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3.3StatisticsStretch

The Inverse Normal Distribution

The inverse normal function works backwards: given a probability, it finds the value of X that produces it. This answers questions like “what mark is the top 10% boundary?”.

22 min The Normal Distribution
Edexcel A Level Maths: 3.3 The Inverse Normal Distribution FunctionCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use the inverse normal function
  • Find a value of X from a probability
  • Handle “greater than” probabilities
  • Apply to percentile questions
1

Working backwards

Given P(X<a)=pP(X < a) = p, the inverse normal function returns aa. Enter the probability (as area to the left), μ\mu and σ\sigma.

P(X<a)=p ⇒ a=InvNorm(p,μ,σ)P(X < a) = p \ \Rightarrow\ a = \text{InvNorm}(p, \mu, \sigma)
From probability to value.
Example 1For X∼N(50,42)X \sim N(50, 4^2), find the value aa such that P(X<a)=0.9P(X < a) = 0.9.
1InvNorm(0.9, 50, 4).
2a≈55.1a \approx 55.1.
Answer
a≈55.1a \approx 55.1

Tip — For “top 10%”, the area to the LEFT is 0.9 — convert before using InvNorm.

Formula recap

a=InvNorm(p,μ,σ)a = \text{InvNorm}(p, \mu, \sigma)
Inverse normal (left area p).
P(X>a)=0.1⇒P(X<a)=0.9P(X > a) = 0.1 \Rightarrow P(X < a) = 0.9
Convert tails.

Common mistakes to avoid

Entering a right-tail probability directly.
InvNorm uses the area to the LEFT — convert first.
Forgetting to enter μ and σ.
The function needs the distribution’s parameters.

Key takeaways

  • Inverse normal: given p, find a with P(X < a) = p.
  • Convert right-tail probabilities to left areas first.
  • Useful for percentile / boundary questions.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The inverse normal function finds:

    • a value from a probability
    • a probability from a value
    • the mean
    • the variance
    Show answer and explanation

    Answer: a value from a probability

    Works backwards.

  2. InvNorm uses the area to the:

    • left
    • right
    • middle
    • either
    Show answer and explanation

    Answer: left

    Cumulative from the left.

  3. For “top 10%”, the left area to enter is:

    • 0.90.9
    • 0.10.1
    • 0.50.5
    • 0.950.95
    Show answer and explanation

    Answer: 0.90.9

    1 − 0.1.

  4. For N(50,42)N(50, 4^2), the value with P(X<a)=0.9P(X<a)=0.9 is about:

    • 55.155.1
    • 5050
    • 4545
    • 5858
    Show answer and explanation

    Answer: 55.155.1

    InvNorm(0.9,50,4).

  5. You must also supply the function with:

    • μ and σ
    • only p
    • n and p
    • the median
    Show answer and explanation

    Answer: μ and σ

    Distribution parameters.

Next step

Check your understanding

Work through questions on The Inverse Normal Distribution. Use the feedback to decide what needs another look.

2 past-paper questions · 7 marks

Practise this topic

Related lessons

All 7 lessons

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