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When a change in one variable brings about a change in another variable?

Causation means that changes in one variable brings about changes in the other; there is a cause-and-effect relationship between variables.

What is a variable that changes as another variable changes?

A dependent variable is a variable whose value will change depending on the value of another variable, called the independent variable. In a scientific experiment, it is the variable being tested, and therefore, it is called the dependent variable.
Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other.

What is a variable that is presumed to cause a change in another variable called?

A variable that is presumed to cause a change in another variable is called: An intervening variable.