Normal

Normal is a versatile term used across disciplines to denote what is typical, standard, or expected in a given context. As an adjective, it describes ordinary or conforming to a set of rules, norms, or averages. What is considered normal can depend on culture, context, and time. In medicine and statistics, a normal value or normal range refers to measurements regarded as typical for healthy individuals of a population.

In geometry, normal describes a direction or line perpendicular to a reference object. A normal vector to a surface at a point is perpendicular to the tangent plane there. The normal line to a curve at a point is perpendicular to the curve’s tangent, and a normal plane is the plane perpendicular to a normal vector.

Normal distribution, or Gaussian distribution, is a continuous, symmetric bell-shaped distribution defined by a mean and standard deviation. It appears as a central model in statistics because many phenomena tend toward normality under the central limit theorem. It underpins standard scores, confidence intervals, and many inferential methods.

Normal subgroup: In abstract algebra, a subgroup N of a group G is normal if it is invariant under conjugation: gNg^{-1} = N for all g in G. Normal subgroups enable the construction of quotient groups and other algebraic objects.

Normal matrix: In linear algebra, a matrix A is normal if it commutes with its conjugate transpose, AA^* = A^*A. Normal matrices include Hermitian, unitary, and real symmetric matrices; they are unitarily diagonalizable, meaning they can be diagonalized by a unitary transformation.

Normality in chemistry: Historically, normality (N) expressed concentration as equivalents per liter. While still encountered in some contexts, it has largely been supplanted by molarity and other concentration measures due to definition complexities in reactions with varying equivalents.