Dependent Variable

A dependent variable is the quantity in an analysis whose value is being explained or predicted, on the assumption that it depends on one or more other factors called independent variables. In a cost formula, the total cost is the dependent variable and the activity that drives it, such as units produced or machine hours, is the independent variable. In graphs it is conventionally placed on the vertical axis and written as $Y$.

Accountants and financial analysts meet dependent variables mainly in regression analysis, cost estimation and forecasting. The same idea runs through statistics, econometrics and the sciences, where an experiment changes an independent variable and observes the effect on a dependent one. Identifying which variable depends on which is the first step in building any of these models.

Think of a taxi fare. The longer the ride, the higher the fare. The fare depends on the distance, so the fare is the dependent variable and the distance is the independent one. Nobody would say the distance depends on the fare.

Businesses ask the same kind of question all the time. How does the electricity bill change when the factory runs more hours? How do sales respond to advertising? In each case the thing you want to predict, the bill or the sales, is the dependent variable, and the thing you think is causing it to change is the independent variable.

The standard linear cost equation used in management accounting is:

• $Y$ is the dependent variable, the total cost for the period.
• $a$ is the intercept, the estimated fixed cost per period.
• $b$ is the slope, the estimated variable cost per unit of activity.
• $X$ is the independent variable, the level of activity.

Suppose a warehouse's monthly handling cost has been estimated as $Y = 12{,}000 + 4.50X$, where $X$ is the number of orders shipped. In a month with 3,000 orders the predicted cost is 12,000 plus 4.50 times 3,000, or $25,500. The handling cost is the dependent variable; the number of orders is what it depends on.

A model is only as good as the link between the dependent variable and the factors chosen to explain it. Accountants test several candidate drivers and look at how closely each moves with the cost, using correlation and the coefficient of determination. A driver that explains 90 percent of the variation in cost is far more useful than one explaining 30 percent. The relationship should also make operational sense: shipping costs should plausibly depend on orders or weight, not on the number of days of rain.

Where one factor is not enough, multiple regression uses several independent variables at once. For example, a delivery company might model fuel cost as the dependent variable with miles driven and number of stops as two independent variables.

• Cost behavior. Splitting mixed costs into fixed and variable parts, using regression or the high-low method, treats the cost as the dependent variable.
• Budgeting. A flexible budget predicts each cost as a dependent variable of actual activity.
• Profit planning. In cost-volume-profit analysis, profit is the dependent variable and sales volume is the independent variable.
• Finance. When estimating a stock's beta, the stock's return is the dependent variable and the market's return is the independent one.

Calling one variable dependent is an assumption about cause, not a proof of it. Statistical tools show only that two series move together; whether the activity truly drives the cost must come from knowledge of the business. Predictions are also reliable only within the relevant range of past data. A cost equation fitted to months with 1,000 to 4,000 orders may say little about a month with 10,000, when extra staff or space would be needed and the cost relationship would change.