDs: between μ - 3σ and μ + 3σ
Worked example
Heights: N(170, 10²).
What % between 160 and 180 cm?
Step by step
- 160 = 170 - 10 = μ - σ. 180 = 170 + 10 = μ + σ
- Asking for percent within 1 SD of mean
- By the rule: 68%
Final answer
68% of people are 160-180 cm tall.
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How the mean and standard deviation shape the curve
μ moves it, σ stretches it: The mean μ sets where the centre (peak) of the bell sits.
The standard deviation σ sets how wide the bell is.
Effect of each parameter
- Larger μ → the whole curve shifts to the right (smaller μ → left).
- Larger σ → the curve is wider and flatter (data more spread out).
- Smaller σ → the curve is narrower and taller (data tightly clustered).
- The area under any normal curve is always 1.
Same shape, different scale: Every normal curve has the same symmetric bell shape — only its centre (μ) and width (σ) change.
Two classes with the same mean but different σ have curves centred at the same place, but one is wider.
AI SL method: You never convert to z-scores in AI SL.
To find exact probabilities you use the GDC directly with μ and σ (covered in Normal Probabilities).
Probabilities from symmetry
The curve is symmetric about μ: Because the bell is symmetric, exactly half the area lies on each side of the mean.
Quick facts you can state without a calculator
- P(X < μ) = 0.5 and P(X > μ) = 0.5.
- P(within 1σ of μ) ≈ 0.68, so each tail beyond 1σ ≈ 0.16.
- P(within 2σ of μ) ≈ 0.95, so each tail beyond 2σ ≈ 0.025.
- P(within 3σ of μ) ≈ 0.997.
Worked example — using symmetry
Test scores are N(60, 10²).
Find the percentage of students scoring more than 60, and the percentage scoring between 50 and 70.
Step by step
- 60 is the mean, so exactly half score above it.
- 50 = μ − σ and 70 = μ + σ, so this is ''within 1 SD''.
Final answer
50% score above 60; about 68% score between 50 and 70.
For any other boundary: If the boundary is not a whole number of standard deviations from μ, use the GDC normalcdf (see Normal Probabilities) — there are no z-tables in AI SL.