Normal distribution and standardisation (IB Math AA HL)

Key ideas

• X ~ N(μ, σ²): the second number is the variance. X ~ N(50, 16) has standard deviation σ = 4.
• About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
• Normal CDF gives P(a < X < b). Inverse normal gives the value with a stated probability below it (check your GDC's tail setting).
• X is continuous, so P(X = a) = 0 and P(X < a) = P(X ≤ a).
• Standardise with z = (x − μ)/σ. When μ or σ is unknown, the GDC cannot work on X itself, so find z from N(0, 1) first.
• With both μ and σ unknown, each probability you are given makes one equation μ + zσ = x. Two of them make simultaneous equations.

Worked example

X ~ N(μ, σ²) with P(X < 30) = 0.2 and P(X > 50) = 0.1. Find μ and σ.

1. P(Z < z₁) = 0.2 gives z₁ = −0.84162 (inverse normal on N(0, 1)).
2. P(X > 50) = 0.1 means P(X < 50) = 0.9, so z₂ = 1.28155.
3. So μ − 0.84162σ = 30 and μ + 1.28155σ = 50.
4. Subtract: 2.12317σ = 20, so σ = 9.42 (3 s.f.), and μ = 30 + 0.84162 × 9.4199 = 37.9 (3 s.f.).

Answer: μ = 37.9 and σ = 9.42 (3 s.f.).

Exam-style HL AA questions

• HL AA Statistics and Probability questions by topic — worked examples and practice questions for every topic in this unit

Do it on your GDC

A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Normal distribution on your calculator
• Try it: normal distribution calculator · inverse normal calculator (free, shows the working)

Maths IA ideas that use normal distribution and standardisation

• Are NBA players getting taller?
• Trapezoidal rule or Simpson's rule? Areas under the normal curve
• Where should I aim on a dartboard?
• Is the euro–dollar exchange rate a random walk?
• IA statistics, step by step: is my data normal?
• All HL AA IA ideas

Related in Statistics and Probability

• Binomial distribution — mean, variance and unknown parameters
• Conditional probability and Bayes' theorem
• Discrete probability distributions — expectation and variance

Revise Normal distribution and standardisation for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Statistics and Probability — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original probability problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S33 · Problem S106 · Problem S34 · Problem S107. All 20 →

Sibling topic in other IB Maths courses