Finding an unknown mean or standard deviation

One unknown

Write (x - μ)/σ = z with the z from the table.

Example: X ~ N(μ, 5²) and P(X > 40) = 0.05. Then (40 - μ)/5 = 1.6449, so μ = 40 - 8.2245 = 31.8 (3 s.f.).

Example: X ~ N(30, σ²) and P(X < 36) = 0.9. Then 6/σ = 1.2816, so σ = 4.68 (3 s.f.).

Two unknowns

P(X < 20) = 0.1 and P(X > 35) = 0.2:

(20 - μ)/σ = -1.2816, so μ - 1.2816σ = 20

(35 - μ)/σ = 0.8416, so μ + 0.8416σ = 35

Subtract: 2.1232σ = 15, so σ = 7.065 and μ = 29.05.

Exam points

• Use z to 4 d.p. from the percentage points table, not 1.28 or 1.282.
• Make sure the sign of z matches the side of the mean (sketch!).
• Show the simultaneous equations and their solution by hand, not just calculator output.

Worked example

X ~ N(μ, σ²), P(X < 20) = 0.1 and P(X > 35) = 0.2. Find μ and σ.

1. μ - 1.2816σ = 20
2. μ + 0.8416σ = 35
3. Subtract: 2.1232σ = 15

Answer: σ = 7.06, μ = 29.1 (3 s.f.)

Check yourself

1. X ~ N(μ, 5²) and P(X > 40) = 0.05. Find μ. Give your answer to 3 significant figures.
   Show answer

   31.8. (40 - μ)/5 = 1.6449, so μ = 40 - 8.2245 = 31.776.

2. X ~ N(30, σ²) and P(X < 36) = 0.9. Find σ. Give your answer to 3 significant figures.
   Show answer

   4.68. 6/σ = 1.2816, so σ = 4.6816.

3. X ~ N(μ, σ²) with P(X < 20) = 0.1 and P(X > 35) = 0.2. Find σ. Give your answer to 3 significant figures.
   Show answer

   7.06. Subtracting the equations: 2.1232σ = 15, σ = 7.0648.

4. X ~ N(μ, σ²) with P(X < 20) = 0.1 and P(X > 35) = 0.2. Find μ. Give your answer to 3 significant figures.
   Show answer

   29.1. μ = 20 + 1.2816 × 7.0648 = 29.054.

5. X ~ N(μ, σ²) with P(X < 12) = 0.3 and P(X > 28) = 0.3. Find μ.
   Show answer

   20. The two tails are equal, so by symmetry μ is halfway: 20.

6. X ~ N(μ, 2²) and P(X < 10) = 0.025. Find μ. Give your answer to 3 significant figures.
   Show answer

   13.9. (10 - μ)/2 = -1.96, so μ = 10 + 3.92 = 13.92.

7. X ~ N(μ, σ²) and P(X > 50) = 0.1. Which equation is correct?
   • (50 - μ)/σ = -1.2816
   • (50 - μ)/σ = 1.2816
   • (50 - μ)/σ = 0.1
   • (μ - 50)/σ = 1.2816
   Show answer

   (50 - μ)/σ = 1.2816. 50 is above the mean with upper tail 0.1, so z = +1.2816.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1
• Pearson Edexcel IAL Statistics S1 (WST01) examiners' report, October 2024

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