Not a teacher, 53-year-old American.
The phrase 'early twenties' would be used to describe someone when you don't know their actual age, but wish to convey a range of ages you think the person MIGHT be. To me - the range is 20-23.
24, 25, & 26 would be 'mid-twenties' (to me).
27, 28 & 29 would be 'late-twenties' (to me).
- For Teachers