🔔 The Normal (or Gaussian) distribution is the famous bell curve. Unlike the uniform, it stretches out forever in both directions — but the action is concentrated in the middle.

It's continuous, like the uniform — x can take any value. But here's a subtlety that matters: for a continuous distribution, the probability of exactly one value is zero. What's P(the response was exactly 4.00000…°)? Zero — there are infinitely many values. So we ask about intervals instead: the probability of landing between x₁ and x₂ is the area under the curve there.

That's why the curve is a probability density function (PDF), often written f(x) — not a mass function. For the discrete PMF we summed masses to 1; for a PDF we integrate the area to 1:

∫ f(x) dx = 1 (area under the whole curve)

The total area under the bell equals 1 — the continuous version of "all the probabilities add up to 1."

The Gaussian is set by exactly two numbers: the mean μ (where the peak sits) and the standard deviation σ (how wide it is) — or its square, the variance σ². Watch which one a formula wants; mixing up σ and σ² is the single most common Gaussian bug.

f(x) = 1/√(2πσ²) · exp( −(x−μ)² / (2σ²) )

The famous PDF. The exp(−(x−μ)²/…) is the bell shape (falls off fast as you leave μ); the 1/√(2πσ²) out front is just the scaling that makes the area exactly 1.

Working example: ask someone to point toward a sound they just heard. Their answers scatter around the true direction in a bell shape — most points near the truth, fewer far off. Below, μ slides that center and σ sets the width. The red curve is the true bell; the teal bars are a histogram of actual samples.

With just a few samples the bars are jagged and barely match the curve. Crank the sample count up and the histogram melts onto the red bell — the law of large numbers, one more time.

🦌 Ilya says: Measurement noise at NMA is almost always modeled as Gaussian. Two numbers — μ and σ — capture it. Get comfortable here and a big chunk of applied stats is already yours.