A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Probability and statistics
CSCA Mathematics · Lesson 16 of 16Statistics and the normal distribution
Statistiques et loi normale · الإحصاء والتوزيع الطبيعي
▶ Watch the video lesson (A7LAM students)
Statistics describes data with a centre and a spread. The normal distribution is the bell curve behind many real measurements.
Key ideas
- Mean x̄ = (x₁ + … + xₙ)/n; the median is the middle value of the sorted data.
- Variance s² = average of (xᵢ − x̄)²; standard deviation s = √s².
- Normal N(μ, σ²): a symmetric bell curve centred at μ.
- About 68% of values lie within μ ± σ and 95% within μ ± 2σ.
Formulas to know
- s² = [(x₁ − x̄)² + … + (xₙ − x̄)²] / n
- Normal: P(X < μ) = P(X > μ) = 0.5
- Data × k: mean × k, variance × k²
Worked example
Find the mean and variance of 2, 4, 4, 4, 5, 5, 7, 9.
- Mean: 40 / 8 = 5
- Squares: 9, 1, 1, 1, 0, 0, 4, 16 → sum 32
- Variance: 32 / 8 = 4, so s = 2
Answer: Mean 5, variance 4
Practice questions
1. Median of 3, 9, 1, 7, 5?
5 ✓ · 7 · 3 · 25
Sorted: 1, 3, 5, 7, 9.
2. X ~ N(100, 15²). P(X > 100) = ?
0.25 · 0.5 ✓ · 0.68 · 1
The curve is symmetric about μ = 100.
3. Every value in a data set is doubled. The variance:
stays the same · doubles · is multiplied by 4 ✓ · is halved
Variance scales with k² = 4.