Normal distribution

The formula

The probability density function:

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

Density peaks at the mean μ and falls off with the squared distance from it, measured in units of the standard deviation σ. The 1/(σ·√(2π)) out front just makes the total area equal 1.

The z-score converts any normal into the standard one (μ = 0, σ = 1):

z = (x − μ) / σ

Read it as "how many sigmas away from the mean is this reading" — one table, one function, covers every normal distribution ever.

There is no closed-form antiderivative. Cumulative probability comes from the error function:

Φ(z) = ½ · (1 + erf( z/√2 ))

Φ(z) is the fraction of the population at or below z sigmas — what a z-table, NORM.DIST in Excel, or erf on any calculator gives you.

The coverage numbers worth memorizing:

±1σ → 68.27%     ±2σ → 95.45%     ±3σ → 99.73%
±1.96σ → 95.00%  ±2.576σ → 99.00%

And the property that makes tolerance math work — independent normals add with their variances, not their sigmas:

X + Y  ~  Normal( μ₁ + μ₂ ,  σ = √(σ₁² + σ₂²) )

That square root is the entire justification for RSS (root-sum-square) tolerance stacking.

Where you meet it

On the bench, characterizing a measurement. Take fifty readings of the same voltage with the same DMM and histogram them — you get the bell. The standard deviation of that pile is your Type A uncertainty, and the k = 2 coverage factor on every calibration certificate is shorthand for "about 95% of a normal distribution."
At the test stand, setting alarm limits. Strain gauge and thermocouple channels ride on Gaussian noise. Limits set at ±3σ trip falsely on 0.27% of samples — which at a 1 kHz scan rate is not rare, it's 2.7 times per second. Whoever sets the limit band without doing that arithmetic learns it during the first hot fire.
At the review board, when someone says "we're at four sigma." Process capability (Cp, Cpk), control charts, and the entire Six Sigma vocabulary are statements about where the spec limits sit on a normal curve fitted to production data. The famous 3.4 defects per million is not the ±6σ tail — it's the one-sided tail beyond 4.5σ, because the methodology assumes the process mean drifts 1.5σ over the long run ^[8].
In the RF chain. Thermal (Johnson) noise has a Gaussian amplitude distribution. Its RMS value is σ, its instantaneous value exceeds 3σ for 0.27% of the time and 4σ for 63 ppm of the time — which is exactly the headroom argument behind ADC clipping specs and noise crest-factor rules of thumb.

How it works

The reason the normal shows up everywhere is the central limit theorem: add up many small, independent random effects and the sum tends toward a normal distribution regardless of what the individual pieces look like. Measurement error is usually dozens of tiny effects — contact resistance, thermal EMFs, quantization, vibration — so the default Gaussian assumption isn't laziness, it's the CLT doing what it's proved to do. Same theorem, run in reverse, is why averaging N readings cuts the noise on the mean by √N.

The whole curve scales with σ. Peak height is 0.3989/σ, the full width at half maximum is 2.355·σ, the inflection points sit exactly at μ ± σ. For a zero-mean Gaussian signal, the RMS value is σ — which is why noise specs quoted in RMS convert directly to tail probabilities.

Now the gotchas, in the order they cost money.

The tails are the lie. The Gaussian model is superb within ±2σ or so and increasingly a matter of faith beyond ±4σ. Real processes have drift, operator changes, temperature cycles, and the occasional dropped part — all of which fatten the tails. When someone computes "that's a one-in-a-billion event" from a normal fit to 200 data points, the honest translation is "we have no data about events that rare." The fit says nothing about a region the sample never visited.

Wrong variable, wrong distribution. The normal is symmetric and runs from −∞ to +∞. Quantities that are bounded or skewed — time to failure, particle counts, VSWR, fatigue life, anything that can't go negative but can go very large — belong to Weibull, lognormal, or Poisson territory. Fitting a normal to failure times and quoting a 3σ life is how a reliability number ends up predicting negative hours.

Small samples don't buy you z. With 10 readings, your estimate of σ is itself noisy, and the ±1.96σ recipe understates the interval. That's what Student's t is for; below roughly 30 samples, use t, not z. Related mistake: the mean of 30 readings is very nearly normal (CLT again) even when the individual readings aren't — normality of the average is not normality of the population.

RSS requires independence. Stacking tolerances by √(Σσᵢ²) assumes the error sources are uncorrelated. Two dimensions machined on the same setup with the same worn tool are not independent, and their errors add linearly, not in quadrature. RSS on correlated errors is the optimistic mistake; worst-case stacking on independent errors is the expensive one.

Before leaning on any of this, check normality the honest way — a probability plot (quantile-quantile), where normal data falls on a straight line and tail trouble shows up immediately. A histogram that "looks bell-shaped" has approved a lot of bad assumptions.

History

The curve entered mathematics as a shortcut for arithmetic nobody could finish. In a Latin pamphlet dated 13 November 1733, Abraham de Moivre — a French Protestant refugee scraping out a living in London coffeehouses — showed that the binomial coefficients for many coin flips are approximated by what we now recognize as the normal curve, work he folded into the 1738 second edition of his Doctrine of Chances ^[1]^[2]. He treated it purely as an approximation trick; the idea of a probability density hadn't been invented yet ^[1].

Pierre-Simon Laplace supplied the machinery. He evaluated the integral ∫ e^(−t²) dt = √π in 1782 — the normalization constant the curve needs — and around 1810 proved the central limit theorem, later consolidated in his 1812 Théorie analytique des probabilités, which explained why sums of random effects head toward this one curve ^[1]^[5].

The name on the curve belongs to Carl Friedrich Gauss, and he earned it sideways. In his 1809 astronomy treatise Theoria motus, Gauss asked which error law makes the arithmetic mean of repeated observations the most probable value of the thing measured — and showed the answer is exactly the exponential-of-a-square law, which in turn justifies least squares fitting ^[1]^[3]^[4]. He had been using least squares privately since his celebrated 1801 recovery of the lost asteroid Ceres ^[3]^[4]. The Irish-American mathematician Robert Adrain published independent derivations at nearly the same moment; almost nobody read them until 1871 ^[1]^[4].

The word "normal" came later. By the 1870s, C. S. Peirce, Francis Galton, and Wilhelm Lexis were independently calling it the normal distribution — Peirce as early as 1873, from studies of measurement error ^[1]^[4]. Karl Pearson, who coined "standard deviation" in 1893 and put the modern σ notation on the curve, adopted "normal" partly to dodge the Gauss-versus-Laplace priority fight — and later admitted the name has the disadvantage of implying every other distribution is abnormal ^[1]^[6]^[7].

Related tools

/tools/thermal-noise-floor — kTB noise power; the amplitude behind that number is Gaussian
/tools/rms-peak — for zero-mean Gaussian noise, RMS is σ, and peak is a probability statement
/tools/crest-factor — Gaussian noise headroom: how many σ of peak you budget before clipping
/tools/snr-enob — noise treated as Gaussian is what makes the SNR-to-bits conversion honest
/tools/adc-resolution — quantization noise riding on Gaussian analog noise sets real resolution
/tools/strain-gauge-bridge — the bench measurement whose scatter this distribution models

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →