Chess sets come in various different sizes. In determining a suitable size for the chess set that you are buying, there are a couple of factors to consider.
If it is for personal use, then the size is only a matter of personal preference. An average size set would be appropriate it’s meant to be used for playing at home.
On the other hand, a small and portable set would be more convenient if you intend to carry it around.
If it is to be used for tournament play, then the size of the chess pieces and boards should follow the U.S. Chess Federation standards.
For the King, it has to measure 3-3/8″ to 4-1/2″ with a base diameter of 40-50% of the height.
For the chess board, a general rule for determining the proper square size is that the width of the base of the King should be 75% of the width of a square.
Another good guideline that you can use to determine the proportion of the size of the square to the chess piece is the USCF specifics:
“The pieces should fit comfortably on the board, being neither too crowded nor too isolated on the squares. The king and queen, for example, should be subject to easy placement on a square without touching any edge. Boards for standard sets should have squares of approximately 2-2 1/2 inches (5.08-6.35cm). One convenient test is that the square formed by the bases of four pawns should be about the same size as any square on their board.”