# [Statistics] Independent vs. Dependent Variables

• statisticsquestion
In summary, the conversation is about determining the independent and dependent variables in a data set that gives the number of miles traveled and travel time for 10 cars' driving assignments. The discussion includes using regression analysis to determine these variables and predicting travel time based on distance traveled. There is confusion about which variable should be considered independent and dependent, with the grader arguing that time is the dependent variable in this scenario. However, the question and the grader's response are both considered poor, as regression analysis cannot determine the type of driver the data comes from and can only show the relationship between time and distance traveled. Overall, it is important to use common sense and context to determine which variable should be considered independent or dependent in a given scenario.
statisticsquestion
Hi everyone I have a quick question about independent and dependent variables.

## Homework Statement

The following data set gives the number of miles traveled, and the travel time in hours for each of the 10 car's driving assignments.

Miles Time
90 9
40 5
90 9
90 6
45 4
75 6
60 7
63 6
85 7
85 6

PART A.
Use regression analysis to determine the independent variable and the dependent variable.

PART E.
Assuming that 75 miles are traveled by a car, use your model to predict the travel time.

## The Attempt at a Solution

I said that the independent variable was time and the dependent variable was miles traveled. My solution was marked incorrect. Here is my justification:

"I was under the impression that distance is always a function of time traveled and therefore travel time ought to be the independent variable because the distance depends on changes in time.
In physics and mathematics, velocity and acceleration are 1st and 2nd derivatives of displacement respectively. This is so because they depend on displacement (distance traveled) with respect to time."

"The dependent variable is the variable that we are trying to predict. So it varies based on the scenario and the problem that needs to be solved. In this case we need to predict travel time (see part e). Hence travel time in this scenario is the dependent variable."

My confusion
I feel like the question could have been elaborated better (explicitly stating which variable was under direct control). Nevertheless, it doesn't make sense to me how I would have to look at the question in part E to determine how to move forward for part A.

Can someone please further elucidate why time is not the independent variable in this instance?

Thanks.

statisticsquestion said:
Use regression analysis to determine the independent variable and the dependent variable.

In simple linear regression, the model used to describe the relationship between a single dependent variable Sat time t as y and a single independent variable say distance traveled as x is
y = a0+ a1x + k.
a0 and a1 are referred to as the model parameters, and k is a probabilistic error term that accounts for the variability in y that cannot be explained by the linear relationship with x.
If the error term were not present, the model would be deterministic; in that case, knowledge of the value of x would be sufficient to determine the value of y.
alternatively a scatter graph of distance and time can be drawn and a mean linar graph can be used for prediction.
Regarding the grader's argument ... i can say that they wanted to predict the time taken , so the dependent variable can be taken to be 'time'.

statisticsquestion said:
My confusion
I feel like the question could have been elaborated better (explicitly stating which variable was under direct control). Nevertheless, it doesn't make sense to me how I would have to look at the question in part E to determine how to move forward for part A.

Can someone please further elucidate why time is not the independent variable in this instance?

A "driving assignment" is some given destination ie drive to bobs business on main street.
The time required to get to that location is dependant on how far away that location is.

A "driving assignment" is not - drive for X hours and let's see how far you get. (unless you consider le Mans 24hr race a driving assignment..)
I think it's fairly reasonable to expect students to determine this via common sense.

However, the question is poor and so is the graders response. Regression analysis cannot determine the dependant and independent variable in this case.
If you are a delivery driver, time is dependant, if you're a Le Mans driver, distance is dependant, regression cannot determine what type of driver the data comes from.
All regression can say is that driving for more time results in greater distance traveled and vice versa.

,

## 1. What is the difference between independent and dependent variables in statistics?

Independent variables are factors that are manipulated or controlled by the researcher in an experiment, while dependent variables are the outcome or response that is measured based on changes in the independent variable. In other words, independent variables cause changes in the dependent variable.

## 2. Can a variable be both independent and dependent?

No, a variable can only be either independent or dependent in a given experiment. However, a variable can be independent in one study and dependent in another, depending on the research question and design.

## 3. How do you identify independent and dependent variables in a study?

To identify the independent and dependent variables, you need to determine what the researcher is manipulating or controlling (independent variable) and what is being measured or observed (dependent variable). The independent variable is typically listed first in a research question or hypothesis, and the dependent variable follows after words like "affect," "influence," or "effect."

## 4. What is the purpose of distinguishing between independent and dependent variables?

Distinguishing between independent and dependent variables is crucial in research as it helps establish cause-and-effect relationships. By manipulating the independent variable and observing changes in the dependent variable, researchers can determine whether the independent variable has a significant impact on the dependent variable.

## 5. Can there be more than one independent or dependent variable in a study?

Yes, there can be multiple independent and dependent variables in a study. This is often the case in complex research studies where multiple factors are being manipulated and measured. It is important to clearly define and control for all independent and dependent variables in order to accurately analyze the results.

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