It is not necessary for the dependent variable to increase by the same amount as the independent variable increases in order for a linear relationship to exist between the independent and the dependent variable. However, it is necessary for the dependent variable (y) to increase by the same amount for each unit of increase in the independent variable (x). If the amount of increase in the dependent variable varies, the line on the graph will not be a straight one and the relationship will not be a linear one.
A direct relationship between two variables graphs as an upsloping line on an x-y coordinate graph. An upsloping line means that when the independent variable (x) increases, the dependent variable (y) also increases; and when the independent variable decreases, the dependent variable also decreases. A linear relationship may be a direct relationship, but it is not required to be a direct relationship. It may also be an inverse relationship, in which the dependent variable changes in the opposite direction from the direction of change in the independent variable.
If the relationship between an independent variable and a dependent variable is linear, it will graph as a straight line on an x-y coordinate graph. For each unit that the x variable changes, a proportional change will take place in the y variable, and this proportional amount of change is a constant amount. A linear function will have the form ? = ax + b, where ? = the predicted value of y on the regression line corresponding to each value of x; a = the slope of the line (the amount by which y increases for each unit of increase in x); b = the y intercept, or the value of y when x is 0; x = the value of x on the x axis that corresponds to the value of y on the regression line.
Just because a linear relationship exists between an independent variable and a dependent variable, that does not necessarily mean that a forecast made using the historical data will be reasonably accurate. In order for a forecast to be reasonably accurate, the historical changes in the two variables need to be correlated and there needs to be some reason for believing that changes in the independent variable are a cause of changes in the dependent variable. Data can be linear without much correlation and without causation being present. And even when the variables appear to be well correlated, there may be no logical causative relationship between them. For example, one might say "As electric bills increase, sales of ice cream increase. Therefore, an increase in electricity use causes an increase in sales of ice cream." Even though electricity usage and ice cream sales might increase and decrease in tandem, there is no support for the statement that an increase in electricity usage causes an increase in sales of ice cream. It is, however, very possible that both variables are being caused by a third factor, such as hot weather. A graph of temperatures on the X axis and ice cream sales on the Y axis would likely indicate a strong correlation along with causation.
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