The idea, step by step
What a normal model describes
A normal distribution is a continuous, symmetric bell-shaped model. Its mean μ locates the centre and its standard deviation σ>0 sets the spread. In the notation X~N(μ,σ²), the second parameter is variance, not standard deviation.
The total area under the density curve is one. Probabilities are areas over intervals. The probability of one exact value is zero in a continuous model, even though intervals near the centre can have high probability.
Standardise and assess the approximation
Use z=(x-μ)/σ to measure how many standard deviations a value lies above or below the mean. About 68%, 95% and 99.7% lie within one, two and three standard deviations in a normal model. These are useful checks, not replacements for a requested precise probability.
A real data set need not be normal simply because a mean and standard deviation have been calculated. Look for symmetry, a single bell-shaped group and suitable context; severe skewness or multiple groups weakens the model.
A common trap
In N(500,16), the standard deviation is 16.
Instead: With this notation, 16 is variance and the standard deviation is √16=4.
See the thinking
Read the notation
Packet masses follow X~N(500,16) in grams. State μ and σ, and give the approximate central 95% interval.
- The variance is 16 g², so σ=4 g.
- Two standard deviations is 8 g.
- The approximate interval is 500±8.
Standardise a score
A normal score model has μ=70 and σ=8. Standardise a score of 82.
- Subtract the mean: 82-70=12.
- Divide by the standard deviation: 12/8=1.5.
- The score is 1.5 standard deviations above the modelled mean.
Check your understanding
For X~N(30,25), what is P(X<30)?
Show answer
0.5.
The mean is the symmetry centre, so half of the area lies below it.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- For X~N(40,9), state the mean and standard deviation.3 marks
- A normal model has μ=100 and σ=15. Standardise x=85.3 marks
- For μ=60 and σ=5, give the approximate central 68% interval.3 marks
- For μ=60 and σ=5, give the approximate central 99.7% interval.3 marks
- In a continuous normal model, find P(X=μ).3 marks
- A histogram has two well-separated peaks. Is a single normal model automatically justified by knowing its mean and SD?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- For X~N(40,9), state the mean and standard deviation.3 marks
The second parameter is variance 9.
Take its positive square root.
μ=40; σ=3. - A normal model has μ=100 and σ=15. Standardise x=85.3 marks
(85-100)/15=-1.
z=-1. - For μ=60 and σ=5, give the approximate central 68% interval.3 marks
Use μ±σ.
55 to 65. - For μ=60 and σ=5, give the approximate central 99.7% interval.3 marks
Use μ±3σ.
45 to 75. - In a continuous normal model, find P(X=μ).3 marks
A single point has zero interval width and area.
This does not mean values near μ are unlikely.
0. - A histogram has two well-separated peaks. Is a single normal model automatically justified by knowing its mean and SD?3 marks
Investigate distinct groups or another distribution.
No. The bimodal shape suggests that one bell-shaped model may be unsuitable.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: statistical models. Original lessons and assessment practice; not an official SACE assessment.
Official curriculum source ↗