Sunday, June 23, 2024

# How well are your incentives working?

Here’s the situation. You’re asking customers to sign up for your gym membership by offering  some discounts. But how well are they working? Let’s find out (working with limited data).

Information on hand is that you made 20 attempts or trials and find that 22% signed up. Your task is to calculate the various chances of success (=sign-up for membership) from various trials. You only have 2 pieces of data to go with 22% success, and 20 tries. And that’s all you’ll need to project and quantify the probability of success if you utilize the concept of Binomial distribution.

We’ll use 2 versions of Binomial functions: one for probability mass function, another for cumulative. They will yield different (even opposing) perspectives but of the same outcomes.

First, we find the percentage of outcomes for each version for each trial. We’ll run it from 0 to 20 trials as per the scenario. To do this, we can use BINOM.DIST() in Excel: one set to FALSE for probability mass function and again set to TRUE (for cumulative) and create two columns which look like this:

Understanding the Table:

Looking at the the Percentage of Outcomes (mass) column, we see that success is 0.69% (not 69%! 0.69%), so about 0.7% chance of 0 sign-ups. And 0.0…% chance of getting 12 or more sign-ups, and no chance at all getting 20 sign-ups.

Looking at the the Percentage of Outcomes (cum.) column, we also see that success is 0.69%, so about 0.7% chance of 0 sign-ups. But the numbers start to look quite different as we go down the Successes rows.

If we chart each column’s values, we get the following and let’s title the charts to differentiate from each other accordingly by the version of Binomial function used.

and the next chart…

Understanding the Charts:

In the Sign-Up Percentage 1 chart, we see chances of success is 4 at best using mass density  function. The probability of getting a specific number of successes goes up as we go to 4 trials, and then going down and it’s very unlikely to get exactly >=11 sign-ups. So, probability of success is 22%*number of trials (20)=4.4, meaning we can expect about 4.4 people to sign up. This is confirmed in chart 1, where 4 matches up with the highest chance of 21.30%.

In the Sign-Up Percentage 2 chart, we see that there’s less than 0.69% chance of getting zero signups, 4.6% of getting 1 or less people to sign up, 15% chance of getting two or less, and then 99.868% chance of getting 10 or less people to sign up. Put another way, >=10 (or 11, depending on the decimal precision you need) signups is very unlikely.

So, How Do We Use this Result?

In summary, you can expect 4.4 people to sign up. However, if there’s more than that, maybe the staff working on signing up customers is really good at it! Or, the discounts are working better than expected!

If it’s less than 4.4, we know something isn’t working quite as well. If the signups are consistently higher than 10, then something abnormal is occurring (we should check the data) because chart 2 tells us that after 10, the success rate almost flattens out. And from chart 1, we see that it’s very unlikely to get >10 signups.

So, no more guess-work!