The normal distribution

AS Level Mathematics · Probability & Statistics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

The normal distribution models a continuous variable, such as height or mass, whose values cluster symmetrically around the mean in a bell-shaped curve. It is written N(μ, σ2): the second number is the variance, so take its square root to get the standard deviation σ. Probabilities are areas under the curve.

To use the tables, standardise: change x into z, the number of standard deviations from the mean. The tables give Φ(z) = P(Z < z) for positive z. Use symmetry for negative z.

A binomial distribution with large n can be approximated by a normal distribution with the same mean and variance, using a continuity correction.

Rules to remember

Common mistake

Dividing by the variance instead of the standard deviation. In N(30, 25) the standard deviation is 5, not 25.

Worked example

X has the distribution N(70, 64). Find P(60 < X < 82), to 3 significant figures.

  1. σ = √64 = 8.
  2. z1 = (60 − 70) ÷ 8 = −1.25 and z2 = (82 − 70) ÷ 8 = 1.5.
  3. Φ(1.5) = 0.9332 and Φ(−1.25) = 1 − Φ(1.25) = 1 − 0.8944 = 0.1056.
  4. P = 0.9332 − 0.1056 = 0.8276.

Answer: 0.828

Practice questions

Try each one, then open the answer.

1. X has the distribution N(μ, 9) and P(X > 50) = 0.1. Find μ, to 3 significant figures.

  1. A
    53.8
  2. B
    46.2
  3. C
    38.5
  4. D
    48.5
Show answer

Answer: B. P(Z > z) = 0.1 gives z = 1.282. Then (50 − μ)/3 = 1.282, so μ = 50 − 3.85 = 46.2. σ is 3 (not 9), and μ must be below 50 since only 10% lies above 50.

2. X is normally distributed with mean 50 and standard deviation 4. Find P(X > 56), to 3 significant figures.

  1. A
    0.933
  2. B
    0.433
  3. C
    0.0668
  4. D
    0.0228
Show answer

Answer: C. z = (56 − 50)/4 = 1.5. P(Z > 1.5) = 1 − Φ(1.5) = 1 − 0.9332 = 0.0668. Remember to subtract from 1 for the upper tail.

3. X has the distribution N(30, 25). Find P(X < 22), to 3 significant figures.

  1. A
    0.945
  2. B
    0.375
  3. C
    0.0668
  4. D
    0.0548
Show answer

Answer: D. The variance is 25, so σ = 5. z = (22 − 30)/5 = −1.6, and P(Z < −1.6) = 1 − Φ(1.6) = 1 − 0.9452 = 0.0548. Using σ = 25 wrongly gives 0.375.

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