04 // Interactive Formula Compendium // STATISTICSf(x) = [ 1 / (σ√(2π)) ] · exp(−½ · ((x − μ) / σ)²)
Normal (Gaussian) Probability Density & Z-Score Formula
The Gaussian probability density function f(x) = (1/(σ√(2π)))·e^(−(x−μ)²/(2σ²)) defines the symmetric bell curve whose integral over ℝ equals 1 and whose z-score z = (x−μ)/σ measures standard deviations from mean μ.
Formula Reference
Main Formula
f(x) = [ 1 / (σ√(2π)) ] · exp(−½ · ((x − μ) / σ)²)
Rearranged Forms
z = (x − μ) / σ
Φ(z) = ½ · [1 + erf(z / √2)]
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| Probability Density f(x) | density | |
| x | Observation x | units |
| mu | Population Mean μ | units |
| sigma | Standard Deviation σ | units |
Inputs & Parameters
[units]
[units]
[units]
Calculated Result (Probability Density f(x))
pdf =0.016131density
Formula Notes & Background
- Normalized via the Gaussian integral ∫_{−∞}^{∞} e^(−u²/2) du = √(2π).
- Maximizes differential Shannon entropy among all continuous real distributions with fixed variance σ².
Step-by-Step Examples
#1: Standardized score x = 115 on a scale with μ = 100 and σ = 15.
x = 115μ = 100σ = 15
z = (115 − 100)/15 = +1.00; f(115) = (1 / (15√(2π))) · e^(−0.5)
→ z = +1.00, f(115) = 0.01613, cumulative percentile Φ(1.0) = 84.13%
#2: Empirical rule coverage within ±1σ, ±2σ, and ±3σ of the mean.
z = ±1, ±2, ±3
erf(z / √2) evaluated at z = 1, 2, 3
→ 68.27% (±1σ), 95.45% (±2σ), 99.73% (±3σ)
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Frequently Asked Questions
Can probability density f(x) ever exceed 1?+
Yes! When σ < 1/√(2π) ≈ 0.3989, peak density f(μ) exceeds 1 because density has units of 1/x; only integrated area (probability) is bounded by 1.
Why does the normal distribution appear everywhere in nature?+
By the Central Limit Theorem, the sum or average of many independent finite-variance random variables converges toward a Gaussian distribution regardless of their individual shapes.
What z-score corresponds to a two-sided 95% confidence interval?+
Exactly z = ±1.95996 (commonly rounded to ±1.96), leaving 2.5% in each tail.