= Solution
A <Gaussian distribution> with mean $\mu$ and <standard deviation> $\sigma$ has <probability density>
$$
\boxed{p(x)=\frac1{\sqrt{2\pi}\sigma}
\exp\!\left[-\frac{(x-\mu)^2}{2\sigma^2}\right].}
$$
Its symmetry, finite <variance> and simple behavior under linear combinations make it useful for repeated measurements. Roughly $68.3\%$, $95.4\%$ and $99.7\%$ of observations lie within one, two and three true standard deviations of the mean. These are coverage probabilities for a specified distribution, not guarantees that every experimental error is normal.
\b[Why it appears.] The <central limit theorem> explains why many sums of small independent finite-variance contributions approach a Gaussian law after centering and scaling, provided no single contribution dominates. Detector <noise>, small mechanical disturbances and multiple electronic fluctuations can combine approximately this way. The theorem does not remove a common calibration bias, and it does not justify assuming normal tails for rare large disturbances or arbitrarily correlated errors. A large-mean <Poisson distribution> also approaches a Gaussian with <variance> equal to its mean, making normal approximations useful for high-count photon or particle data; low counts require the discrete model.
A classical particle at thermal equilibrium in a one-dimensional <harmonic potential> has a Gaussian <position> distribution directly from the <Boltzmann distribution>:
$$
p(x)\propto e^{-kx^2/(2k_BT)},\qquad
\langle x^2\rangle=\frac{k_BT}{k}.
$$
Thus a Gaussian law can arise from the <energy> dependence of equilibrium <probability>, rather than only from an informal appeal to many independent errors.
\b[Precision of repeated measurements.] For $N$ independent observations with the same mean and <variance>, the <sample mean> has <variance> $\sigma^2/N$. For Gaussian observations this mean is exactly Gaussian:
$$
\boxed{\sigma_{\overline x}=\frac\sigma{\sqrt N}.}
$$
Consequently quadrupling the number of independent readings halves the random <standard error>. When the true <variance> is unknown, a small-sample <confidence interval> for a normal mean uses a <Student t-distribution> with $N-1$ <statistical degrees of freedom>; replacing that by a normal quantile can overstate precision. Shared <systematic errors> and correlations do not decrease by the same averaging rule.
\b[Combining results and fitting models.] Independent Gaussian estimates $x_i$ of one quantity, with known <variances> $\sigma_i^2$, combine through the inverse-variance weighted mean
$$
\widehat\mu=\frac{\sum_i x_i/\sigma_i^2}{\sum_i1/\sigma_i^2},\qquad
\operatorname{Var}(\widehat\mu)=\frac1{\sum_i1/\sigma_i^2}.
$$
For a model $y_i=g_i(\theta)+\epsilon_i$ with independent zero-mean <Gaussian noise>, maximizing the <likelihood> is equivalent to minimizing the weighted least-squares statistic
$$
\chi^2(\theta)=\sum_i\frac{[y_i-g_i(\theta)]^2}{\sigma_i^2}.
$$
For a correct linear model with known independent Gaussian errors and $p$ fitted parameters of full rank, the minimized statistic has a <chi-squared distribution> with $N-p$ degrees of freedom. Nonlinear fits only inherit that familiar approximation under additional regularity and adequate-data assumptions. Known correlated Gaussian errors require the full inverse <covariance matrix> in the quadratic form.
These properties support <confidence intervals>, consistency checks, calibration and error propagation. They also give practical diagnostics: inspect residuals for trends, changing spread, correlations or outliers before trusting a Gaussian fit. \b[Gaussian models connect physical fluctuations to quantitative uncertainty, but their assumptions must be checked rather than inferred from the shape of a convenient bell curve.]
Back to article page
