Rigour is the word for strictness, severity or harshness, as in the rigours of a hard winter. Rigor, in our data, is recorded only as a surname of Spanish origin, common in the Philippines. If you mean strictness or thoroughness, rigour is the word you want.
The core difference
Rigour is a noun meaning strictness, severity or harshness; rigor is recorded as a personal surname, not as a variant spelling of it.
- Rigour — strictness, severity or harshness: The rigours of the climate kept most visitors away in winter.
- Rigor — a surname of Spanish origin, found predominantly in the Philippines: Mr. Rigor greeted his neighbours on the way to church.
How to tell them apart
It is easy to expect rigor and rigour to work like colour and color, differing only by an extra letter. Our dictionary data tells a different story: rigor is entered as a surname, most often found in the Philippines, while rigour carries the familiar senses of strictness, harshness and inflexibility.
So when you are reaching for the word that means demanding standards, thorough method, or harsh conditions, rigour is the one to use. Rigor, on this evidence, belongs to a name rather than to that meaning.