Statistics / Probability

Normal Distribution & Variance

Variance (σ²) controls how spread out values fall around the mean (μ = 0)

σ² = σ =

Peak height =  ·  ≈68% of values fall within ±σ =

Key intuition

  • Variance σ² > 0 always. It's the average squared distance from the mean. Squaring eliminates negatives.
  • Larger σ² → flatter, wider bell → values more spread out, more uncertainty.
  • Smaller σ² → taller, narrower bell → values cluster tightly around the mean.
  • σ² = 0 only if the variable is a constant. σ² < 0 is mathematically impossible.
  • The area under every curve = 1. Total probability is always 100%, just differently shaped.