Normal: Height Distribution

Probability · Axiom Academy

EXAMPLE Normal: Height Distribution Using the normal distribution to calculate probabilities and find percentiles Excellent work! You've completed this normal distribution example. Here's what we learned: Standardization: Convert any normal variable X ~ N(μ, σ²) to standard normal Z ~ N(0, 1) using Z = (X - μ)/σ. Probability Calculations: P(X > 185) ≈ 6.7% found by standardizing to Z = 1.5 and using Φ(1.5) ≈ 0.933. Range Probabilities: P(160 < X < 180) ≈ 68.3% found by P(Z < 1) - P(Z < -1) = 0.841 - 0.159. Percentiles: The 90th percentile is found by solving Φ(z) = 0.90 to get z ≈ 1.28, then x = 170 + 10(1.28) = 182.8 cm. The normal distribution is fundamental in statistics and appears throughout natural and social sciences!

This is the written version of the interactive lesson above. See the full Probability course.