Origins
The noun and adjective 'mean' in its mathematical and philosophical sense — the middle value, the average, the point equidistant between extremes — arrived in English from a completely different direction than either its verb sibling or its adjective sibling. Where the verb 'mean' (to intend) comes from an old Germanic root for thought, and the adjective 'mean' (unkind) descends from a PIE root for exchange, this 'mean' comes through French from the very heart of classical Latin's vocabulary for spatial and conceptual centrality.
The direct English source is Old French 'meien' (also 'moyen,' still the modern French word for average or middle), which itself comes from Latin 'mediānus,' an adjective derived from 'medius' (middle, in the middle). Latin 'medius' is one of the best-attested words in all of Indo-European linguistics: it descends directly from PIE *médʰyos (middle, midmost), a reconstruction supported by cognates in nearly every branch of the family.
Sanskrit 'madhya' (middle), Greek 'mésos' (middle, the source of 'Mesopotamia' — the land between the rivers — and 'Mesoamerica'), Old Irish 'mid,' Welsh 'mid,' Old Church Slavonic 'meždu' (between), and Gothic 'midjis' all derive from *médʰyos with entirely regular sound correspondences. The reconstruction is one of the most secure in PIE linguistics, and the persistence of the root across such geographically and temporally dispersed languages testifies to its centrality in the conceptual vocabulary of the Proto-Indo-European speakers.
French Influence
Latin 'medius' produced an enormous family of English words, almost all arriving through learned borrowing or Old French. 'Medium' (the middle substance or intermediate agent), 'median' (the middle value in a sorted sequence, or the central dividing strip of a road), 'mediate' (to occupy a middle position between parties), 'medieval' (from 'medium aevum,' the middle age), 'Mediterranean' (from 'medius' + 'terra,' the sea in the middle of the land), 'mediocre' (from 'mediocris,' literally at a middle height — a mediocre person reaches only halfway up the mountain of achievement), and 'immediate' (from 'immediatus,' with nothing in the middle — hence without delay) all share this ancestor.
The English word 'middle' itself is from Old English 'middel,' from Proto-Germanic *midjaz, the Germanic reflex of the same PIE root — so 'middle' and 'mean' (average) are doublets at the deepest level, one a native Germanic inheritance and one a Latin borrowing, both expressing the same ancient concept of centrality.
In mathematics and statistics, 'mean' specifically denotes an average calculated by summing values and dividing by their number — the arithmetic mean, as distinguished from the median (the middle value by position) and the mode (the most frequent value). The terminology became established in English mathematical writing during the seventeenth century, as algebra and statistics developed as formal disciplines. The philosophical sense — the 'golden mean,' the Aristotelian 'mesotēs' (middleness) as the virtuous path between extremes — predates the mathematical usage and reflects the same Latin 'medius' mediated through Scholastic philosophy.
Latin Roots
Aristotle's doctrine of the mean ('to meson' in Greek) was the ethical principle that virtue lies between the deficiency and the excess of any quality: courage is the mean between cowardice and recklessness, generosity the mean between miserliness and prodigality. This concept was central to medieval European ethical thought, transmitted through Latin translations of Aristotle where 'mesotēs' was regularly rendered as 'mediocritas' or 'medium.' The phrase 'golden mean' (from the Latin 'aurea mediocritas' of Horace's Odes) entered English in the sixteenth century and has since become a stand-alone cultural phrase, divorced from its Latin origin but still carrying the Aristotelian sense of balanced moderation.
The three-way collision of 'mean' (to intend), 'mean' (unkind/common), and 'mean' (average) in English is extraordinary even by English standards. Three entirely separate etymological families — Germanic *mēnijan (to intend), Germanic *gamainiz (common), and Latin 'mediānus' (middle) — produced three words that converged on identical spelling and pronunciation through independent sound changes. No other common English word is quite so radically polysemous through pure etymological accident.