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The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects. They are used to distinguish between two types of quantities being considered, separating them into those available at the start of a process and those being created by it, where the latter (dependent variables) are dependent on the former (independent variables).^{[citation needed]}

Use in Mathematics

In traditional calculus, a function is defined as a relation between two terms called variables because their values vary. Call the terms, for example, x and y. If every value of x is associated with exactly one value of y, then y is said to be a function of x. It is customary to use x for what is called the "independent variable," and y for what is called the "dependent variable" because its value depends on the value of x.^[1] Therefore, y = x² means that y, the dependent variable, is the square of x, the independent variable.^[1][2]

The most common way to denote a "function" is to replace y, the dependent variable, by f(x), where f is the first letter of the word "function." Thus, y = f(x) = x² means that y, a dependent variable, a function of x, is the square of x. Also, in this form, the expression is called an "explicit" function of x, contrasted with x² − y = 0, which is called an "implicit" function.^[1]

Use in Statistics

In statistics, in an experiment, the dependent variable is the event studied and expected to change when the independent variable is changed.^[2]

In the design of experiments, independent variables are those whose values are controlled or selected by the person experimenting (experimenter) to determine its relationship to an observed phenomenon (the dependent variable). In such an experiment, an attempt is made to find evidence that the values of the independent variable determine the values of the dependent variable (that which is being measured). The independent variable can be changed as required, and its values do not represent a problem requiring explanation in an analysis, but are taken simply as given. The dependent variable on the other hand, usually cannot be directly controlled.^{[citation needed]}

Controlled variables are also important to identify in experiments. They are the variables that are kept constant to prevent their influence on the effect of the independent variable on the dependent. Every experiment has a controlling variable, and it is necessary to not change it, or the results of the experiment won't be valid.^{[citation needed]}

In summary:

• Independent variables answer the question "What do I change?".
• Dependent variables answer the question "What do I observe?".
• Controlled variables answer the question "What do I keep the same?".

Alternative terminology in statistics

In statistics, the dependent/independent variable terminology is used more widely than just in relation to controlled experiments. For example the data analysis of two jointly varying quantities may involve treating each in turn as the dependent variable and the other as the independent variable. However, for general usage, the pair "response variable" and "explanatory variable" is preferable as quantities treated as "independent variables" are rarely statistically independent.^[3]

Depending on the context, an independent variable is also known as a "predictor variable," "regressor," "controlled variable," "manipulated variable," "explanatory variable," "exposure variable," and/or "input variable." A dependent variable is also known as a "response variable," "regressand," "measured variable," "observed variable," "responding variable," "explained variable," "outcome variable," "experimental variable," and/or "output variable."^{[citation needed]}

Examples

• If one were to measure the influence of different quantities of fertilizer on plant growth, the independent variable would be the amount of fertilizer used (the changing factor of the experiment). The dependent variables would be the growth in height and/or mass of the plant (the factors that are influenced in the experiment) and the controlled variables would be the type of plant, the type of fertilizer, the amount of sunlight the plant gets, the size of the pots, etc. (the factors that would otherwise influence the dependent variable if they were not controlled).^{[citation needed]}

• In a study of how different doses of a drug affect the severity of symptoms, a researcher could compare the frequency and intensity of symptoms (the dependent variables) when different doses (the independent variable) are administered, and attempt to draw a conclusion.^{[citation needed]}

• In measuring the acceleration of a vehicle, time is usually the independent variable and speed is the dependent variable. This is because when taking measurements, times are usually predetermined, and the resulting speed of the vehicle is recorded at those times. As far as the experiment is concerned, the speed is dependent on the time. Since the decision is made to measure the speed at certain times, time is the independent variable.^{[citation needed]}

• In measuring the amount of color removed from beetroot samples at different temperatures, the dependent variable would be the amount of pigment removed, because it is depending on the temperature (which is the independent variable).^{[citation needed]}

• In sociology, in measuring the effect of education on income or wealth, the dependent variable could be a level of income or wealth measured in monetary units (United States Dollars for example), and an independent variable could be the education level of the individual(s) who compose(s) the household (i.e. academic degrees).^{[citation needed]}

References

At Wikiversity, you can learn about: Independent variable

At Wikiversity, you can learn about: Dependent variable