Practice Maths

The Normal Distribution

Key Terms

The normal distribution N(μ, σ²) is symmetric and bell-shaped, centred at μ
Approximately 68% of data lies within 1σ of μ; 95% within 2σ; 99.7% within 3σ
The standard normal Z ~ N(0, 1) is obtained by the transformation Z = (X − μ) / σ
P(X < x) is found using CAS: normCdf(−∞, x, μ, σ)
P(a < X < b) = normCdf(a, b, μ, σ) on CAS
The total area under the normal curve = 1; the curve never touches the x-axis
Key formulas:
Z-score:   Z = (X − μ) / σ

68–95–99.7 rule:
P(μ − σ < X < μ + σ) ≈ 0.68
P(μ − 2σ < X < μ + 2σ) ≈ 0.95
P(μ − 3σ < X < μ + 3σ) ≈ 0.997

By symmetry: P(X < μ) = P(X > μ) = 0.5
Worked Example 1: X ~ N(50, 9). Find P(X < 53).

Method 1 (Z-score):
Z = (53 − 50) / 3 = 1
P(X < 53) = P(Z < 1) ≈ 0.8413

Method 2 (CAS):
normCdf(−1099, 53, 50, 3) = 0.8413
Worked Example 2: X ~ N(100, 225). Find P(85 < X < 115).

σ = √225 = 15
Z1 = (85 − 100)/15 = −1;   Z2 = (115 − 100)/15 = 1
P(85 < X < 115) = P(−1 < Z < 1) ≈ 0.6827
(Or by CAS: normCdf(85, 115, 100, 15) = 0.6827)
Hot Tip: Always check whether the question gives σ (standard deviation) or σ² (variance). In N(μ, σ²) notation, the second parameter is variance. For CAS, you always enter the standard deviation σ.

The Shape and Properties of the Normal Distribution

The normal distribution is the most important continuous probability distribution in statistics. Its probability density function creates the famous bell-shaped curve. Any normally distributed variable X is described by two parameters: the mean μ (centre of the bell) and the variance σ² (spread of the bell). We write X ~ N(μ, σ²).

μ μ−σ μ+σ μ−2σ μ+2σ 68% 95%

Key properties of the normal distribution:

  • Perfectly symmetric about the mean μ
  • The mean, median and mode are all equal to μ
  • The curve extends to ±∞ but approaches zero rapidly
  • Total area under the curve = 1
  • Increasing σ makes the curve wider and flatter; decreasing σ makes it taller and narrower

The 68–95–99.7 Rule

For any normal distribution X ~ N(μ, σ²):

  • About 68% of values lie within 1 standard deviation of the mean: P(μ−σ < X < μ+σ) ≈ 0.6827
  • About 95% of values lie within 2 standard deviations: P(μ−2σ < X < μ+2σ) ≈ 0.9545
  • About 99.7% of values lie within 3 standard deviations: P(μ−3σ < X < μ+3σ) ≈ 0.9973

This rule allows quick mental estimates without CAS. For example, if X ~ N(70, 16) (so σ = 4), then about 95% of values lie between 62 and 78.

The Standard Normal Distribution and Z-Scores

The standard normal distribution Z ~ N(0, 1) has mean 0 and standard deviation 1. Any normal distribution can be converted to the standard normal using the standardisation formula:

Z = (X − μ) / σ

The Z-score tells you how many standard deviations a value is from the mean. A Z-score of +2 means the value is 2 standard deviations above the mean; Z = −1.5 means 1.5 standard deviations below.

Example: Heights of adults are normally distributed with μ = 175 cm and σ = 8 cm. A person is 191 cm tall. Their Z-score is (191 − 175)/8 = 2. This person is 2 standard deviations above average.

Finding Probabilities Using CAS

On your CAS calculator:

  • P(X < b): normCdf(−1099, b, μ, σ)
  • P(X > a): normCdf(a, 1099, μ, σ)
  • P(a < X < b): normCdf(a, b, μ, σ)

Important: Enter σ (standard deviation) in CAS, not σ² (variance). If given X ~ N(50, 16), then μ = 50 and σ = √16 = 4.

Strategy: Always draw and shade a diagram before calculating. This helps you see whether to use 1−P or whether the answer should be greater or less than 0.5, avoiding sign errors.

Using Symmetry

Because the normal distribution is symmetric about μ:

  • P(X > μ + k) = P(X < μ − k) for any k > 0
  • P(X < a) + P(X > a) = 1, so P(X > a) = 1 − P(X < a)
  • P(Z < −z) = P(Z > z) = 1 − P(Z < z)

These symmetry properties reduce the number of distinct calculations needed.

Mastery Practice

See Answers ➔
  1. X ~ N(20, 4). Find: (a) μ and σ   (b) P(X < 22) using CAS   (c) P(X > 20)
  2. For X ~ N(50, 100), use the 68–95–99.7 rule to find: (a) P(40 < X < 60)   (b) P(X > 70)
  3. Standardise the following values from X ~ N(30, 25): (a) x = 35   (b) x = 22   (c) x = 30
  4. X ~ N(0, 1). Use CAS to find: (a) P(Z < 1.5)   (b) P(Z > −0.8)   (c) P(−1 < Z < 2)
  5. The heights of a group of adults are normally distributed with mean 172 cm and standard deviation 6 cm. Find the probability that a randomly selected adult is:
    • (a) Shorter than 165 cm
    • (b) Taller than 180 cm
    • (c) Between 166 cm and 178 cm tall
  6. X ~ N(45, 64). Find P(38 < X < 55). Show your working including the Z-scores used.
  7. Explain why P(X < μ) = 0.5 for any normal distribution. Use a diagram to support your explanation.
  8. A normal distribution has P(X < 70) = 0.8413 and σ = 5. Use the Z-score equation to find the mean μ.
  9. The mass of avocados at a supermarket is normally distributed with μ = 280 g and σ = 20 g.
    • (a) An avocado is chosen at random. Find the probability its mass is between 260 g and 320 g.
    • (b) Avocados under 250 g are sold at a discount. What percentage of avocados are sold at a discount?
    • (c) If 500 avocados are in stock, how many would you expect to be over 310 g?
  10. Two factories produce light bulbs. Factory A produces bulbs with lifetimes X ~ N(1200, 10000) hours and Factory B produces bulbs with lifetimes Y ~ N(1150, 3600) hours. An individual bulb is selected at random from each factory.
    • (a) Find P(X > 1300) and P(Y > 1300).
    • (b) Which factory is more likely to produce a bulb that lasts over 1300 hours? Justify your answer.
    • (c) Find P(Y < 1000). What does this suggest about Factory B’s consistency?