How do you find the relationship between independent and dependent variables?

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To understand the concept of independent and dependent variables, one should understand the meaning of variables. Variables are defined as the properties or kinds of characteristics of certain events or objects.

Independent variables are variables that are manipulated or are changed by researchers and whose effects are measured and compared. The other name for independent variables is Predictor(s). The independent variables are called as such because independent variables predict or forecast the values of the dependent variable in the model.

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The other variable(s) are also considered the dependent variable(s). The dependent variables refer to that type of variable that measures the affect of the independent variable(s) on the test units. We can also say that the dependent variables are the types of variables that are completely dependent on the independent variable(s). The other name for the dependent variable is the Predicted variable(s). The dependent variables are named as such because they are the values that are predicted or assumed by the predictor / independent variables. For example, a student’s score could be a dependent variable because it could change depending on several factors, such as how much he studied, how much sleep he got the night before he took the test, or even how hungry he was when he took it. Usually when one is looking for a relationship between two things, one is trying to find out what makes the dependent variable change the way it does.

Let us identify independent and dependent variables in the following cases:
In the case of a linear model, we have the general equation as:

Here, Y is the variable dependent on X, therefore, X, is an independent variable.

Similarly, in cases of the regression model, we have

Here, the regressors, ßij (j=1, p) are the independent variables and the regressands Yi are the dependent variables.

Independent variables are also called “regressors,“ “controlled variable,” “manipulated variable,” “explanatory variable,” “exposure variable,” and/or “input variable.” Similarly, dependent variables are also called “response variable,” “regressand,” “measured variable,” “observed variable,” “responding variable,” “explained variable,” “outcome variable,” “experimental variable,” and/or “output variable.”

A few examples can highlight the importance and usage of dependent and independent variables in a broader sense.

If one wants to measure the influence of different quantities of nutrient intake on the growth of an infant, then the amount of nutrient intake can be the independent variable, with the dependent variable as the growth of an infant measured by height, weight or other factor(s) as per the requirements of the experiment.

If one wants to estimate the cost of living of an individual, then the factors such as salary, age, marital status, etc. are independent variables, while the cost of living of a person is highly dependent on such factors. Therefore, they are designated as the dependent variable.

In the case of time series analysis, forecasting a price value of a particular commodity is again dependent on various factors as per the study. Suppose we want to forecast the value of gold, for example. In this case the seasonal factor can be an independent variable on which the price value of gold will depend.

In the case of a poor performance of a student in an examination, the independent variables can be the factors like the student not attending classes regularly, poor memory, etc., and these will reflect the grade of the student. Here, the dependent variable is the test score of the student.

An independent variable is the variable that is changed or controlled in a scientific experiment to test the effects on the dependent variable.

A dependent variable is the variable being tested and measured in a scientific experiment.

The dependent variable is 'dependent' on the independent variable. As the experimenter changes the independent variable, the effect on the dependent variable is observed and recorded.

Independent vs Dependent Variable

• There can be many variables in an experiment, but the two key variables that are always present are the independent and dependent variable.
• The independent variable is the one that the researcher intentionally changes or controls.
• The dependent variable is the factor that the research measures. It changes in response to the independent variable or depends upon it.

Independent and Dependent Variable Examples

For example, a scientist wants to see if the brightness of light has any effect on a moth being attracted to the light. The brightness of the light is controlled by the scientist. This would be the independent variable. How the moth reacts to the different light levels (distance to light source) would be the dependent variable.

As another example, say you want to know whether or not eating breakfast affects student test scores. The factor under the experimenter's control is the presence or absence of breakfast, so you know it is the independent variable. The experiment measures test scores of students who ate breakfast versus those who did not. Theoretically, the test results depend on breakfast, so the test results are the dependent variable. Note that test scores are the dependent variable, even if it turns out there is no relationship between scores and breakfast.

For another experiment, a scientist wants to determine whether one drug is more effective than another at controlling high blood pressure. The independent variable is the drug, while patient blood pressure is the dependent variable. In some ways, this experiment resembles the one with breakfast and test scores. However, when comparing two different treatments, such as drug A and drug B, it's usual to add another variable, called the control variable. The control variable, which in this case is a placebo that contains the same inactive ingredients as the drugs, makes it possible to tell whether either drug actually affects blood pressure.

How to Tell the Variables Apart

The independent and dependent variables may be viewed in terms of cause and effect. If the independent variable is changed, then an effect is seen in the dependent variable. Remember, the values of both variables may change in an experiment and are recorded. The difference is that the value of the independent variable is controlled by the experimenter, while the value of the dependent variable only changes in response to the independent variable.

Remembering Variables With DRYMIX

When results are plotted in graphs, the convention is to use the independent variable as the x-axis and the dependent variable as the y-axis. The DRY MIX acronym can help keep the variables straight:

D is the dependent variable
R is the responding variable
Y is the axis on which the dependent or responding variable is graphed (the vertical axis)

M is the manipulated variable or the one that is changed in an experiment
I is the independent variable
X is the axis on which the independent or manipulated variable is graphed (the horizontal axis)

Independent vs Dependent Variable Key Takeaways

• The independent and dependent variables are the two key variables in a science experiment.
• The independent variable is the one the experimenter controls. The dependent variable is the variable that changes in response to the independent variable.
• The two variables may be related by cause and effect. If the independent variable changes, then the dependent variable is affected.

Sources

• Carlson, Robert (2006). A concrete introduction to real analysis. CRC Press, p.183.
• Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9.
• Edwards, Joseph (1892). An Elementary Treatise on the Differential Calculus (2nd ed.). London: MacMillan and Co.
• Everitt, B. S. (2002). The Cambridge Dictionary of Statistics (2nd ed.). Cambridge UP. ISBN 0-521-81099-X.
• Quine, Willard V. (1960). "Variables Explained Away". Proceedings of the American Philosophical Society. American Philosophical Society. 104 (3): 343–347. 

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Helmenstine, Todd. "Difference Between Independent and Dependent Variables." ThoughtCo. https://www.thoughtco.com/independent-and-dependent-variables-differences-606115 (accessed December 20, 2022).

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