Practice Maths

Using the Normal Distribution

Key Terms

Inverse normal
Given a probability p, find the value x such that P(X < x) = p
CAS: invNorm(p, μ, σ) gives the value x with P(X < x) = p
To find μ with known σ: use Z = (x − μ)/σ and the known Z-score
To find σ with known μ: rearrange Z = (x − μ)/σ for σ
To find both μ and σ: set up two equations from two known probabilities and solve simultaneously
Inverse Normal (CAS):
If P(X < x) = p, then x = invNorm(p, μ, σ)

Finding unknowns:
Given P(X < x1) = p1:   Z1 = invNorm(p1, 0, 1)
Then: x1 = μ + Z1σ

With two equations:
x1 = μ + Z1σ   and   x2 = μ + Z2σ
Worked Example 1 (Inverse Normal): X ~ N(60, 25). Find a such that P(X < a) = 0.9.

a = invNorm(0.9, 60, 5) ≈ 66.41

Or using Z-scores: P(Z < z) = 0.9 ⇒ z ≈ 1.282
a = μ + zσ = 60 + 1.282 × 5 ≈ 66.41
Worked Example 2 (Finding μ and σ): X is normally distributed. P(X < 50) = 0.2 and P(X < 70) = 0.9. Find μ and σ.

From P(X<50) = 0.2: Z1 = invNorm(0.2, 0, 1) ≈ −0.8416
From P(X<70) = 0.9: Z2 = invNorm(0.9, 0, 1) ≈ 1.2816
Equations: 50 = μ − 0.8416σ … (i)
70 = μ + 1.2816σ … (ii)
(ii) − (i): 20 = 2.1232σ ⇒ σ ≈ 9.42
μ = 50 + 0.8416 × 9.42 ≈ 57.93
Hot Tip: For "find μ and σ" problems, always convert the probabilities to Z-scores FIRST using the standard normal, then write equations x = μ + zσ. Solving two such equations simultaneously gives both unknowns cleanly.

Inverse Normal Problems

So far we have found probabilities given values of x. The inverse problem asks: given a probability, what is the value of x? This is called an inverse normal calculation.

Example: Exam scores are distributed as X ~ N(65, 100). The top 10% of students receive an A. What is the minimum score to get an A?

We want x such that P(X > x) = 0.10, i.e., P(X < x) = 0.90.
x = invNorm(0.90, 65, 10) ≈ 77.8. So students need at least 78 marks for an A.

Percentiles and quantiles: The p-th percentile is the value below which p% of the data falls.

  • Median = 50th percentile = μ (for any normal distribution)
  • Lower quartile Q1 = 25th percentile = invNorm(0.25, μ, σ)
  • Upper quartile Q3 = 75th percentile = invNorm(0.75, μ, σ)

Finding μ When σ Is Known

If you know σ and are given a probability condition, use the Z-score to find μ:

  1. Find the Z-score corresponding to the given probability.
  2. Write the equation: x = μ + zσ
  3. Solve for μ

Example: X ~ N(μ, 16). P(X < 30) = 0.75. Find μ.
Z = invNorm(0.75, 0, 1) ≈ 0.6745
30 = μ + 0.6745 × 4
30 = μ + 2.698
μ ≈ 27.30

Finding σ When μ Is Known

If you know μ and are given a probability condition, use the Z-score to find σ:

  1. Find the Z-score corresponding to the given probability.
  2. Write the equation: x = μ + zσ
  3. Solve for σ

Example: X ~ N(100, σ²). P(X < 120) = 0.9. Find σ.
Z = invNorm(0.9, 0, 1) ≈ 1.2816
120 = 100 + 1.2816σ
20 = 1.2816σ
σ ≈ 15.61

Finding Both μ and σ

With two probability conditions, you can find both μ and σ simultaneously. This requires solving two linear equations.

  1. Convert each probability condition to a Z-score.
  2. Write two equations: x1 = μ + z1σ and x2 = μ + z2σ
  3. Subtract to eliminate μ and solve for σ
  4. Back-substitute to find μ
Exam tip: When solving for μ and σ, always verify your answers by substituting back and checking both probability conditions. A small arithmetic error can be caught this way.

Applications of the Normal Distribution

The normal distribution models many real-world phenomena: heights, weights, test scores, measurement errors, manufacturing tolerances, and financial returns. Key application types include:

  • Quality control: Finding what proportion of items meet a specification
  • Percentile problems: Finding cut-off scores or values
  • Working backwards: Finding parameters from observed probabilities

Mastery Practice

See Answers ➔
  1. X ~ N(40, 9). Find: (a) the value a such that P(X < a) = 0.95   (b) the value b such that P(X > b) = 0.2
  2. X ~ N(50, 16). Find the values of the lower quartile (Q1) and upper quartile (Q3).
  3. X ~ N(μ, 25). Given that P(X < 45) = 0.8413, find μ.
  4. X ~ N(70, σ²). Given that P(X < 80) = 0.9772, find σ.
  5. The mass of packets of cereal is normally distributed. The 10th percentile is 490 g and the 90th percentile is 510 g. Find the mean and standard deviation.
  6. X is normally distributed with P(X < 25) = 0.3 and P(X < 40) = 0.85. Find μ and σ, giving your answers to 2 decimal places.
  7. A machine produces bolts whose diameter X ~ N(10, 0.04) mm. Bolts are acceptable if their diameter is between 9.5 mm and 10.5 mm. What percentage of bolts are acceptable?
  8. IQ scores are modelled by X ~ N(100, 225). Mensa membership requires an IQ in the top 2% of the population. Find the minimum IQ score required for Mensa membership.
  9. A factory makes two types of components. Type A has lengths normally distributed with mean 50 mm and standard deviation 2 mm. Type B has lengths normally distributed with mean 80 mm and standard deviation 3 mm. Components are assembled end-to-end (total length = A + B).
    • (a) Find the probability that a Type A component is shorter than 48 mm.
    • (b) Find the minimum length exceeded by 95% of Type B components.
    • (c) If the specification requires Type A components between 47 mm and 53 mm, and Type B components between 75 mm and 85 mm, what percentage of each type meets the specification?
  10. The heights of students at a school are normally distributed. 20% of students are shorter than 160 cm and 15% are taller than 180 cm.
    • (a) Set up two equations in μ and σ using Z-scores.
    • (b) Solve to find μ and σ (to 2 decimal places).
    • (c) Find the probability that a randomly chosen student is between 165 cm and 175 cm tall.