The Gaussian curve and the central limit theorem

What just happened?

A single die roll gives a uniform distribution — six outcomes 1–6 with equal chance 1/6. But when you sum many such independent rolls, the sum distribution stops being flat and ever more closely approximates the characteristic “bell” — the normal distribution (the Gaussian curve). The key point: this happens regardless of what distribution the terms had.

The central limit theorem (CLT)

When some quantity is a sum of many small, independent random factors, its distribution tends to the normal one. The normal distribution has the density:

where $\mu$ is the mean (center of the bell), and $\sigma$ is the standard deviation (width of the bell). For the sum of n dice we compute them exactly: $\mu = 3.5\,n$ and $\sigma = \sqrt{\tfrac{35}{12}\,n}$.

The invariant: the 68–95–99.7 rule

For the normal distribution a constant fraction of results falls within successive intervals around the mean — regardless of μ and σ:

The panel on the right shows how the actual fraction of sum results (computed from the exact distribution) converges to these values as n grows. This is a practical tool for interpreting the standard deviation.

A bit of history

The normal distribution was described mathematically by Carl Friedrich Gauss (1809), studying measurement errors in astronomy — hence the “Gaussian curve”. The general central limit theorem was proved by Pierre-Simon Laplace (1810). The CLT is one of the pillars of statistics: it explains why the “bell” appears everywhere in nature and measurements — people’s height, measurement errors, test scores — because each of these quantities is a sum of many small, independent influences.

Why you need this for the exam:

The normal distribution and the 68–95–99.7 rule let you quickly estimate how “typical” a given result is: we measure distance from the mean in units of σ. The trick: when a problem mentions a “sum/average of many trials”, expect a normal distribution — even if a single trial was completely different.

Remember: the sum of many independent random events gives a normal distribution (the Gaussian curve) — regardless of the distribution of the terms. About 68% of results fall within ±1σ, about 95% within ±2σ, around the mean μ.