The Normal Distribution: Anatomy of the Bell Curve

The formula behind the bell

The normal (Gaussian) distribution is completely described by two numbers: a mean μ, which sets where the curve is centred, and a standard deviation σ, which sets how spread out it is. Its probability density function is the formula below — note the curve is never zero anywhere, it just decays extremely fast away from μ, thanks to the squared term inside the exponential.

f(x) = 1 / (σ√(2π))  ·  e^( −(x−μ)² / (2σ²) )
   μ = mean (location)      σ = standard deviation (spread)

Standardising: every Gaussian is the same shape

Any normal distribution can be converted to the standard normal, with μ = 0 and σ = 1, by the substitution z = (x − μ) / σ. That single transformation is why statistics tables only ever need to tabulate one curve — every Gaussian, no matter its mean or spread, is literally the same shape stretched and shifted, and z tells you how many standard deviations a value sits from its own mean.

Why the CDF has no closed form

The cumulative distribution function — the probability of landing at or below some value x — is the running integral of the bell curve, and that integral cannot be written using ordinary elementary functions. It’s instead expressed through the error function, erf, which is itself defined as an integral and evaluated numerically: Φ(z) = ½[1 + erf(z/√2)]. Every statistics table, z-score calculator and software library is, underneath, just a precomputed or numerically integrated version of this one function.

The 68–95–99.7 rule

Because the shape is fixed once standardised, fixed percentages of the distribution always fall within fixed numbers of standard deviations of the mean: about 68% within ±1σ, about 95% within ±2σ, and about 99.7% within ±3σ. This empirical rule is why a result more than two or three standard deviations from the mean is treated as notably unusual across an enormous range of fields, from quality control to experimental physics — it’s a direct, memorable consequence of the exponential decay in the PDF.

Why this particular shape shows up everywhere

Two separate mathematical reasons converge on the same curve. The Central Limit Theorem shows that averages of many independent random quantities tend toward a Gaussian shape regardless of the shape of the original data, which is why so many naturally aggregated measurements — heights, measurement errors, exam scores — look approximately normal. Separately, among all probability distributions with a given mean and variance, the Gaussian is the one with maximum entropy — the least additional structure or assumption baked in beyond just fixing the average and spread — which is a strong theoretical reason to expect it whenever only the mean and variance of a process are known or constrained.

Frequently asked questions

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