Normality and normalcy are different spellings of the same word. Normality is centuries older, though, and many usage authorities consider it the superior form. Nouns ending in -cy are usually derived from adjectives ending in -t—for example, pregnancy from pregnant, complacency from complacent, hesitancy from hesitant—while adjectives ending in -l usually take the -ity suffix. Normalcy is unique in flouting this convention.
Normalcy was popularized in the early 20th century thanks to President Warren G. Harding’s “return to normalcy” campaign slogan (though the word did exist before then), and language authorities have been unable to stamp it out.
Examples
Most edited publications prefer normality—for example:
Sunday saw Cairo begin to return to a kind of normality. [Los Angeles Times]
He said the company was focusing on restoring normality. [Wall Street Journal]
Since being released from all restrictions, Sihali’s life has at least acquired some semblance of normality. [The Guardian]
But some (including some of the same publications that usually use normality) have no qualms about using normalcy at least some of the time—for example:
It’s an attempt to retain normalcy in his life. [Boston Globe]
These days, I imagine, he would be diagnosed with ADHD and medicated into normalcy. [Sydney Morning Herald]
And while the government hailed what it called a return to normalcy, the protesters vowed that there was no turning back. [New York Times]
Ngram
This Ngram charts the use of normalcy and normality in English-language books published from 1800 to 2000.
The graph limited to American English is roughly the same. Normalcy is a little less common in British English.
