Gaussian (Normal) Distribution

Equivalent forms

\int_{-\infty}^{\infty} p(x)\,dx = 1

P(|x-\mu|<\sigma) \approx 0.68

The unique distribution that maximizes entropy for a fixed mean and variance — nature's least-committal guess.

Symbol	Name	SI unit	Dimension	Range
$\mu$	Mean Center of the distribution	1	1	-3 – 3
$\sigma$	Standard deviation Width of the bell	1	1	0.3 – 2.5
$x$	Value Random variable	1	1	-6 – 6
$p$	Probability densityoutput Height of the curve at x	1	1	0 – 1.4

Unit systems

Symbol	SI	CGS	Imperial
$x$	1	1	1
$\mu$	1	—	—
$\sigma$	1	—	—

Where it holds

Defined for all real x; models sums of many small independent effects. Poor for heavy-tailed or bounded data.

Dimensional analysis

p (dim 1/x)

e^{-(x-\mu )^{2}/2\sigma ^{2}}/(\sigma \sqrt{2}\pi )

Exponent is dimensionless (x,

\mu

,

\sigma

share units); prefactor

1/\sigma

carries

1/x so \int p dx

is dimensionless.

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Discovery

Gauss & Laplace · 1809

De Moivre found the curve approximating binomials in 1733; Gauss derived it in 1809 from his theory of measurement errors while tracking the asteroid Ceres, and Laplace tied it to the Central Limit Theorem. It became the default model of noise across all of science.

Try this

Heights, measurement errors, thermal velocities, exam scores — wildly different things pile up into the same bell. Why does nature keep drawing this one curve?

Drag the mean to slide the bell and the width to fatten or pinch it; the area under the curve stays locked at 1 no matter what.

Research status: stable

Real-world applications

Modeling measurement noise and instrument error
Maxwell–Boltzmann velocity components in gases
Diffusion / heat-kernel solutions
Statistical hypothesis testing and confidence intervals

Common misconceptions

All bell-shaped data is Gaussian — many humps are Student-t, logistic, or Cauchy with fatter tails.
68% of data always lies within $one \sigma$ — only for truly normal data.
The peak height is the probability — it's a density; probability is area.

Experimental verification

Galton's bean machine (quincunx) physically stacks beads into a bell; thermal (Maxwell–Boltzmann) velocity components are Gaussian, verified by molecular-beam experiments.

Derivation

Require a density depending only on

x-\mu

, factorizing across independent coordinates, and rotationally symmetric in several dimensions.

These force

f(x)\propto e^{-\alpha x^{2}}

; normalizing

\int f=1

with variance

\sigma ^{2}

fixes

\alpha =1/(2\sigma ^{2})

and the prefactor.

Limiting cases

\sigma \to 0

⟶ spike

at \mu

The bell narrows toward a Dirac delta — certainty.

\sigma

large⟶ flat, broad humpHigh uncertainty; density spreads thin.

x = \mu

⟶ peak

= 1/(\sigma \sqrt{2}\pi )

Maximum density sits at the mean.

What if…

What if I add two independent Gaussians?

You get another Gaussian: means add and variances add. The family is closed under convolution.

What if the data is bounded (0 to 1)?

The Gaussian leaks outside the range; use a Beta or truncated normal instead.

1

Probability within one sigma

Given ·

\mu:: 0
\sigma:: 1

Find · P(-1<x<1)

Steps

Integrate the standard normal from -1 to 1
Use the error function $erf(1/\sqrt{2})$
$\approx 68.3$ %

Result ·

\approx 0.6827

2

Peak height for σ = 2

Given ·

\sigma:: 2

Find · max density

Steps

Peak $= 1/(\sigma \sqrt{2}\pi )$
$\sigma \sqrt{2}\pi = 2\cdot 2.5066$
$\approx 0.199$

Result ·

1/(2\sqrt{2}\pi ) \approx 0.199

Central Limit Theorem→Gamma Function→information entropy shannonrandom walk