6.2.1—Normal distribution

For X~N(μ,σ²), z=(x−μ)/σ converts a value to the standard normal scale. Interval and tail probabilities follow from the standard CDF.

Sketch the region, standardise both bounds and use symmetry or complements consistently. Keep the distinction between variance σ² and standard deviation σ.

If μ=50 and σ=8, P(42<X<58)=P(−1<Z<1)≈0.6827.

Using σ² in the denominator produces a dimensionally wrong z-score.