A variable is considereddependent if it depends on (or is hypothesized to depend on) anindependent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by amathematical function), on the values of other variables. Independent variables, on the other hand, are not seen as depending on any other variable in the scope of the experiment in question.[a] Rather, they are controlled by the experimenter.
In mathematics, afunction is a rule for taking an input (in the simplest case, a number or set of numbers)[2] and providing an output (which may also be a number or set of numbers).[2] A symbol that stands for an arbitrary input is called anindependent variable, while a symbol that stands for an arbitrary output is called adependent variable.[3] The most common symbol for the input isx, and the most common symbol for the output isy; the function itself is commonly writteny =f(x).[3][4]
It is possible to have multiple independent variables or multiple dependent variables. For instance, inmultivariable calculus, one often encounters functions of the formz =f(x,y), wherez is a dependent variable andx andy are independent variables.[5] Functions with multiple outputs are often referred to asvector-valued functions.
Inmathematical modeling, the relationship between the set of dependent variables and set of independent variables is studied.[citation needed]
In the simplestochasticlinear modelyi = a + bxi +ei the termyi is theith value of the dependent variable andxi is theith value of the independent variable. The termei is known as the "error" and contains the variability of the dependent variable not explained by the independent variable.[citation needed]
With multiple independent variables, the model isyi = a + bxi,1 + bxi,2 + ... + bxi,n +ei, wheren is the number of independent variables.[citation needed]
In statistics, more specifically inlinear regression, ascatter plot of data is generated withX as the independent variable andY as the dependent variable. This is also called a bivariate dataset,(x1,y1)(x2,y2) ...(xi,yi). The simple linear regression model takes the form ofYi = a + Bxi +Ui, fori = 1, 2, ... ,n. In this case,Ui, ... ,Un are independent random variables. This occurs when the measurements do not influence each other. Through propagation of independence, the independence ofUi implies independence ofYi, even though eachYi has a different expectation value. EachUi has an expectation value of 0 and a variance ofσ2.[6]Expectation ofYi Proof:[6]
The line of best fit for thebivariate dataset takes the formy =α +βx and is called the regression line.α andβ correspond to the intercept and slope, respectively.[6]
In anexperiment, the variable manipulated by an experimenter is something that is proven to work, called an independent variable.[7] The dependent variable is the event expected to change when the independent variable is manipulated.[8]
Indata mining tools (formultivariate statistics andmachine learning), the dependent variable is assigned arole astarget variable (or in some tools aslabel attribute), while an independent variable may be assigned a role asregular variable[9] or feature variable. Known values for the target variable are provided for the training data set andtest data set, but should be predicted for other data. The target variable is used insupervised learning algorithms but not in unsupervised learning.
Depending on the context, an independent variable is sometimes called a "predictor variable", "regressor", "covariate", "manipulated variable", "explanatory variable", "exposure variable" (seereliability theory), "risk factor" (seemedical statistics), "feature" (inmachine learning andpattern recognition) or "input variable".[10][11]Ineconometrics, the term "control variable" is usually used instead of "covariate".[12][13][14][15][16]
"Explanatory variable" is preferred by some authors over "independent variable" when the quantities treated as independent variables may not be statistically independent or independently manipulable by the researcher.[17][18] 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.[11][17][18]
Depending on the context, a dependent variable is sometimes called a "response variable", "regressand", "criterion", "predicted variable", "measured variable", "explained variable", "experimental variable", "responding variable", "outcome variable", "output variable", "target" or "label".[11] In economics endogenous variables are usually referencing the target.
"Explained variable" is preferred by some authors over "dependent variable" when the quantities treated as "dependent variables" may not be statistically dependent.[19] If the dependent variable is referred to as an "explained variable" then the term "predictor variable" is preferred by some authors for the independent variable.[19]
An example is provided by the analysis of trend in sea level byWoodworth (1987). Here the dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available. The primary independent variable was time. Use was made of a covariate consisting of yearly values of annual mean atmospheric pressure at sea level. The results showed that inclusion of the covariate allowed improved estimates of the trend against time to be obtained, compared to analyses which omitted the covariate.
| independent | dependent |
| input | output |
| regressor | regressand |
| predictor | predicted |
| explanatory | explained |
| exogenous | endogenous |
| manipulated | measured |
| exposure | outcome |
| feature | label or target |
A variable may be thought to alter the dependent or independent variables, but may not actually be the focus of the experiment. So that the variable will be kept constant or monitored to try to minimize its effect on the experiment. Such variables may be designated as either a "controlled variable", "control variable", or "fixed variable".
Extraneous variables, if included in aregression analysis as independent variables, may aid a researcher with accurate response parameter estimation,prediction, andgoodness of fit, but are not of substantive interest to thehypothesis 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 thedependent variable. If included in a regression, it can improve thefit of the model. If it is excluded from the regression and if it has a non-zerocovariance with one or more of the independent variables of interest, its omission willbias the regression's result for the effect of that independent variable of interest. This effect is calledconfounding oromitted variable bias; in these situations, design changes and/or controlling for a variable statistical control is necessary.
Extraneous variables are often classified into three types:
In modelling, variability that is not covered by the independent variable is designated by and is known as the "residual", "side effect", "error", "unexplained share", "residual variable", "disturbance", or "tolerance".
