Normal distribution
Learning intention
Model continuous data with a normal distribution and connect probabilities, z-values and inverse-normal boundaries.
The core idea
A normal model is determined by its mean μ and standard deviation σ. It is symmetric about μ, and probabilities are areas under the curve.
Standardizing with z = (x − μ)/σ measures how many standard deviations x lies from the mean. Use a cumulative probability for an interval or tail, and inverse normal when the area is known but a boundary is not.
Key definitions and formulas
- Model: X ~ N(μ, σ²).
- Standard score: z = (x − μ)/σ.
- Symmetry: P(X < μ − a) = P(X > μ + a).
- Inverse normal: area → boundary value.
Worked example 1 — foundation
For X ~ N(50, 6²), find the z-value of x = 62.
- Identify μ = 50 and σ = 6.
- Substitute z = (62 − 50)/6.
- Simplify 12/6.
z = 2, so 62 is two standard deviations above the mean.
Worked example 2 — exam-style
For X ~ N(80, 10²), describe how to find P(70 < X < 92).
- Write the probability statement with both bounds.
- Use normal CDF with lower 70, upper 92, mean 80 and standard deviation 10.
- Keep full calculator precision before rounding.
- Check that the result is plausible from a sketch centered at 80.
The required result is the area between 70 and 92, not either tail.
Worked example 3 — HL reasoning
The 90th percentile of X ~ N(μ, 4²) is 31.12. Find μ given z₀.₉ = 1.28.
- Use x = μ + zσ.
- Substitute 31.12 = μ + 1.28(4).
- Subtract 5.12.
μ = 26.
GDC checkpoint
Common IB mistake
Variance and standard deviation are not interchangeable. In X ~ N(μ, σ²), the second parameter is variance, but a calculator commonly asks for σ.
Practice checks
Use the feedback to refine your method, not only your final answer.
Summary
- Write the model and probability event first.
- Standardize with z = (x − μ)/σ.
- Use CDF for areas and inverse normal for boundaries.
- Sketch and estimate before accepting calculator output.
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