Monthly Total Running Average Rates

Unfortunately, the desired result isn't possible in beast mode because each record will fall into exactly one category, which in this case appears to be the month. If a record is used in the January line point, it isn't also part of the group of records used to calculate the February line point. 

You could create a dataset that behaves how you'd like using a dataflow, however. You might try something like this:

Transform 1: months

/* Selects distinct month values from the data for use in the next transform */

SELECT

    DISTINCT LAST_DAY(`date`) AS `Month`

FROM

    your_table

ORDER BY

    `Month`

Transform 2: running_averages

/* Uses a sub query to calculate averages from the beginning of the year through the month of each record in the months table */

SELECT
    `MONTH`
    , (SELECT

            SUM(yt.`conversions`) / SUM(yt.`opportunities`)

       FROM 

           your_table yt

       WHERE

           LAST_DAY(yt.`start`) <= m.`Month`

           AND LAST_DAY(yt.`start`) >= STR_TO_DATE(CONCAT(YEAR(m.`Month`),'-01-01'),'%Y-%m-%d')) AS `Running Average

FROM
months m

You'll need to swap out your own column and table names, of course, but hopefully this helps you get started.

This code is set up to give you Jan through Dec metrics. Can you clarify what your criteria are for what period should be included in the rolling average, e.g. instead of for the year, do you want to see the last 6 months?