What is the X in Xmas?
"intend, have in mind," Old English mænan "to mean, intend, signify; tell, say; complain, lament," from West Germanic *mainijan (cf. Old Frisian mena "to signify," Old Saxon menian "to intend, signify, make known," Dutch menen, German meinen "think, suppose, be of the opinion"), from PIE *meino "opinion, intent" (cf. Old Church Slavonic meniti "to think, have an opinion," Old Irish mian "wish, desire," Welsh mwyn "enjoyment"), perhaps from root *men "think" (see mind (n.)). Conversational question you know what I mean? attested by 1834.
"calculate an arithemtical mean," 1882, from mean (n.).
"lowquality," c.1200, "shared by all," from imene, from Old English gemæne "common, public, general, universal, shared by all," from ProtoGermanic *gamainiz "possessed jointly" (cf. Old Frisian mene, Old Saxon gimeni, Middle Low German gemeine, Middle Dutch gemene, Dutch gemeen, German gemein, Gothic gamains "common"), from PIE *komoini "held in common," a compound adjective formed from collective prefix *ko "together" (ProtoGermanic *ga) + *moin, suffixed form of PIE root *mei "to change, exchange" (see mutable). Cf. second element in common (adj.), a word with a sense evolution parallel to that of this word.
Of things, "inferior, secondrate," from late 14c. (a secondary sense in Old English was "false, wicked"). Notion of "soso, mediocre" led to confusion with mean (n.). Meaning "inferior in rank or status" (of persons) emerged early 14c.; that of "ordinary" from late 14c.; that of "stingy, nasty" first recorded 1660s; weaker sense of "disobliging, pettily offensive" is from 1839, originally American English slang. Inverted sense of "remarkably good" (i.e. plays a mean saxophone) first recorded c.1900, perhaps from phrase no mean _______ "not inferior" (1590s, also, "not average," reflecting further confusion with mean (n.)).
"occupying a middle or intermediate place," mid14c., from AngloFrench meines (plural), Old French meien, variant of moiien "mid, medium, common, middleclass" (12c., Modern French moyen), from Late Latin medianus "of the middle," from Latin medius "in the middle" (see medial (adj.)). Meaning "intermediate in time" is from mid15c. Mathematical sense is from late 14c.
"that which is halfway between extremes," early 14c., from Old French meien "middle, means, intermediary," noun use of adjective from Latin medianus "of or that is in the middle" (see mean (adj.2)). Oldest sense is musical; mathematical sense is from c.1500. Some senes reflect confusion with mean (adj.1). This is the mean in by no means (late 15c.).
mean (mēn)
n.
Something having a position, quality, or condition midway between extremes; a medium.
A number that typifies a set of numbers, such as a geometric mean or an arithmetic mean.
The average value of a set of numbers.
Occupying a middle or intermediate position between two extremes.
Intermediate in size, extent, quality, time, or degree; medium.
mean (mēn)

In statistics, an average of a group of numbers or data points. With a group of numbers, the mean is obtained by adding them and dividing by the number of numbers in the group. Thus the mean of five, seven, and twelve is eight (twentyfour divided by three). (Compare median and mode.)
Excellent; wonderful; classy, wicked: This girl has already proved she can play a mean game of tennis/ And Wheelright had a great, mean ear for dialect (1900+ Black)
Related Termslean and mean, shake a wicked calf
in mathematics, a quantity that has a value intermediate between those of the extreme members of some set. Several kinds of mean exist, and the method of calculating a mean depends upon the relationship known or assumed to govern the other members. The arithmetic mean, denoted x, of a set of n numbers x1, x2, , xn is defined as the sum of the numbers divided by n: