Normal distribution: z score and P(X ≤ a)
Every normal distribution reduces to the standard normal via z; for negative z, Φ(−z) = 1 − Φ(z).
Worked example
A measurement is normally distributed with μ = 120 and σ = 15. What is P(X ≤ 102.45)?
Worked solution
- 1.z = (x − μ) / σ = (102.45 − 120) / 15 = -1.17
- 2.P(X ≤ 102.45) = Φ(-1.17) = 0.1210 (table)
Typical mistake: For negative z, remember Φ(−z) = 1 − Φ(z).
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task 1 of 3Normal distribution
Task
A measurement is normally distributed with μ = 50 and σ = 12. What is P(X ≤ 68.36)?
z value (2 decimal places)
P (4 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- 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
Typical mistake: For negative z, remember Φ(−z) = 1 − Φ(z).
Frequently asked questions
What is the typical mistake in "Normal distribution: z score and P(X ≤ a)"?
For negative z, remember Φ(−z) = 1 − Φ(z).
How do I read negative z values?
Tables usually list positive z only: Φ(−z) = 1 − Φ(z).