Sorry to hear it wasn't the right one in Liverpool. Don't give up! It is best to get hold of the film of the parish register if possible, as the IGI is not always complete or reliable - if you look at freebmd there were rather a lot of of William Balls getting married around that time, not all recorded in the IGI.
I will look at 1871 for you tomorrow when I will have more time. You can probably get free access to the IOM census data at your nearest City library if you are in the UK, as they have usually have a licence for Ancestry. Just phone and check before you travel, and you may need to book a time. They are very helpful. Or you can pay per view on Ancestry. I expect libraries in big cities in other countries might have the same licence, perhaps someone else on the message board knows more about this.
Will post again when I've had another look at the census data tomorrow.
Susan