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Dependent and independent variables

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).

Simplified example

The independent variable is typically the variable representing the value being manipulated or changed and the dependent variable is the observed result of the independent variable being manipulated. For example concerning nutrition, the independent variable of daily vitamin C intake (how much vitamin C one consumes) can influence the dependent variable of life expectancy (the average age one attains). Over some period of time, scientists will control the vitamin C intake in a substantial group of people. One part of the group will be given a daily high dose of vitamin C, and the remainder will be given a placebo pill (so that they are unaware of not belonging to the first group) without vitamin C. The scientists will investigate if there is any statistically significant difference in the life span of the people who took the high dose and those who took the placebo (no dose). The goal is to see if the independent variable of high vitamin C dosage has a correlation with the dependent variable of people's life span. The designation independent/dependent is clear in this case, because if a correlation is found, it cannot be that life span has influenced vitamin C intake, but an influence in the other direction is possible.

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. Therefore, y = x^2 means that y, the dependent variable, is the square of x, the independent variable.

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) = x^2 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 x^2 - y = 0, which is called an "implicit" function.

Use in statistics

Controlled experiments

In a statistics experiment, the dependent variable is the event studied and expected to change whenever the independent variable is altered.

In the design of experiments, an independent variable's values are controlled or selected by the experimenter to determine its relationship to an observed phenomenon (i.e., 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. 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."

In addition, some special types of statistical analysis use terminology more relevant to the specific context. For example reliability theory uses the term exposure variable for what would otherwise be an explanatory or dependent variable, and medical statistics may use the term risk factor.


  • 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 doses of a drug affect the severity of computer programming, a static variable is a variable that has been allocated statically— whose lifetime extends across the entire run of the program. This is in contrast to the more ephemeral automatic variables (local variables), whose storage is allocated and deallocated on the call stack; and in contrast to objects whose storage is dynamically allocated.

    In many programming languages, such as Pascal, all local variables are automatic and all global variables are allocated statically. In these languages, the term "static variable" is generally not used, since "local" and "global" suffice to cover all the possibilities.

    C and related languages

    In the C programming language (and its close descendants such as C++ and Objective-C), staticis a storage class (not to be confused withclasses in object-oriented programming), along with auto(for automatic variables),register(a variant ofauto) and extern(for static variables and functions which are visible in other source files). Every variable in a C program has exactly one of these storage classes.

    Every C variable declared as static has static lifetime and is not visible outside its own translation unit. There are other effects depending on where the variable is defined:

    • Static global variables: variables declared as static at the top level of a source file (outside any function definitions) are visible throughout that file ("file scope").
    • Static local variables: variables declared asstatic inside a function are statically allocated while having the same scope as automatic local variables. Hence whatever values the function puts into its static local variables during one call will still be present when the function is called again.
    • static member variables: in C++, member variables declared as static inside class definitions are class variables (shared between all class instances, as opposed to instance variables).

Random variable

In probability and statistics, a random variable or stochastic variable is a variable whose value is not known. Its possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the potential values of a quantity whose already-existing value is uncertain (e.g., as a result of incomplete information or imprecise measurements). Intuitively, a random variable can be thought of as a quantity whose value is not fixed, but which can take on different values; a probability distribution is used to describe the probabilities of different values occurring. Realizations of a random variable are called random variates.

Random variables are usually real-valued, but one can consider arbitrary types such as boolean values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions, and processes. The term random elementis used to encompass all such related concepts. A related concept is thestochastic process, a set of indexed random variables (typically indexed by time or space).


Real-valued random variables (those whose range is the real numbers) are used in the sciences to make predictions based on data obtained from scientific experiments. In addition to scientific applications, random variables were developed for the analysis of games of chance and stochastic events. In such instances, the function that maps the outcome to a real number is often the identity function or similarly trivial function, and not explicitly described. In many cases, however, it is useful to consider random variables that are functions of other random variables, and then the mapping function included in the definition of a random variable becomes important. As an example, the square of a random variable distributed according to a standard normal distribution is itself a random variable, with a chi-square distribution. One way to think of this is to imagine generating a large number of samples from a standard normal distribution, squaring each one, and plotting a histogram of the values observed. With enough samples, the graph of the histogram will approximate the density function of a chi-square distribution with one degree of freedom.

Another example is the sample mean, which is the average of a number of samples. When these samples are independent observations of the same random event they can be called independent identically distributed random variables. Since each sample is a random variable, the sample mean is a function of random variables and hence a random variable itself, whose distribution can be computed and properties determined.

One of the reasons that real-valued random variables are so commonly considered is that the expected value (a type of average) and variance (a measure of the "spread", or extent to which the values are dispersed) of the variable can be computed.

There are two types of random variables: discrete and continuous. A discrete random variable maps outcomes to values of a countable set (e.g., the integers), with each value in the range having probability greater than or equal to zero. A continuous random variable maps outcomes to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive. A random variable can be "mixed", with part of its probability spread out over an interval like a typical continuous variable, and part of it concentrated on particular values like a discrete variable. These classifications are equivalent to the categorization of probability distributions.

The expected value of random vectors, random matrices, and similar aggregates of fixed structure is defined as the aggregation of the expected value computed over each individual element. The concept of "variance of a random vector" is normally expressed through a covariance matrix. No generally-agreed-upon definition of expected value or variance exists for cases other than just discussed.


The possible outcomes for one coin toss can be described by the state space \Omega = {heads, tails}. We can introduce a real-valued random variable Y as follows:

Y(\omega) = \begin{cases} 1, & \text{if} \ \ \omega = \text{heads} ,\\ 0, & \text{if} \ \ \omega = \text{tails} . \end{cases}

If the coin is equally likely to land on either side then it has a probability mass function given by:

\rho_Y(y) = \begin{cases}\frac{1}{2},& \text{if }y=1,\\

\frac{1}{2},& \text{if }y=0.\end{cases}

From Yahoo Answers

Question:If 3 plants are planted at the same time and added water at different times what are the constant variable, manipulated variable, responding variable of a plant?

Answers:Th constant variable would be the time planted, manipulated variable would be the times water, and responding variable would be the growth of the plant.

Question:when it is just a number, like the population mean? For example a simple random sample of size 50 produced x=938.5; how can the number 938.5 be a random variable?"

Answers:Let's say you consider the population of the United States. You want to know what their average age is. That means you ask every person in the population their age and then calculate the average. There is only 1 possible answer. There is nothing random about it. But a sample mean is different. Say you don't have enough time and money to ask every single person in the US their age. So you select 100 random people and you average their ages and it comes out to 35. That is a sample mean. But later in the day if you select another 100 random people and average their age, it might be 29.5. That is a sample mean. You ask another 100 random people and average their age, you might get 36.3. That is a sample mean. Therefore sample mean is not deterministic. It depended on who you sampled which was random.

Question:just the definition of a variable, not of a dependent or independent variable

Answers:a variable is any value which is not a constant, that is, it can change.

Question:Please give me a definition of a variable used in VB

Answers:A variable is a location in memory where a value can be stored during the execution of a Visual Basic application. Visual Basic variables are assigned names by the programmer when they are declared so that they can easily be referenced in other places in the application code.

From Youtube

Variable Definition in Scala :Short video that explains how to define a new variable in Scala. The video explains the difference between a variable we define using var and a variable we define using val.

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