All subjects · Mathematics · WST01 (S1)

Inverse normal: finding values from probabilities

Sometimes the probability is given and you need the value of X. Work backwards: find z from the table, then use x = μ + zσ.

The percentage points table

The table of percentage points gives z for P(Z > z) = p for common p. Use these values to 4 decimal places, as examiners require: for example P(Z > z) = 0.05 gives z = 1.6449, 0.1 gives 1.2816, 0.2 gives 0.8416, 0.025 gives 1.9600, 0.01 gives 2.3263.

For other probabilities use the Φ table in reverse and take the nearest z.

Method

1. Sketch: is the value above or below the mean? Which tail has the given area?

2. Find z (negative if the value is below the mean).

3. Solve (x - μ)/σ = z, so x = μ + zσ.

Example: X ~ N(80, 102), P(X < a) = 0.95. Upper tail 0.05 gives z = 1.6449, so a = 80 + 16.449 = 96.4.

Lower tail values

For P(X < c) = 0.1, the value is below the mean, so z = -1.2816 and c = 80 - 12.816 = 67.2. Quartiles of a normal distribution use z ≈ ±0.67 (from Φ(0.67) = 0.7486, the nearest table value to 0.75), so IQR ≈ 2 × 0.67σ.

Worked example

X ~ N(80, 102). Find b such that P(X > b) = 0.2.

  1. P(Z > z) = 0.2 gives z = 0.8416
  2. b = 80 + 0.8416 × 10

Answer: b = 88.42, so 88.4 (3 s.f.)

Key idea: Sketch, find z from the percentage points table (4 d.p.), use x = μ + zσ; z is negative below the mean.

Check yourself

  1. X ~ N(80, 102). Find a such that P(X < a) = 0.95. Give your answer to 3 significant figures.
    Show answer

    96.4. z = 1.6449, a = 80 + 1.6449 × 10 = 96.449.

  2. X ~ N(80, 102). Find b such that P(X > b) = 0.2. Give your answer to 3 significant figures.
    Show answer

    88.4. z = 0.8416, b = 80 + 8.416 = 88.416.

  3. X ~ N(80, 102). Find c such that P(X < c) = 0.1. Give your answer to 3 significant figures.
    Show answer

    67.2. z = -1.2816, c = 80 - 12.816 = 67.184.

  4. Find z such that P(Z > z) = 0.025.
    Show answer

    1.96. From the percentage points table, z = 1.9600.

  5. Find z such that P(-z < Z < z) = 0.9. Give your answer to 4 decimal places.
    Show answer

    1.6449. Each tail is 0.05, so z = 1.6449.

  6. Lengths L cm have L ~ N(20, 32). Estimate the upper quartile of L. Give your answer to 3 significant figures.
    Show answer

    22.0. P(Z < z) = 0.75 gives z ≈ 0.67 (table) or 0.6745 (exact); Q3 = 20 + 0.6745 × 3 = 22.02.

  7. X ~ N(80, 102) and P(X < c) = 0.3. Which is true?
    • c is greater than 80
    • c is less than 80, so its z value is negative
    • c = 80 × 0.3
    • c = 80 + 0.3 × 10
    Show answer

    c is less than 80, so its z value is negative. Less than half the area lies below c, so c is below the mean.

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Last updated 7 November 2026. Spotted a mistake? Tell us.