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ccount log in personal tools donate create account log in contents move to sidebar hide top 1 in pure mathematics 2 in modeling and statistics 3 synonyms toggle synonyms subsection 3 1 antonym pairs 4 other variables 5 examples 6 see also 7 notes 8 references toggle the table of contents dependent and independent variables 32 languages العربية беларуская বাংলা català deutsch esperanto español euskara فارسی suomi gaeilge עברית हिन्दी kreyòl ayisyen magyar ido italiano 日本語 한국어 मगही nederlands polski português русский simple english svenska தமிழ் türkçe українська tiếng việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from independent variable for dependent and independent random variables see independence probability theory concept in mathematical modeling statistical modeling and experimental sciences a variable is considered dependent if it depends on or is hypothesized to depend on an independent variable dependent variables are the outcome of the test they depend on by some law or rule e g by a mathematical function 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 single variable calculus a function is typically graphed with the horizontal axis representing the independent variable and the vertical axis representing the dependent variable 1 in this function y is the dependent variable and x is the independent variable in pure mathematics edit in mathematics a function 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 an independent variable while a symbol that stands for an arbitrary output is called a dependent variable 3 the most common symbol for the input is x and the most common symbol for the output is y the function itself is commonly written y f x 3 4 it is possible to have multiple independent variables or multiple dependent variables for instance in multivariable calculus one often encounters functions of the form z f x y where z is a dependent variable and x and y are independent variables 5 functions with multiple outputs are often referred to as vector valued functions in modeling and statistics edit in mathematical modeling the relationship between the set of dependent variables and set of independent variables is studied citation needed in the simple stochastic linear model y i a b x i e i the term y i is the i th value of the dependent variable and x i is the 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 citation needed with multiple independent variables the model is y i a b x i 1 b x i 2 b x i n e i where n is the number of independent variables citation needed in statistics more specifically in linear regression a scatter plot of data is generated with x as the independent variable and y as the dependent variable this is also called a bivariate dataset x 1 y 1 x 2 y 2 x i y i the simple linear regression model takes the form of y i a b x i u i for i 1 2 n in this case u i u n are independent random variables this occurs when the measurements do not influence each other through propagation of independence the independence of u i implies independence of y i even though each y i has a different expectation value each u i has an expectation value of 0 and a variance of σ 2 6 proof of the expectation value of y i 6 e y i e α β x i u i α β x i e u i α β x i displaystyle operatorname e y_ i operatorname e alpha beta x_ i u_ i alpha beta x_ i operatorname e u_ i alpha beta x_ i the line of best fit for the bivariate dataset takes the form y α βx and is called the regression line α and β correspond to the intercept and slope respectively 6 in an experiment 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 synonyms edit depending on the context an independent variable is sometimes called a predictor variable regressor covariate manipulated variable explanatory variable exposure variable see reliability theory risk factor see medical statistics feature in machine learning and pattern recognition or input variable 9 10 in econometrics the term control variable is usually used instead of covariate 11 12 13 14 15 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 10 in economics endogenous variables are usually referencing the target antonym pairs edit antonym pairs independent dependent input output regressor regressand predictor predicted explanatory explained exogenous endogenous manipulated measured exposure outcome feature label or target 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 16 17 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 10 16 17 explained variable is preferred by some authors over dependent variable when the quantities treated as dependent variables may not be statistically dependent 18 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 18 in data mining tools for multivariate statistics and machine learning the dependent variable is called target variable or in some tools as label attribute while an independent variable may be called a regular variable 19 or feature variable other variables edit 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 are candidate independent variables which may be included in a regression analysis to 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 controlling for a variable are 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 blocking variables or 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 time of day in modelling variability that is not covered by the independent variable is designated by e i displaystyle e_ i and is known as the residual side effect error unexplained share residual variable disturbance or tolerance examples edit effect of fertilizer on plant growths 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 sugar added in a coffee the taste varies with the amount of sugar added in the coffee here the sugar is the independent variable while the taste is the dependent variable sea level change an example is provided by the analysis of trend in sea level by woodworth 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 see also edit abscissa and ordinate blocking statistics latent and observable variables mediation statistics observed value notes edit even if the existing dependency is invertible e g by finding the inverse function when it exists the nomenclature is kept if the inverse dependency is not the object of study in the experiment references edit hastings nancy baxter 1998 workshop calculus guided exploration with review vol 2 springer science business media p 31 1 2 carlson robert a concrete introduction to real analysis crc press 2006 p 183 1 2 stewart james 2011 1 1 calculus cengage learning anton howard irl c bivens and stephen davis calculus single variable john wiley sons 2012 section 0 1 larson ron and bruce edwards calculus cengage learning 2009 section 13 1 1 2 3 dekking frederik michel 2005 a modern introduction to probability and statistics understanding why and how springer isbn 1 85233 896 2 oclc 783259968 variables random house webster s unabridged dictionary random house inc 2001 page 534 971 isbn 0 375 42566 7 dodge y 2003 the oxford dictionary of statistical terms oup isbn 0 19 920613 9 entry for independent variable 1 2 3 dodge y 2003 the oxford dictionary of statistical terms oup isbn 0 19 920613 9 entry for regression gujarati damodar n porter dawn c 2009 terminology and notation basic econometrics fifth international ed new york mcgraw hill p 21 isbn 978 007 127625 2 wooldridge jeffrey 2012 introductory econometrics a modern approach fifth ed mason oh south western cengage learning pp 22 23 isbn 978 1 111 53104 1 last john m ed 2001 a dictionary of epidemiology fourth ed oxford up isbn 0 19 514168 7 everitt b s 2002 the cambridge dictionary of statistics 2nd ed cambridge up isbn 0 521 81099 x woodworth p l 1987 trends in u k mean sea level marine geodesy 11 1 57 87 bibcode 1987marge 11 57w doi 10 1080 15210608709379549 1 2 everitt b s 2002 cambridge dictionary of statistics cup isbn 0 521 81099 x 1 2 dodge y 2003 the oxford dictionary of statistical terms oup isbn 0 19 920613 9 1 2 ash narayan sah 2009 data analysis using microsoft excel new delhi isbn 978 81 7446 716 4 english manual version 1 0 archived 2014 02 10 at the wayback machine for rapidminer 5 0 october 2013 wikiversity has learning resources about independent variable wikiversity has learning resources about dependent variable v t e differential equations classification operations differential operator notation for differentiation ordinary partial differential algebraic integro differential fractional linear non linear holonomic attributes of variables dependent and independent variables homogeneous nonhomogeneous coupled decoupled order degree autonomous exact differential equation on jet bundles relation to processes difference discrete analogue stochastic stochastic partial delay solutions existence uniqueness picard lindelöf theorem peano existence theorem carathéodory s existence theorem cauchy kowalevski theorem solution topics wronskian phase portrait phase space lyapunov stability asymptotic stability exponential stability rate of convergence series solutions integral solutions numerical integration dirac delta function solution methods inspection substitution separation of variables method of undetermined coefficients variation of parameters integrating factor integral transforms euler method finite difference method crank nicolson method runge kutta methods finite element method finite volume method galerkin method perturbation theory examples list of named differential equations list of linear ordinary differential equations list of nonlinear ordinary differential equations list of nonlinear partial differential equations mathematicians isaac newton gottfried wilhelm leibniz leonhard euler jacob bernoulli émile picard józef maria hoene wroński ernst lindelöf rudolf lipschitz joseph louis lagrange augustin louis cauchy john crank phyllis nicolson carl david tolmé runge martin kutta sofya kovalevskaya retrieved from https en wikipedia org w index php title dependent_and_independent_variables oldid 1374195356 categories design of experiments regression analysis mathematical terminology independence probability theory hidden categories articles with short description short description matches wikidata all articles with unsourced statements articles with unsourced statements from february 2024 articles with unsourced statements from november 2019 webarchive template wayback links this page was last edited on 10 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