Independence for a country, organization or branch of government is the same as self-rule, as opposed to being ruled by someone else. Establishing such independence sometimes implies a violent break-out or spin-off, and is the reason for recurring celebration by the winner.
The date of independence ported into many articles from the CIA World Factbook is defined as follows: for most countries, [..] date that sovereignty was achieved [..]. For the other countries, the date given may not represent "independence" in the strict sense, but rather some significant nationhood event such as the traditional founding date or the date of unification, federation, confederation, establishment, fundamental change in the form of government, or state succession.
In mathematics and sciences, independence is a relationship between two variables meaning that a change in one of them doesn't imply a change in the other.