normaldistributionrelated

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped. It is completely described by two parameters: the mean μ and the standard deviation σ, with density f(x) = (1/(σ√(2π))) exp(-(x−μ)^2/(2σ^2)). The distribution is fully determined by μ and σ; μ sets the center, σ sets the spread. The standard normal distribution is the special case μ = 0, σ = 1, and the standard score z = (x − μ)/σ maps any normal variable to Z.

The normal distribution has several key properties: symmetry about μ, unimodality, and tails that decline exponentially. Its moments are E[X] = μ and Var(X) = σ^2, with skewness 0 and excess kurtosis 0. The inflection points are at μ ± σ. The cumulative distribution function, Φ(z), has no elementary closed form, but is tabulated and implemented in software.

Relation to statistics and data analysis: many statistical methods assume normality of errors or latent variables; under the central limit theorem, sums of independent random effects tend toward normality. Practically, the normal model underpins confidence intervals, hypothesis tests, regression inference, and ANOVA when assumptions hold. Assessment of normality uses Shapiro–Wilk or Kolmogorov–Smirnov tests, Q–Q plots, and transformations such as Box–Cox when needed.

History and use: the distribution is named after Carl Friedrich Gauss and is sometimes called the Gaussian distribution. It arises naturally as a model of measurement error and of natural phenomena due to the central limit principle. It remains a foundational tool in statistics, probability, and data analysis, with robust computational support in virtually all statistical software and libraries.