Which is the best day of the month to regularly invest on?

I always try to encourage 80-20 Investor members to start discussions or ask questions of me and other members via Chatterbox. Sometimes I will receive a question that warrants its own research piece in order to answer it. Below is a question I was asked by an 80-20 Investor member:

I know you like to analyse things. If you are making a monthly contribution.  What would be the best day of the month to invest on?

So true to form, I decided to carry out some analysis to see if there is an answer to the question. While it may seem a simple question it actually requires a lot of data crunching.

Firstly I assumed that an investor decided to regularly invest in the FTSE 100. By keeping the underlying investment the same any variations in total return will be as a result of the differing time periods and investment methodologies.