Normal distribution: z score, probability, quantile
Any normal distribution reduces to the standard normal with z = (x − μ) / σ and can be read off the Φ table.
Formulas
- z = (x − μ) / σ
- P(X ≤ a) = Φ(z)
- P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
- Φ(−z) = 1 − Φ(z); quantile: x = μ + z · σ
Your turn
Three problems on this topic. Comma or point, fractions like 1/6 work too.
Task
The fill volume is normal with μ = 500 ml and σ = 20 ml. Which volume do 97.5% of packs fall below? (z = 1.960)
x (2 decimal places)
Comma or point; fractions like 1/6 work too.
Worked example
A measurement is normally distributed with μ = 70 and σ = 10. What is P(X ≤ 55.20)?
Worked solution
- 1.z = (x − μ) / σ = (55.20 − 70) / 10 = -1.48
- 2.P(X ≤ 55.20) = Φ(-1.48) = 0.0694 (table)
Typical mistake: For negative z, remember Φ(−z) = 1 − Φ(z).
The fill volume is normal with μ = 1000 ml and σ = 20 ml. Which volume do 97.5% of packs fall below? (z = 1.960)
Worked solution
- 1.x = μ + z · σ = 1000 + 1.960 · 20 = 1039.20
Typical mistake: Confusing z with the probability p and calculating p·σ.
Frequently asked questions
How do I read negative z values?
Tables usually list positive z only: Φ(−z) = 1 − Φ(z).