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The assumption of independence has two related meanings in statistical testing, one relating to the independence of tests and another to the independence of variables.

A definition

Two events are independent if knowing the outcome of one event tells us nothing about other. For example, whether the next car I see is read or blue is independent of whether or not I will win the lottery.

Events are dependent when they are not independent. The events of whether it rains and whether a person is carrying an umbrella are dependent.

Note that a conclusion that events are not independent neither implies that one even causes another nor implies that the relationship between the events is strong. By contrast, a conclusion of independence suggests an absence of a causal relationship and the absence of any relationship.

The independence of variables

Variables are said to be independent if knowledge of the values of one variable provides no indication as to the values of another. For example, height and gender are not independent, because knowledge of a person's gender gives a clue as to their likely height.

The null hypothesis of most statistical tests can be viewed as being one of independence.

The independence of separate statistical tests

In the context of multiple testing (i.e., conducting multiple tests of statistical significance), tests are said to be independent if there is no necessary mathematical relationship between the results of one test and the results of another.

The following two tests are independent:

  • A test of the difference between the heights of men versus women.
  • A test of the difference between average price paid for cereal of blue eyed versus brown eyed people.

By contrast, the following two tests are dependent:

  • A test of the difference between the heights of men versus women.
  • A test of the difference between the weight of men versus women.

These tests are dependent because height and weight are logically related and thus if men are taller than women it follows that they also be heavier.