Normal Distribution & Standardization

Compute P(a≤X≤b) for any N(μ,σ²) and see how standardization Z=(X-μ)/σ turns every normal into N(0,1). Build the reflex behind every z-table lookup.

P.02 / NORMAL DISTRIBUTION

Normal Distribution & Standardization

The standard normal was fixed at μ=0, σ=1. Real data has any mean and spread. Move μ and σ to wield the normal as a tool. Standardization maps it back to Z, so any normal connects to the standard normal.

The general version of the standard normal is N(μ, σ²). μ sets the center, σ sets the spread. Slide the parameters and the curve glides; the probability of falling inside [a, b] (pink area) updates live.
That pink area IS the "percentage" you hear in the news. Say adult male heights are N(170, 36) (mean 170cm, σ=6cm). What share falls in 165–175cm? Set μ=170, σ=6, then a=165, b=175 — you get ≈ 59.6%. Test scores, measurement errors, IQ — anything roughly normal gets its "X% of people in this range" from exactly this area.
Tip: drag directly on the graph to move the a/b bounds — whichever handle is closest follows your finger.

f(x) = (1 / √(2πσ²)) · exp( −(x−μ)² / 2σ² )

μ (mean) = 0

σ (std dev) = 1

Interval [a,b] : a = -1

b = 1

P(a ≤ X ≤ b)—

z-score (a)—

z-score (b)—

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