Average Due Date (Average Billing Date)

A trader who owes several separate bills, each for a different amount and each falling due on a different date, may prefer to settle everything with one single payment on one single date instead of paying each bill separately as it falls due. The Average Due Date (also called the Average Billing Date) is the one date on which this single combined payment can be made without either party losing or gaining interest, compared with paying every bill exactly on its own separate due date.

The method: pick any convenient base date (most simply, the earliest of the given due dates), and measure the number of days from the base date to each bill's own due date. Each bill's amount is then a "weight," and the average due date is found by taking the amount-weighted average of these day-counts — exactly the same weighted-average idea already used for the arithmetic mean of grouped data in the Class 11 Statistics chapters of this syllabus, here applied to dates and amounts instead of frequencies and class-marks.

A larger bill pulls the average due date closer to its own due date — exactly as a larger frequency pulls a statistical mean closer to its own class-mark — while a bill due on the base date itself contributes $t_i=0$ to the weighted sum (it still counts in the total amount $\sum a_i$, but adds nothing to the numerator). …