Updated 12 September 2026

Normal distribution

New to the notation? Start with the connected foundation for the underlying definitions and a worked example.

The normal — or Gaussian — distribution shows up everywhere averages do. Two numbers fully describe it: the mean $\mu$ tells you where it centres, the standard deviation $\sigma$ tells you how wide it spreads.

What to notice

• Shifting $\mu$ slides the whole curve left or right without changing shape.
• Shrinking $\sigma$ makes the bell taller and narrower — the total area is always 1, so height and width trade off.
• Resampling redraws the sample. The histogram wiggles, but it hugs the PDF more tightly as you raise $n$. That convergence is a preview of the Law of Large Numbers.

Why it matters

The Central Limit Theorem says the sum of many independent small effects tends toward a normal, even if none of the individual effects are normal themselves. This can justify a normal approximation for some averages. It does not make every measurement or error distribution normal; independence, tail behavior, and the question being asked still matter.

Compare the assumptions and event probabilities in normal vs Student’s t, or use the distribution chooser.

Explore how distributions connect to distinguish squared normals, exponentiation, and random-variance mixtures. The distribution-reading guide connects density areas to CDFs and quantiles.

Turn an observation into an area

For X with mean 1 and standard deviation 2, an observation x=3 has z=(3−1)/2=1. Its lower-tail probability is about 0.841345. The 95th percentile is 1+2×1.644854≈4.289707. The computation controls keep the standardized probability and the value in original units distinct.

A quantile is a boundary, not the percentage of observations exactly equal to it. In a continuous model every individual value has zero probability. Use an interval or a tail, and remember that the chart’s visible window omits a small amount of mass.

A sum is not an average

For n independent Normal(μ,σ²) observations, the sum is Normal(nμ,nσ²), while the average is Normal(μ,σ²/n). With four independent Normal(1,4) observations, the sum has mean 4 and standard deviation 4; the average has mean 1 and standard deviation 1. Independence is what removes covariance terms from these variance calculations.

The population need not become normal as the sample grows. For example, Bernoulli observations always remain zero or one; under suitable conditions it is their standardized average that approaches a normal law. Heavy-tailed Cauchy observations do not even have the finite variance required by the classical CLT. A smooth histogram alone is not a normality assumption.

Reference

NIST: normal distribution describes the standardized density and cumulative probabilities. The CLT lesson distinguishes normal observations from approximate normality of averages.

For two measurements, the multivariate normal lesson connects correlation to conditional distributions and linear combinations.