Ratio + Quantity I am trying to create a metric which would tell us how good or bad a client is. Right now I am using a simple ratio: 
$$
\frac{\text{number of bills paid on time}}{\text{number of bills}}.
$$ This give me a percentage and it essentially tell us if a client always pays it bill on time ($0$ = never, $1$ = always). For instance, a client who paid only one bill on time is as good as a client who paid 20 bills on time with this metric.
The problem with this metric is that it doesn’t capture the number of bills the consumer had to pay. Does anybody has an idea on how I could capture these two information (paid on time + number of bills) into one single number?
 A: Before anyone can help you with this, there are some questions you have to decide. Do you want a higher number for a client who is 1 out of 1, or a client who is 19 out of 20? If 19/20 is better than 1/1, how about 18/20? 17/20? how bad does it have to be to be worse than 1/1? 
Here's one possible answer: $${1+{\rm\ paid\ on\ time}\over2+{\rm\ number\ of\ bills}}$$ It has the following properties: 


*

*It's always (strictly) between 0 and 1. 

*If ${\rm paid\ on\ time}=0$ then it approaches zero as number of bills increases. 

*If all bills are paid on time then it approaches 1 as number of bills increases. 
One bill, paid on time, gets evaluated as $.666\dots$, same as 13 bills out of 19 paid on time. If 13 out of 19 isn't as good as 1 out of 1, then you'll have to make some adjustment to the formula, replace the 1 and the 2 I put in there with some other numbers. In fact, I encourage you to experiment a bit, try different numbers in place of my 1 and 2, and see whether you can get a formula that matches your feelings about various situations. 
