Independent variable

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Independent variable

The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and statistics

Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
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).

raditional calculus, a function is defined as a relation between two terms called variable

Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
s because their values vary.

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Use in Mathematics

In traditional calculus, a function is defined as a relation between two terms called variable

Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
s 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. Therefore, means that y, the dependent variable, is the square of x, the independent variable.

The most common way to denote a "function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
" is to replace y, the dependent variable, by , where f is the first letter of the word "function." Thus, 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 , which is called an "implicit" function.

Use in Statistics

Controlled experiments

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

In the design of experiments

Design of experiments

Design of experiments, or experimental design, is the design of all information-gathering exercises where variation is present, whether under the full control of the experimenter or not....
, independent variables are those whose values are controlled or selected by the person experimenting (experimenter) to determine its relationship to an observed

Observation

Observation is either an activity of a living being , consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments....
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.

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.

"Extraneous variables" are those that might affect the relationship between the independent and dependent variables. Extraneous variables are usually not theoretically interesting. They are measured in order for the experimenter to compensate for them. For example, an experimenter who wishes to measure the degree to which caffeine intake (the independent variable) influences explicit recall for a word list (the dependent variable) might also measure the participant's age (extraneous variable). She can then use these age data to control for the uninteresting effect of age, clarifying the relationship between caffeine and memory.

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?"
Extraneous variables answer the question "What uninteresting variables might mediate the effect of the IV on the DV?"

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.

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."

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).

In a study of how different dose
Effective dose

An effective dose in pharmacology is the amount of medication that produces a therapeutic response in 50% of the people taking it, sometimes also called ED-50....
s of a drug
Medication

A pharmaceutical drug, also referred to as medicine or medicament, can be loosely defined as any substance intended for use in the diagnosis, cure, mitigation, treatment, or prevention of disease....
affect the severity of symptom
Symptom

A symptom is a departure from normal function or feeling which is noticed by a patient, indicating the presence of disease or abnormality. A symptom is subjective, observed by the patient, and not measured....
s, 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.

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.

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).

In sociology
Sociology

Sociology is a branch of the social sciences that uses systematic methods of Empiricism and critical theory to develop and refine a body of knowledge about human social structure and activity, sometimes with the goal of applying such knowledge to the pursuit of social welfare....
, in measuring the effect of education
Education

File:Inukshuk Monterrey 1.jpgEducation can be seen as a product or a process and considered in a broad sense or a technical sense. According to philosophy of education George F....
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).