From normal +‎ -cy. The original sense was mathematical. The broader sense was popularized in the United States by its use by President Warren G. Harding.[1]

• (General American) IPA^(key): /ˈnɔɹməlsi/

• Audio (Southern England):

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normalcy (countable and uncountable, plural normalcies)

1. (mathematics) The state of being geometrically normal, at a right angle.
2. (US) The state of being normal; the fact of being normal; normality.
   • 1920 May 14, Warren G. Harding, “Presidential Speeches: Warren G. Harding Presidency: May 14, 1920: Readjustment”, in Miller Center‎^[2], University of Virginia:

     America's present need is not heroics, but healing; not nostrums, but normalcy; not revolution, but restoration; not agitation, but adjustment; not surgery, but serenity; not the dramatic, but the dispassionate; not experiment, but equipoise; not submergence in internationality but sustainment in triumphant nationality.

   • 1928, Dorothy Parker, “The Searched Soul”, in Sunset Gun, New York: Boni & Liveright, page 36:

     When so I ponder, here apart, / What shallow boons suffice my heart, / What dust-bound trivia capture me, / I marvel at my normalcy.

   • 1981, William Irwin Thompson, The Time Falling Bodies Take to Light: Mythology, Sexuality and the Origins of Culture, London: Rider/Hutchinson & Co., page 193:

     The homosexual dismisses heterosexual love as a distasteful bondage to normalcy and bourgeois domestication, but the Platonic lover of the soul is dismissing all sexuality as bondage to the physical world.

   • 2001, Jeffery Veen, The Art & Science of Web Design, →ISBN, page 104:

     I’ve been consistently awed by how good designers can create normalcy out of chaos; they can clearly communicate ideas through the organizing and manipulating of words and pictures.

Although sometimes used, normalcy is less common than normality in American English. It is very rarely used in the UK, Canada, Australia, and New Zealand. It is frequent in India and Zimbabwe, however.

• normalhood, normality (UK, US), normaldom, normalness; see also Thesaurus:normality

• abnormalcy
• abnormality

• normal
• normality
• normally
• normalness

• Dictionary.com
• Merriam-Webster Online