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The normal distribution
Lesson 49 of 75
Challenge

Inverse normal problems

Objective. Recover values or parameters from probabilities.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Inverse normal problems work backwards from a known probability to find the corresponding x-value or an unknown parameter, using the inverse of Φ, written Φ⁻¹. Given P(X < x) = p, find z = Φ⁻¹(p) from tables, then convert back using x = μ + zσ. For X ~ N(100, 225) with P(X < x) = 0.95, z = 1.645, so x = 100 + 1.645(15) ≈ 124.7. When σ is unknown, rearrange the standardisation formula instead to solve for σ directly.

Worked example

Given X ~ N(40, 64) and P(X < x) = 0.9332, find x, an inverse normal problem, using Φ⁻¹(0.9332) = 1.5.

64) and P(X < x) = 0
Illustrated card showing 64) and P(X < x) = 0
  1. 1Identify μ = 40 and σ = 8, since σ² = 64
  2. 2Read the z-score from the inverse normal value: z = 1.5
  3. 3Apply x = μ + zσ
  4. 4Compute x = 40 + 1.5(8) = 52

Answer: x = 52

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Common mistake. Students often apply Φ instead of Φ⁻¹ when a probability is already given and an x-value is required, reversing the correct direction of the lookup. Always use Φ⁻¹ to convert a probability into a z-score before finding x.

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