The normal distribution · Grade 12
In Grade 12 you learn: standardising with z.
Many measurements, such as heights, masses and exam marks, follow a symmetrical bell-shaped curve called the normal distribution. It is described by its mean μ and standard deviation σ, written X ~ N(μ, σ²). Probabilities are areas under the curve.
Key facts
- The curve is symmetrical about the mean, so mean = median = mode.
- The total area under the curve is 1.
- Standardise with z = (x − μ)/σ. Then Z ~ N(0, 1).
- About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
- P(X > x) = 1 − P(X < x) and P(a < X < b) = P(X < b) − P(X < a).
- By symmetry, P(Z < −z) = 1 − P(Z < z).
- For a continuous variable, P(X = x) = 0. So < and ≤ give the same answer.
Worked examples
Example 1
X ~ N(50, 4²). Find P(X < 56), to 4 decimal places.
Answer: 0.9332
Example 2
X ~ N(50, 4²). Find P(46 < X < 55), to 4 decimal places.
Answer: 0.7357