A statistical analysis of marketing promotions

Journal of the Academy of Business and Economics, March, 2004 by John S. Croucher

ABSTRACT

This paper examines three types of common marketing promotions and their possible outcomes. The first of these concerns a scratch card give-away where customers can win cash or prizes simply by making purchases. The second involves a television game show where the promoter attempts to prolong viewer interest by changing the rules, but in doing so inadvertently make it easier for the contestant to win. Finally, a well-known type of competition is examined where a company includes a set of items in their product and makes it tempting for the consumer to purchase more of the product just to collect all items. In each case a statistical analysis is undertaken so that any promoter who undertakes such an activity will know what to expect.

1. INTRODUCTION

There is an endless supply of promotional ideas that organisations can use to assist them in obtaining increased sales. These can range from the simple to the complex, but in all cases a little time spent analysing the possible effects can provide big dividends and save headaches down the track. In particular, how can it be determined if a promotion has somehow 'gone wrong' and is costing a great deal more money than was first thought? Is it possible that the rules of the competition are imprecise or not what was intended? Is it conceivable that some consumers have been able to take an unfair advantage of the promotion to exploit vast amounts of money?

The only way to determine whether a promotion is running smoothly is to perform an analysis on the expected outcomes, including a calculation on the best and worst possible scenarios for the promoter. In this way it should be easy to tell about how much the promotion will cost and if it is going according to plan. The following sections consider three typical situations and give an idea of the type of statistical investigation that is required.

2. THE SCRATCH CARD PROMOTION

This type promotion endeavours to entice the customer to spend more than they otherwise would on a particular product or in an establishment. It involves a 'give-away' of cash or prizes by means of rewarding the customer with a free scratch lottery ticket provide they spend a minimum amount on other goods or services. This ticket has the possibility of the reward.

In a typical promotion, suppose that a customer has to spend $x in order to receive a free ticket and will receive one free ticket for each whole multiple of $x that is spent. Each ticket has y scratch panels that are each covered by obscuring silver paint. Under exactly k of these panels is the word WIN and under the remaining (n-k) panels are the words NO WIN. The customer must scratch exactly k of these panels and to receive their reward of $y must scratch only the k WIN panels. Otherwise they get nothing. The number of ways in which k WIN panels can be distributed among n possible panels is:

(1) Number of possible combinations = [sup.n][C.sub.k]

It is assumed that the tickets are printed such that there is an equal number of each of these combinations. For example, if N tickets are printed in total, from (1) there would be N/[sup.n][C.sub.k] of each combination printed.

Clearly for the competition to be viable, x > y/[sup.n][C.sub.k] or else the promoter is going to lose a great deal of money. The probability of a customer winning on a particular ticket is 1/[sup.n][C.sub.k] since there is no strategy available for playing.

With a reward of $y, the expected return (ER) in $ to the customer on a single ticket is:

(2) ER = y/[sup.n][C.sub.k]

so that the expected total payout (ETP) in $ by the promoter on N tickets is:

(3) ETP = N(ER) = Ny/[sup.n][C.sub.k]

The total amount spent by the customers in obtaining these N tickets is $Nx. If it can be assumed that customers might not have otherwise made the purchase to obtain these tickets, the profit (in $) on the promotion is (Nx - Ny/[sup.n][C.sub.k]) so that:

(4) Percentage profit to the promoter = 100 x {1 - [y/x([sup.n][C.sub.k])]} %

In this case the outcome is binomial (either the customer either wins $y or gets nothing), it can easily be shown that the variance 0-2 of the outcome on a single game is:

(5) [[sigma].sup.2] = [y.sup.2]([sup.n][C.sub.k] - 1] [([sup.n][C.sub.k]).sup.2]

Since there are a total of N tickets printed, from (3) and (4) the total payout for the promoter follows an approximate normal distribution with parameters:

(6) [micro] = ETP = Ny/[sup.n][C.sub.k]

(7)[[sigma].sup.2] = [Ny.sup.2]([sup.n][C.sub.k] - 1]) [([sup.n][C.sub.k]).sup.2]

The approximation to normal is better as N becomes larger, but since this would naturally be the case in any promotion it should be very close.

2.1 Example

A club runs a promotion in which they print N = 50,000 tickets, each with six covered silver panels. The customer, who must spend $5 at the gambling tables to receive a ticket, scratches exactly three panels on the card, and if they reveal the three WIN panels they receive $10. Otherwise they receive nothing.