Gaussian (Normal) Distribution

The Gaussian, or normal, distribution is the single most important continuous law in physics, and it earns that status three times over. Kinematically it is a shape: the exponential of a downward parabola, the symmetric bell whose density $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$ is fixed completely by two numbers — the mean $\mu$ that locates its peak and the variance $\sigma^2$ that sets its width. Structurally it is an algebraic fixed point: the Gaussian is its own Fourier transform up to scaling, so its characteristic function is again Gaussian, and this closure means a sum of independent normals is normal, with means and variances simply adding. And underneath both it is a statement about information: among all distributions with a given mean and variance the Gaussian has maximum entropy, the least committal choice, which is why it appears whenever a system is constrained by nothing more than its first two moments. These three faces converge on the reason the bell is everywhere. The central limit theorem promotes the Gaussian from one distribution among many to the universal attractor of them all: average enough independent finite-variance random quantities — molecular velocities, measurement errors, diffusive steps, photon counts — and the fluctuations forget every detail of their origin and become Gaussian. This article treats the one-dimensional case in full; the generalisation to correlated vectors is the multivariate Gaussian, and the machinery that manufactures the bell from sums is the central limit theorem — both essential neighbours, recalled here by name.

Key takeaways

The normal distribution $N(\mu,\sigma^2)$ has density $f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\big(-\frac{(x-\mu)^2}{2\sigma^2}\big)$ — a symmetric bell centred at the mean $\mu$ with width set by the standard deviation $\sigma$ , and $\sigma$ is the distance from the peak to the inflection point.
It is fixed by exactly two parameters: the mean $\mu$ (location) and the variance $\sigma^2$ (spread). Its cumulant generating function is a pure quadratic, so every cumulant beyond the second vanishes.
Its characteristic function $\varphi(t)=\exp(i\mu t-\tfrac12\sigma^2 t^2)$ is itself Gaussian, which makes it closed under convolution: sums of independent Gaussians are Gaussian, with means and variances adding.
The central limit theorem makes it universal — the standardised sum of many independent finite-variance contributions converges to $N(0,1)$ regardless of the summands' own shape.
Coverage follows the 68–95–99.7 rule: $\approx68.27\%$ of the area lies within $\pm1\sigma$ , $\approx95.45\%$ within $\pm2\sigma$ , and $\approx99.73\%$ within $\pm3\sigma$ ; the particle-physics discovery bar of $5\sigma$ is a one-sided tail of $2.87\times10^{-7}$ .
Among all laws with fixed mean and variance the Gaussian maximises entropy, $H=\tfrac12\ln(2\pi e\sigma^2)$ — the reason it governs thermal noise, diffusion, and error analysis.
Key idea: A Gaussian is the exponential of a quadratic — completely fixed by its first two moments — and it is closed under both convolution (adding independent variables) and Fourier transform, which together explain its universality through the central limit theorem.

Where this lives today

The Gaussian is the quiet default of modern quantitative science, and its sigma language runs through the biggest experiments. In particle physics the discovery of the Higgs boson in July 2012 was announced only once the signal reached $5\sigma$ over background — the one-sided tail of $2.87\times10^{-7}$ computed above — because a Gaussian model of statistical fluctuations sets that bar. In gravitational-wave astronomy the LIGO detectors fight Gaussian noise floors, and the quantum limit of that noise is now beaten with squeezed light, a Gaussian quantum state whose position–momentum fluctuations are redistributed below the symmetric minimum-uncertainty bell. Across machine learning, Gaussian processes treat the normal density as the tractable building block of uncertainty, and the diffusion models behind modern image generators literally run the heat equation backwards, denoising Gaussian noise into structure. In metrology and finance the same curve underwrites confidence intervals and risk estimates — with the standing caveat, learned the hard way in 2008, that real market tails are heavier than Gaussian, exactly the finite-variance failure mode the warning above flags.

The physical picture

Read the density backwards, from the exponent out. The core is $-(x-\mu)^2/2\sigma^2$ : a downward parabola in $x$ , centred at $\mu$ . Exponentiate it and you get a curve that is symmetric about its peak, falls off faster than any power law yet more gently than a step, and has the characteristic gentle shoulders and rapidly vanishing tails that are the signature of the $-x^2$ upstairs. The prefactor $1/(\sigma\sqrt{2\pi})$ does one job only — it scales the total area to one. Because area is conserved, a narrower bell is a taller bell: halve $\sigma$ and the peak height $1/(\sigma\sqrt{2\pi})$ doubles. The single most useful geometric fact is that $\sigma$ is not an abstract width but a visible landmark — it is exactly the distance from the peak to the inflection point, where the curvature of the bell reverses sign. The first figure makes that landmark, the peak height, and the 68–95–99.7 coverage one picture.

Figure

The standard normal N(μ, σ²): peak height 1/(σ√2π), inflection points at μ ± σ where the curvature reverses, and the 68–95–99.7 rule — 68.27% of the area lies within ±1σ, 95.45% within ±2σ, and 99.73% within ±3σ

Why should this particular shape dominate physics rather than some other? Because it is the fixed point of averaging. Add many independent small nudges, each with finite variance, and the total forgets the shape of every individual nudge — only the summed mean and summed variance survive, and what is left is Gaussian. Run the averaging again and nothing changes: the bell reproduces itself. That self-reproduction under summation is the central limit theorem seen as a dynamical system, and the second figure shows it happening — a flat box distribution, averaged with copies of itself, marching through a triangle to a bell in only a handful of steps.

Figure

The central limit theorem in action: averaging independent uniform variables, the density flattens from a box (n = 1) through a triangle (n = 2) to a bell (n = 8). For any finite-variance summands the standardized sum converges to N(0, 1), while means add and variances add along the way

The core equations

The density itself is the object everything else describes. Two parameters enter — a location $\mu$ and a scale $\sigma$ — and no others; a Gaussian carries no shape information beyond its centre and its width.

f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \qquad (1)

where: $x$ — the continuous random variable; $\mu$ — mean, the location of the peak, equal to $\mathbb{E}[X]$ ; $\sigma$ — standard deviation, the width, with $\sigma>0$ ; $\sigma^2$ — variance, $\mathrm{Var}(X)$ . Reading it: the peak sits at $x=\mu$ with height $1/(\sigma\sqrt{2\pi})$ , and the curve has dropped to $e^{-1/2}\approx0.607$ of that height one standard deviation out, at the inflection points $x=\mu\pm\sigma$ . Probabilities are areas under this curve, and because the Gaussian integral has no elementary antiderivative they are written through the error function. Standardising by $z=(x-\mu)/\sigma$ reduces every normal to the single standard normal $N(0,1)$ , whose cumulative distribution is $\Phi$ .

\Phi(z)=\frac{1}{2}\left[1+\operatorname{erf}\!\left(\frac{z}{\sqrt{2}}\right)\right],\qquad \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t \qquad (2)

where: $\Phi(z)$ — CDF of the standard normal $N(0,1)$ ; $z=(x-\mu)/\sigma$ — the standardised variable, "how many sigma from the mean"; $\operatorname{erf}$ — the error function. Reading it: one table of $\Phi$ (or $\operatorname{erf}$ ) serves every Gaussian in physics, because all of them collapse onto $N(0,1)$ after the shift-and-scale $z=(x-\mu)/\sigma$ . The characteristic function is the Fourier transform of the density, and it is where the Gaussian's algebraic magic lives. It is again a Gaussian in the conjugate variable, a quadratic in the exponent with the sign of the $t^2$ term flipped.

\varphi(t)=\mathbb{E}\!\left[e^{itX}\right]=\exp\!\left(i\mu t-\tfrac{1}{2}\sigma^2 t^2\right) \qquad (3)

where: $\varphi(t)$ — characteristic function, $\int e^{itx}f(x)\,\mathrm{d}x$ ; $t$ — the Fourier conjugate variable; $i$ — the imaginary unit. Reading it: because $\log\varphi(t)=i\mu t-\tfrac12\sigma^2 t^2$ is exactly quadratic, the cumulant generating function terminates at second order — the mean and variance are the only two cumulants, and everything else about the distribution follows from them. The moments confirm what the quadratic exponent promises: the distribution is perfectly symmetric and has the reference peakedness against which all other distributions are graded.

\mathbb{E}[X]=\mu,\quad \mathrm{Var}(X)=\sigma^2,\quad \text{skewness}=0,\quad \text{excess kurtosis}=0 \qquad (4)

where: skewness — the third standardised moment, zero here by symmetry; excess kurtosis — the fourth standardised moment minus 3, zero for a Gaussian by definition of the reference. Reading it: "skewness $=0$ " and "excess kurtosis $=0$ " are not coincidences but the visible fingerprint of a cumulant generating function that stops at order two. The central moments themselves have a clean closed form: all odd ones vanish, and the even ones are powers of the variance weighted by double factorials — the same integers that count the pairings in Wick's theorem.

\mu_{2n}=\sigma^{2n}(2n-1)!!,\qquad \mu_{2n+1}=0 \qquad (5)

where: $\mu_k=\mathbb{E}[(X-\mu)^k]$ — the $k$ -th central moment; $(2n-1)!!=1\cdot3\cdot5\cdots(2n-1)$ — the double factorial. Reading it: the first few are $\mu_2=\sigma^2$ , $\mu_4=3\sigma^4$ , $\mu_6=15\sigma^6$ ; the sequence $1,3,15,105,\dots$ is exactly the number of ways to pair up $2n$ objects, which is why Gaussian integrals in field theory reduce to sums over pairings. The closure property that makes the Gaussian a fixed point of summation is a one-line consequence of equation (3): multiply two Gaussian characteristic functions and the exponents add.

N(\mu_1,\sigma_1^2)\ast N(\mu_2,\sigma_2^2)=N(\mu_1+\mu_2,\;\sigma_1^2+\sigma_2^2) \qquad (6)

where: $\ast$ — convolution, the density of a sum of two independent variables. Reading it: adding independent Gaussians adds their means and their variances (never their standard deviations) — the rule behind error propagation and the seed of the central limit theorem. Finally, two working numbers every spectroscopist and experimentalist keeps at hand: the full width at half maximum, and the coverage of the sigma bands.

\mathrm{FWHM}=2\sqrt{2\ln 2}\,\sigma\approx 2.3548\,\sigma,\qquad P(|X-\mu|<k\sigma)=\{0.6827,\,0.9545,\,0.9973\}_{k=1,2,3} \qquad (7)

where: $\mathrm{FWHM}$ — the width of the curve at half its peak height; $k$ — the number of standard deviations. Reading it: the FWHM is about $2.35\sigma$ , wider than the naive $2\sigma$ , and the three coverage numbers are the everyday meaning of one-, two-, and three-sigma confidence.

Derivation ANormalisation by the Gaussian integral

The goal is to prove that the prefactor $1/(\sigma\sqrt{2\pi})$ in equation (1) is forced — that it is the unique constant making the total probability equal one — and to do so by evaluating the Gaussian integral $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\,\mathrm{d}x$ with the polar-coordinate trick, since the integrand has no elementary antiderivative and the one-dimensional integral cannot be done head-on. Establishing $\int e^{-ax^2}\mathrm{d}x=\sqrt{\pi/a}$ once fixes the normalisation of every Gaussian in the article.

Step 1. Shift by $u=x-\mu$ (which leaves the integral unchanged, the domain being the whole line) so that we need only evaluate $I=\int_{-\infty}^{\infty}e^{-u^2/2\sigma^2}\,\mathrm{d}u$ and show it equals $\sigma\sqrt{2\pi}$ . Why: translating to the peak removes $\mu$ from the problem — normalisation cannot depend on where the bell is centred, only on its width.

Step 2. Square the integral and rename the dummy variable in the second copy, turning a stubborn 1-D integral into a 2-D one over the whole plane, $I^2=\int_{-\infty}^{\infty}\!\int_{-\infty}^{\infty}e^{-(x^2+y^2)/2\sigma^2}\,\mathrm{d}x\,\mathrm{d}y$ . Why: the exponent $x^2+y^2$ is rotationally symmetric, so the plane is begging for polar coordinates — the trick works precisely because a product of two Gaussians is a function of radius alone.

Step 3. Switch to polar coordinates $(r,\theta)$ , where the area element carries a factor $r$ : $I^2=\int_0^{2\pi}\!\int_0^{\infty}e^{-r^2/2\sigma^2}\,r\,\mathrm{d}r\,\mathrm{d}\theta$ . Why: that Jacobian factor $r$ is exactly the derivative structure needed to make the radial integral elementary — $r\,e^{-r^2/2\sigma^2}$ is a perfect derivative, which the bare 1-D integrand was not.

Step 4. Do the angular integral ( $2\pi$ ) and the radial integral ( $[-\sigma^2 e^{-r^2/2\sigma^2}]_0^\infty=\sigma^2$ ) to get $I^2=2\pi\sigma^2$ , hence $I=\sigma\sqrt{2\pi}$ ; dividing $f$ by $I$ gives unit area. Why: this pins the prefactor uniquely — any other constant would make the total probability differ from one, which is not allowed for a density. *As a quick check: setting $\sigma=1/\sqrt2$ collapses $I=\sigma\sqrt{2\pi}$ to $\int e^{-x^2}\mathrm{d}x=\sqrt\pi$ , recovering the general identity $\int e^{-ax^2}\mathrm{d}x=\sqrt{\pi/a}$ .*

Go deeper · Derivation B · 1 minThe characteristic function and closure under convolution

Derivation BThe characteristic function and closure under convolution

The goal is to compute the Fourier transform of the Gaussian density, show that the answer $\varphi(t)=\exp(i\mu t-\tfrac12\sigma^2 t^2)$ is again a Gaussian, and then harvest the most consequential corollary in the subject — that a sum of independent Gaussians is Gaussian with means and variances adding, the mechanism the central limit theorem exploits. The method is to complete the square inside the exponential and use that a complex contour shift leaves the normalised Gaussian integral equal to one.

Step 1. Insert the density into $\varphi(t)=\int e^{itx}f(x)\,\mathrm{d}x$ and substitute $y=x-\mu$ , so the exponent becomes $it\mu+ity-y^2/2\sigma^2$ under a prefactor $1/(\sigma\sqrt{2\pi})$ . Why: peeling off the mean as an overall phase $e^{it\mu}$ isolates the genuinely quadratic part in $y$ , which is what completing the square will absorb.

Step 2. Complete the square in $y$ : write $ity-y^2/2\sigma^2=-\tfrac{1}{2\sigma^2}(y-i\sigma^2 t)^2-\tfrac12\sigma^2 t^2$ , so the whole exponent is $i\mu t-\tfrac12\sigma^2 t^2$ plus a shifted perfect square. Why: the cross term $ity$ that carries the transform variable is entirely swallowed into the shifted square, leaving behind exactly the constant $-\tfrac12\sigma^2 t^2$ that will become the new Gaussian's exponent.

Step 3. The remaining integral $\frac{1}{\sigma\sqrt{2\pi}}\int e^{-(y-i\sigma^2 t)^2/2\sigma^2}\,\mathrm{d}y$ equals one: shifting the integration contour by the imaginary amount $i\sigma^2 t$ crosses no singularities of the entire integrand, so it reduces to the normalised real Gaussian. Why: analyticity lets the complex shift be undone for free, which is the deep reason the Gaussian is essentially its own Fourier transform.

Step 4. Collect the pieces to get $\varphi(t)=\exp(i\mu t-\tfrac12\sigma^2 t^2)$ , then apply it to a sum $Z=X+Y$ of independents: $\varphi_Z=\varphi_X\varphi_Y=\exp(i(\mu_1+\mu_2)t-\tfrac12(\sigma_1^2+\sigma_2^2)t^2)$ , so $Z\sim N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$ by uniqueness. Why: independence turns the expectation of a product into a product of expectations, and multiplying exponentials adds exponents — means and variances add, standard deviations do not. *As a consistency check: the added-variance rule reproduces the general identity $\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$ for uncorrelated variables; expanding $\varphi(t)=1+i\mu t+\tfrac12(i\mu)^2t^2+\cdots$ recovers $\mathbb{E}[X]=\mu$ and $\mathbb{E}[X^2]=\mu^2+\sigma^2$ .*

Go deeper · Derivation C · 1 minThe Gaussian as the maximum-entropy distribution

Derivation CThe Gaussian as the maximum-entropy distribution

The goal is to show that the bell shape is not merely convenient but forced by an information principle: among all densities on the line with prescribed mean $\mu$ and variance $\sigma^2$ , the one maximising the differential entropy $H=-\int f\ln f\,\mathrm{d}x$ is exactly $N(\mu,\sigma^2)$ . This is why the Gaussian arises whenever a system is constrained by nothing more than its first two moments, and the tool is the calculus of variations with Lagrange multipliers for the constraints.

Step 1. Build the constrained functional $\mathcal{L}=-\int f\ln f\,\mathrm{d}x-\lambda_0\int f\,\mathrm{d}x-\lambda_1\int x f\,\mathrm{d}x-\lambda_2\int(x-\mu)^2 f\,\mathrm{d}x$ , with three multipliers for normalisation, mean, and variance. Why: entropy alone is unbounded above; the constraints are what make a genuine maximum exist, and each multiplier is the price of enforcing one moment.

Step 2. Set the functional derivative to zero: $\delta\mathcal{L}/\delta f=-\ln f-1-\lambda_0-\lambda_1 x-\lambda_2(x-\mu)^2=0$ . Why: at the stationary point $\ln f$ must be a quadratic polynomial in $x$ — the constraints reach up to second order, so the optimal log-density can only be second order.

Step 3. Exponentiate to get $f(x)=\exp(-1-\lambda_0-\lambda_1 x-\lambda_2(x-\mu)^2)$ — necessarily the exponential of a quadratic, which is a Gaussian shape. Why: this is the punchline in one line — maximising entropy under two-moment constraints cannot yield anything but a bell, no matter what the underlying physics is.

Step 4. Fix the multipliers by imposing the constraints: symmetry about $\mu$ forces $\lambda_1=0$ , and matching the variance gives $\lambda_2=1/2\sigma^2$ , reproducing $f=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$ exactly. Why: the two surviving multipliers are just the mean and variance in disguise, confirming that fixing those two numbers fixes the whole distribution. *Substituting the optimal $f$ back into the entropy functional gives $H=\tfrac12\ln(2\pi e\sigma^2)$ ; for $\sigma=1$ this evaluates to $H=\tfrac12\ln(2\pi e)=1.4189$ nats, confirming the extremum is the Gaussian.*

Limiting cases

The Gaussian sits at the confluence of many other laws — it is the large-parameter limit of the workhorse discrete distributions and it degenerates cleanly when its width is driven to zero.

Case	Result	Meaning
Binomial → Gaussian (de Moivre–Laplace)	$\binom{n}{k}p^k q^{n-k}\xrightarrow{n\to\infty}\frac{1}{\sqrt{2\pi npq}}\exp\!\big(-\frac{(k-np)^2}{2npq}\big)$	For large $n$ at fixed $p$ , the binomial approaches a Gaussian of mean $np$ and variance $npq$ — the historical first appearance of the bell, and a special case of the CLT.
Poisson → Gaussian	$\frac{\lambda^k e^{-\lambda}}{k!}\xrightarrow{\lambda\to\infty}\frac{1}{\sqrt{2\pi\lambda}}\exp\!\big(-\frac{(k-\lambda)^2}{2\lambda}\big)$	For a large mean count $\lambda$ the Poisson law becomes Gaussian with mean and variance both $\lambda$ — why photon-counting and shot-noise statistics look Gaussian at high flux.
Zero-variance limit	$\lim_{\sigma\to0^+}N(\mu,\sigma^2)=\delta(x-\mu)$	As the width shrinks the bell becomes a Dirac spike at $\mu$ — a deterministic value; the Gaussian is a nascent delta function, the standard smooth regulariser.
Standardisation	$Z=(X-\mu)/\sigma\sim N(0,1)$	Every Gaussian is the standard normal after a shift and rescale; all probability tables reduce to one.

Common trapthe misconception that any average, or any bell-looking data, must be Gaussian

The central limit theorem is powerful but conditional, and two conditions are routinely forgotten. First, it demands finite variance: heavy-tailed summands such as Cauchy-distributed variables never converge to a Gaussian — averaging Cauchy variables gives back a Cauchy of the same width, because the variance integral diverges and there is no $\sigma^2$ to add. Second, the summands must be effectively independent (or weakly dependent); strongly correlated contributions can converge to entirely different limit laws. Two smaller bookkeeping traps: do not confuse the standard deviation $\sigma$ with the variance $\sigma^2$ (they carry different units and the wrong one wrecks every propagation), and do not read the full width at half maximum as $\sigma$ — it is $\mathrm{FWHM}=2\sqrt{2\ln2}\,\sigma\approx2.355\,\sigma$ , wider by more than a factor of two. A bell-shaped histogram is suggestive, not proof: only finite variance plus near-independence licenses the normal law.

Worked examples

1 · Sigma bands, tail probabilities, and the 5σ bar (fully worked)

A calibrated voltage reference outputs $V\sim N(5.000\text{ V},(0.020\text{ V})^2)$ . (i) What fraction of readings fall in the window $4.960$ – $5.040$ V? (ii) What is the probability a single reading exceeds $5.050$ V? (iii) A physicist claims a systematic drift once a reading strays $5\sigma$ from the mean; how far is that, and how rare is it one-sided?

Step 1. Identify the sigma multiples. The window $4.960$ – $5.040$ V is $\mu\pm0.040$ V, and since $\sigma=0.020$ V this is precisely $\mu\pm2\sigma$ . The two-sigma coverage is fixed by the error function: $P(|X-\mu|<2\sigma)=\operatorname{erf}(2/\sqrt2)=\operatorname{erf}(\sqrt2)=\mathbf{0.9545}$ , so about 95.45% of readings land in the window.

Step 2. Standardise the one-sided query. For $V>5.050$ V the z-score is $z=(5.050-5.000)/0.020=2.5$ . The upper-tail probability is $1-\Phi(2.5)=1-0.99379=\mathbf{0.00621}$ , so a reading above $5.050$ V occurs about 0.62% of the time, roughly one in 161 readings.

Step 3. Evaluate the $5\sigma$ claim. Five standard deviations is $5\times0.020=\mathbf{0.100}$ V, i.e. a reading beyond $5.100$ V (or below $4.900$ V). The one-sided tail is $1-\Phi(5)=\mathbf{2.87\times10^{-7}}$ — about one chance in $3.5$ million. This is exactly the particle-physics discovery threshold: the $5\sigma$ convention exists because $2.87\times10^{-7}$ is small enough to survive the many-trials scrutiny of a real experiment. (Counting both tails doubles it to $5.7\times10^{-7}$ .)

2 · Propagating error through a sum (variances add, not sigmas)

Two components are measured independently as $a=12.00\pm0.05$ cm and $b=7.00\pm0.12$ cm, each uncertainty a one-sigma Gaussian. Report the total length $s=a+b$ and the gap $d=a-b$ with their uncertainties.

Step 1. Combine the central values directly: $s=12.00+7.00=19.00$ cm and $d=12.00-7.00=5.00$ cm. The means simply add or subtract.

Step 2. Combine the variances, never the standard deviations. By equation (6) the variance of an independent sum (or difference) is the sum of variances: $\sigma_s^2=\sigma_a^2+\sigma_b^2=0.05^2+0.12^2=0.0025+0.0144=0.0169\ \text{cm}^2$ , so $\sigma_s=\sqrt{0.0169}=\mathbf{0.13}$ cm. Subtraction gives the same combined variance, so $\sigma_d=0.13$ cm too.

Step 3. Report and sanity-check. $s=\mathbf{19.00\pm0.13\text{ cm}}$ and $d=\mathbf{5.00\pm0.13\text{ cm}}$ . Note the naive "add the errors" answer $0.05+0.12=0.17$ cm overstates the uncertainty — because independent errors partly cancel, the Pythagorean combination $0.13$ cm is correct, and the larger component ( $0.12$ ) dominates the budget.

3 · Diffusion as a spreading Gaussian

A fluorescent dye molecule in water has diffusion constant $D=5.0\times10^{-10}\ \mathrm{m^2/s}$ . Starting from a point, where is it likely to be after $t=10\text{ s}$ ?

Step 1. The heat-kernel solution of the diffusion equation is a Gaussian of zero mean and variance $\sigma^2=2Dt$ (one dimension). Here $\sigma^2=2\times(5.0\times10^{-10})\times10=1.0\times10^{-8}\ \mathrm{m^2}$ , so the root-mean-square displacement is $\sigma=\sqrt{1.0\times10^{-8}}=1.0\times10^{-4}\ \mathrm{m}=\mathbf{0.10\text{ mm}}$ .

Step 2. Apply the sigma bands to the spatial spread. The probability the molecule lies within $\pm0.20$ mm of the start is $P(|x|<2\sigma)=\mathbf{95.45\%}$ , and within $\pm0.30$ mm is $99.73\%$ . The bell has literally become the position probability cloud.

Step 3. Read the physics off the scaling. Because $\sigma\propto\sqrt{t}$ , the spread grows only as the square root of time — to double the RMS displacement to $0.20$ mm the molecule must diffuse for four times as long ( $40$ s). This $\langle x^2\rangle=2Dt$ is the Einstein relation, and its slow $\sqrt{t}$ growth is why diffusion is a hopeless way to move anything across a room.

Physics of the portrait

The normal distribution is where three centuries of probability, astronomy, and physics converge on one curve — approximated first as a computational shortcut for the binomial, elevated by the analysis of astronomical errors, and finally proved universal.

Origins · 1 minCarl Friedrich Gauss and how this was discovered

Carl Friedrich Gauss

1777–1855 · least squares, geodesy, and the theory of measurement errors

The bell carries Gauss's name, but its story begins two generations before him. Abraham de Moivre, working on games of chance around 1733, found that the binomial for large $n$ is closely approximated by the exponential of a negative square — the first appearance of the shape, buried in a problem about coin tosses. Pierre-Simon Laplace developed the curve far more deeply from the 1770s, tying it to the analysis of errors and stating an early central limit theorem. Gauss's decisive contribution came in 1809, in the Theoria Motus on celestial orbits: seeking the error distribution for which the arithmetic mean is the most probable value, he was led uniquely to the normal law, and with it founded the method of least squares that still underlies essentially all data fitting — which is why the distribution and the fitting method share his name. The curve's universality was not secured until the central limit theorem was made rigorous — Aleksandr Lyapunov gave general conditions in 1901 — but the intuition was already Gauss's: when many small independent errors accumulate, only their mean and spread survive, and the bell is what remains. From the geodetic survey of Hanover to the least-squares reduction of every modern experiment, Gauss turned a computational curiosity into the governing law of measurement itself.

History

1733

— de Moivre approximates the binomial distribution by the exponential of a negative square

1774

— Laplace begins his analytic study of the curve of observational errors

1809

— Gauss derives the normal law from the principle of least squares in Theoria Motus

1810

— Laplace states and applies an early central limit theorem

1901

— Lyapunov proves the central limit theorem under general finite-variance conditions

Going further

Texts. Stone & Goldbart, Mathematics for Physics, chapter on probability and statistics · Arfken & Weber, Mathematical Methods for Physicists, 7th ed., probability chapter · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 30 (probability) · Hassani, Mathematical Physics, on distributions and the delta-function limit · Byron & Fuller, Mathematics of Classical and Quantum Physics, on functionals and variational methods (for the maximum-entropy derivation). Every formula here is standard and independently checkable. Key numerical anchors: the Gaussian integral $\int e^{-ax^2}\mathrm{d}x=\sqrt{\pi/a}$ ; the coverage values $\operatorname{erf}(1/\sqrt2)=0.6827$ , $\operatorname{erf}(\sqrt2)=0.9545$ , $\operatorname{erf}(3/\sqrt2)=0.9973$ ; $\mathrm{FWHM}=2\sqrt{2\ln2}\,\sigma=2.3548\sigma$ ; the central-moment rule $\mu_{2n}=\sigma^{2n}(2n-1)!!$ giving $\mu_4=3\sigma^4$ , $\mu_6=15\sigma^6$ ; the maximum entropy $H=\tfrac12\ln(2\pi e\sigma^2)$ ; and the one-sided $5\sigma$ tail $1-\Phi(5)=2.87\times10^{-7}$ .

Prerequisites. Random Variables and Probability Distributions · Expectation, Variance and Moments · Cumulative Distribution Functions and Change of Variables · Moment-Generating and Characteristic Functions. Leads to. Central Limit Theorem · Multivariate Gaussian Distribution · Laws of Large Numbers and Convergence of Random Variables · Poisson Distribution.

Ask the tutor: Ask me to walk you through why the sum of independent Gaussians is Gaussian using characteristic functions, or to compute any sigma-band or tail probability with the error function step by step.

Why does the normalisation have a √2π in it?

The $\sqrt{2\pi}$ is the value of the bare Gaussian integral, not an arbitrary decoration. Working through the polar-coordinate trick, $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\mathrm{d}x=\sigma\sqrt{2\pi}$ , so dividing the exponential by exactly that factor makes the total area one. The $2$ comes from the $\tfrac12$ in the standard exponent $-(x-\mu)^2/2\sigma^2$ , and the $\pi$ is the fingerprint of the rotational symmetry exploited when the one-dimensional integral is squared into a two-dimensional one over the plane. Change the convention (drop the $\tfrac12$ ) and the constant changes to $\sqrt\pi$ , but $2\pi$ is standard because it makes $\sigma^2$ the variance directly.

What is the difference between σ, σ², and the FWHM?

They are three different measures of the same spread, and mixing them is a classic error. The variance $\sigma^2$ is the mean squared deviation, $\mathbb{E}[(X-\mu)^2]$ ; it is the quantity that adds when independent variables combine, and it carries squared units. The standard deviation $\sigma=\sqrt{\sigma^2}$ is its square root, in the same units as $x$ , and is the natural "one-sigma" width — the distance from the peak to the inflection point. The full width at half maximum, $\mathrm{FWHM}=2\sqrt{2\ln2}\,\sigma\approx2.355\sigma$ , is the total width where the curve has dropped to half its peak height, the quantity a spectroscopist reads off a line profile. So a Gaussian line of $\sigma=1$ has FWHM $\approx2.355$ , not $2$ .

Does the central limit theorem mean everything is Gaussian?

No, and believing so is dangerous. The theorem says the standardised sum of many independent, finite-variance contributions approaches a Gaussian — three conditions, each of which can fail. Heavy-tailed quantities with infinite variance (Cauchy, and many financial returns) never converge to a normal; strongly correlated contributions can converge to different limit laws; and a single dominant term prevents any averaging effect. The theorem is about aggregates, not individuals: an individual measurement, a single particle's energy, or a raw waiting time need not be Gaussian at all. What the theorem does guarantee is that when a quantity is genuinely built from many small independent pieces — thermal noise, diffusion, measurement error — the bell is the right and often unavoidable model.

Why is the Gaussian its own Fourier transform?

Because the Fourier transform trades a differential operator for multiplication, and the Gaussian is the eigenfunction that balances the two. Completing the square in the transform integral shows the transform of $e^{-x^2/2\sigma^2}$ is again a Gaussian, now of width $1/\sigma$ — narrow in position becomes wide in frequency, the reciprocity behind the uncertainty principle. Up to that width inversion and a constant, the shape is preserved. This self-duality is why the characteristic function of a Gaussian is Gaussian, why a Gaussian wave packet stays Gaussian under free evolution, and why the harmonic-oscillator ground state — symmetric between position and momentum — is a Gaussian.

Where does the Gaussian show up in quantum mechanics?

Most fundamentally as the ground state of the harmonic oscillator: $\psi_0(x)\propto e^{-m\omega x^2/2\hbar}$ is a Gaussian, so the position probability $|\psi_0|^2$ is $N(0,\hbar/2m\omega)$ . This state saturates the Heisenberg bound, $\sigma_x\sigma_p=\hbar/2$ — the minimum-uncertainty state is Gaussian, a quantum echo of the maximum-entropy property. Coherent and squeezed states (Gaussians with the uncertainty redistributed between $x$ and $p$ ) are the workhorses of quantum optics and the basis of noise reduction in gravitational-wave detectors. More broadly, Gaussian path integrals — where the action is quadratic — are the only ones that can be done exactly.

Why do the higher moments follow a double factorial?

Because the cumulant generating function $\log\varphi(t)=i\mu t-\tfrac12\sigma^2 t^2$ stops at second order, every higher moment is built purely from the second cumulant $\sigma^2$ , and the combinatorics of assembling a moment of order $2n$ from copies of a single variance is exactly the number of ways to pair $2n$ objects — which is $(2n-1)!!=1\cdot3\cdot5\cdots(2n-1)$ . So $\mu_4=3\sigma^4$ , $\mu_6=15\sigma^6$ , $\mu_8=105\sigma^8$ , and so on, while all odd moments vanish by symmetry. Physicists meet this as Wick's theorem: a Gaussian expectation of a product of fields equals the sum over all pairings of two-point functions, the computational backbone of perturbative quantum field theory.