In an experiment, the parts being studied can be classed as either dependent or independent variables. The dependent variable is the possible outcome of the experiment; the effect. It depends on what happens to other variables in the experiment. The dependent variable is basically the part that you are changing. For example, if you want to know how much light a plant needs to grow, the amount of growth is the dependent. If you wanted to see if a plant would grow better in hot or cold areas your independent variable would be the temperature of the air.
The independent variable is the variable that you have control over, what you can choose and manipulate. It is usually what you think will affect the dependent variable. In some cases, you may not be able to manipulate the independent variable. It may be something that is already there and is fixed, something you would like to evaluate with respect to how it affects something else, the independent variable like color, kind, time. As an example, you are interested in how stress affects heart rate in humans. Your independent variable would be the stress and the dependent variable would be the heart rate. You can directly manipulate stress levels in your human subjects and measure how those stress levels change heart rate.
In mathematics, the dependent variable changes with the independent variable, hence the name (it depends on the independent variable).
In calculus, a function is a map whose action is specified on variables. Take x and y to be two variables. A function f may map x to some expression in x. Assigning gives a relation between y and x. If there is some relation specifying y in terms of x, then y is known as a dependent variable (and x is an independent variable). A dependent variable is also known as a "response variable", "regressand", "measured variable", "responding variable", "explained variable", "outcome variable", "experimental variable", and "output variable".
If the independent variable is referred to as an "explanatory variable" then the term "response variable" is preferred by some authors for the dependent variable.
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