### Independent variable

**
Variables used in an experiment or modelling can be divided into three types: "****dependent variable**", "**independent variable**", or other. The "dependent variable" represents the output or effect, or is tested to see if it is the effect. The "independent variables" represent the inputs or causes, or are tested to see if they are the cause. Other variables may also be observed for various reasons.

## Contents

## Use

### Calculus

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 $y\; =\; f(x)$ 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).

### Statistics

In a statistics experiment, the dependent variable is the event studied and expected to change whenever the independent variable is altered.^{[1]}

### Modelling

In mathematical modelling, the dependent variable is studied to see if and how much it varies as the independent variables vary. In the simple stochastic linear model $y\_i\; =\; a\; +\; bx\_i\; +\; e\_i\backslash $ the term $y\_i$ is the *i* ^{th} value of the dependent variable and $x\_i$ is *i* ^{th} value of the independent variable. The term $e\_i$ is known as the "error" and contains the variability of the dependent variable not explained by the independent variable.

With multiple independent variables, the expression is: $y\_i\; =\; a\; +\; bx\_1\; +\; bx\_2\; +\; ...\; +\; bx\_n\; +\; e\_i\backslash $, where *n* is the number of independent variables.

### Simulation

In simulation, the dependent variable is changed in response to changes in the independent variables.

## Statistics synonyms

### Independent variable

An independent variable is also known as a "predictor variable", "regressor", "controlled variable", "manipulated variable", "explanatory variable", "exposure variable" (see reliability theory), "risk factor" (see medical statistics), "feature" (in machine learning and pattern recognition) or an "input variable."^{[2]}^{[3]}

"**Explanatory variable**" is preferred by some authors over "independent variable" when the quantities treated as "independent variables" may not be statistically independent.^{[4]}^{[5]}

Independent variable(s) may be of these kinds: continuous variable(s), binary/dichotomous variable(s), nominal categorical variable(s), ordinal categorical variable(s), among others.

### Dependent 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".^{[3]}

If the independent variable is referred to as an "explanatory variable" (see above) then the term "response variable" is preferred by some authors for the dependent variable.^{[3]}^{[4]}^{[5]}

## Other variables

A variable may be thought to alter the dependent or independent variables, but may not actually be the focus of the experiment. So that variable will be kept constant or monitored to try to minimise its effect on the experiment. Such variables may be called a "controlled variable" or "control variable" or "extraneous variable".

Extraneous variables, if included in a regression as independent variables, may aid a researcher with accurate response parameter estimation, prediction, and goodness of fit, but are not of substantive interest to the hypothesis under examination. For example, in a study examining the effect of post-secondary education on lifetime earnings, some extraneous variables might be gender, ethnicity, social class, genetics, intelligence, age, and so forth. A variable is extraneous only when it can be assumed (or shown) to influence the dependent variable. If included in a regression, it can improve the fit of the model. If it is excluded from the regression and if it has a non-zero covariance with one or more of the independent variables of interest, its omission will bias the regression's result for the effect of that independent variable of interest. This effect is called confounding or omitted variable bias; in these situations, design changes and/or statistical control is necessary.

Extraneous variables are often classified into three types:

- Subject variables, which are the characteristics of the individuals being studied that might affect their actions. These variables include age, gender, health status, mood, background, etc.
- Experimental variables are characteristics of the persons conducting the experiment which might influence how a person behaves. Gender, the presence of racial discrimination, language, or other factors may qualify as such variables.
- Situational variables are features of the environment in which the study or research was conducted, which have a bearing on the outcome of the experiment in a negative way. Included are the air temperature, level of activity, lighting, and the time of day.

In quasi-experiments, differentiating between dependent and other variables may be downplayed in favour of differentiating between those variables that can be altered by the researcher and those that cannot. Variables in quasi-experiments may be referred to as "extraneous variables", "subject variables", "experimental variables", "situational variables", "pseudo-independent variables", "ex post facto variables", "natural group variables" or "non-manipulated variables".

In modelling, variability that is not covered by the explanatory variable is designated by $e\_i$ and is known as the "residual", "side effect", "error", "unexplained share", "residual variable", or "tolerance".

## Examples

- Effects of vitamin C on life span

- In a study whether taking vitamin C pills daily make people live longer, researchers will dictate the vitamin C intake of a group of people over time. One part of the group will be given vitamin C pills daily. The other part of the group will be given a placebo pill. Nobody in the group knows which part they are in. The researchers will check the life span of the people in both groups. Here, the dependent variable is the life span and the independent variable is a binary variable for the use or non-use of vitamin C.

- Effect of fertilizer on plant growth

- In a study measuring the influence of different quantities of fertilizer on plant growth, the independent variable would be the amount of fertilizer used. The dependent variable would be the growth in height or mass of the plant. 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.

- Effect of drug dosage on symptom severity

- 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 when different doses are administered. Here the independent variable is the dose and the dependent variable is the frequency/intensity of symptoms.

- Effect of temperature on pigmentation

- In measuring the amount of color removed from beetroot samples at different temperatures, temperature is the independent variable and amount of pigment removed is the dependent variable.

- Effect of education on wealth

- In sociology, in measuring the effect of education on income or wealth, the dependent variable is level of income/wealth and the independent variable is the education level of the individual.

## References

has learning materials about Independent variable |

has learning materials about Dependent variable |