Probability

Normal Distribution & Variance

Variance ($\sigma^2$) controls how spread out values fall around the mean ($\mu = 0$)

$\sigma^2$ = σ =

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

Key intuition

  • Variance $`\sigma^2 > 0`$ always. It's the average squared distance from the mean. Squaring eliminates negatives.
  • Larger $`\sigma^2`$ → flatter, wider bell → values more spread out, more uncertainty.
  • Smaller $`\sigma^2`$ → taller, narrower bell → values cluster tightly around the mean.
  • $`\sigma^2 = 0`$ only if the variable is a constant. $`\sigma^2 < 0`$ is mathematically impossible.
  • The area under every curve = 1. Total probability is always 100%, just differently shaped.