Normal Distribution Calculations

Probability · Axiom Academy

LESSON Normal Distribution Calculations Computing probabilities and finding percentiles using Z-scores To find the probability that X falls in an interval [a, b], we convert both endpoints to Z-scores and use the property that P(a < X < b) = P(Z < z_b) - P(Z < z_a). The difference of two cumulative probabilities gives the interval probability. Example: If X ~ N(100, 15²), find P(90 < X < 110). Convert: z₁ = -0.67, z₂ = 0.67, giving P ≈ 0.50. A percentile is a value below which a given percentage of observations fall. To find the pth percentile, we work backwards: find the Z-score z_p such that P(Z ≤ z_p) = p, then convert back using x = μ + z_p · σ. Example: The 95th percentile of N(100, 15²) is 100 + 1.645(15) ≈ 124.7. Sometimes we need P(X > a) rather than P(X ≤ a). Use the complement rule: P(X > a) = 1 - P(X ≤ a). For symmetric distributions like the normal, P(Z > z) = P(Z < -z), which can simplify calculations.

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