The reasoning: properties of the normal distribution
- Symmetric, bell‑shaped, unimodal; mean = median = mode.
- Defined by μ and σ; total area = 1.
- Empirical rule: μ ± 1σ ≈ 68.3%; ± 1.96σ = 95%; ± 2.58σ = 99%; ± 3σ ≈ 99.7%.
- Standard normal: μ = 0, σ = 1; Z = (x − μ)/σ.
- Skewness = 0; kurtosis = 3 (mesokurtic).