A standard normal Z ~ N(0,1). What fraction of mass lies within ±1, ±2, and ±3 standard deviations, and what is E[|Z|]?

How to answer

≈68% within ±1σ, ≈95% within ±2σ, ≈99.7% within ±3σ (the 68-95-99.7 rule). E[|Z|] = √(2/π) ≈ 0.80. These are memorize-cold facts used to sanity-check probabilities and confidence intervals on the fly.

Key idea: 68-95-99.7; mean absolute deviation of a standard normal is √(2/π).