medianen

Medianen is the plural form of the term median in several languages, and in mathematical usage it refers to two related concepts: the statistical median and the geometric medians of a triangle.

In statistics, the median is a measure of central tendency that identifies the middle value of a data set when the values are arranged in order. For a data set with n values, if n is odd the median is the ((n+1)/2)-th value; if n is even, the median is the average of the two central values. The median is robust to outliers and skewed distributions, making it a preferred summary statistic for non-symmetric data. The population median is the value m for which the distribution function F satisfies F(m) = 0.5. In practice, the sample median is computed by sorting the data or by using efficient selection algorithms, particularly for large streams.

Geometrically, the medians of a triangle are the segments joining each vertex to the midpoint of the opposite side. All three medians intersect at a single point, the centroid, which divides each median in a 2:1 ratio from the vertex toward the opposite side. The medians partition the triangle into six smaller triangles of equal area. The length of a median can be computed from the triangle’s side lengths via Apollonius’s theorem: m_a^2 = (2b^2 + 2c^2 − a^2)/4, where a, b, c are the side lengths and m_a is the median to side a.

Beyond these contexts, the term appears in various fields to denote central-line concepts or central tendency, depending on the domain.