Normal Distribution & Standardization
What percent of men stand between 165 and 175 cm? About 59.6%. Move μ and σ — and a 'percentage' shows up as visible area.
Normal Distribution — Shaping Mean & Spread
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 follow N(170, 36) (mean 170cm, σ=6cm). What share falls in 165–175cm?
Standardize and compute z-scores — you get ≈ 59.6%.
Test scores, measurement errors, IQ — anything roughly normal gets its "X% of people in this range" from exactly this area.
The sliders below use a standardized scale (μ=0, σ=1 range) so you can feel the same principle.
Tip: drag directly on the graph to move the a/b bounds — whichever handle is closest follows your finger.
- Step 1: Keep μ=0, σ=1, set a=−1, b=1 → ~68.3%. That's "±1σ covers ~70%."
- Step 2: Shrink σ to 0.5 → the pink area for the same [−1, 1] explodes. Less spread = almost everyone is in range.
- Step 3: Slide μ to 2 → the whole curve shifts. Same a,b, but area changes dramatically.
- Step 4: Drag directly on the graph → the nearest a/b boundary follows your finger.
// Formula used here
Each part
• (x − μ)²: squared distance from mean → creates the symmetric bell shape
• ÷ 2σ²: larger σ means slower exponential decay → wider, flatter curve
• exp(…): drops toward zero for extreme values → "rare events are rare"
• 1/√(2πσ²): adjusts height so total area always equals 1
Relation to standard normal
• Plug in μ=0, σ=1 and you recover φ(z) = e−z²/2/√(2π)
• General normal = standard normal stretched horizontally by σ and shifted by μ
// Common misconceptions
σ measures how much individual data points vary. Estimation stability is captured by SE = σ/√n. Even with σ = 100, if n = 10,000 then SE = 1 — the estimate is quite stable.
Heights and measurement errors are close to normal, but income (right-skewed), click-through rates (piled near 0), and failure times (exponential) are not. Many real distributions are far from normal.
If a distribution is normal, these three coincide. But the converse doesn't hold — for example, the symmetric Beta(2,2) distribution also has mean = mode = median. Note that the uniform distribution has equal mean and median, but since every value is equally likely, no unique mode exists.
// Shapes you'll meet again
Around the normal distribution, the same standardize-and-look-up flow keeps reappearing.
- The z-transform to table-lookup flow: for X ~ N(50, 10²), looking at X ≥ 65 reshapes into z = (65−50)/10 = 1.5. "Translate a general normal onto the z-scale" is the same motion that returns whenever probabilities are in play
- The shape of upper/lower α-points: x = μ + z_α·σ. The "center + multiplier × spread" three-layer structure shows up here too
- Additivity of normals: with independent X ~ N(μ₁,σ₁²) and Y ~ N(μ₂,σ₂²), X+Y ~ N(μ₁+μ₂, σ₁²+σ₂²). Means add, variances add (standard deviations don't) — this pairing is its shape
- The standardization picture: z = (x−μ)/σ. "Measure the gap from center in units of spread" is the same picture that carries over into t-statistics and test statistics
To touch only the N(0,1) base in isolation, see Standard Normal.