Chapter 1
Relationships
1.2 Mathematical Models
1.2.2 Independent and Dependent Variables
In some relationships between varying quantities, it may be that one of the quantities determines the values of the other, in which case we say the second quantity “depends on” the first. When this is true, the quantity that depends on the other is called the dependent variable; the other is called the independent variable. For example, because the stopping distance for your car depends on the speed you are traveling, speed is the independent variable and stopping distance is the dependent variable.
There are three other ways the words “independent” and “dependent” may be used to describe related variables:
First, there may be a loose association, in which case either variable may be considered independent and the other dependent. We saw an example of this with the basketball statistics in the previous section. There appears to be an association between number of points scored and number of fouls committed, but neither points nor fouls causes the other. It makes no difference which variable is called independent and which dependent.
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Second, there may be a more definite relationship in which one of the variables is naturally independent. In the per capita cigarette consumption example on the preceding page, there is a definite number representing the per capita consumption for each year. The date on the calendar does not cause cigarette smoking, nor does a certain level of smoking cause a year to happen. Nevertheless, we speak of time as the independent variable and consumption as the dependent variable. We could label the variables the other way around, but that would not seem natural.
Third, it is possible that a value of each of the variables determines — even causes — a corresponding value of the other. For example, the amount people are willing to pay for bonds is related to the effective rate of return on the bonds. At times the rate might determine the price, but sometimes the price also determines the rate. We might declare either variable to be independent and the other dependent as we set up a description of this relationship.
