Probability and Variance in game

[Matsume]

I don't have any recorded data, but before patch 2.1 went live I crafted over 10K HQ two star food recipes. At the time I was interested in converting my crafting AF into materia, so I didn't use the 100% HQ rotation because of obvious stat limitations when your gear is full of tier I materia for SB purposes. The skill rotation I was using gave me on average an 85% HQ rate (+/- ~% conditions permitting). I can confidently say that more often than not after completing 100 or 200 synths that the actual HQ result was within 1% of the expected result. As for materia melding, I don't even know... all I know is I have some horrible luck.

[Shinkyo]

As for materia melding, I don't even know... all I know is I have some horrible luck.

Materia melding is pretty awful... I tend to think that I'm just unlucky but I've had several cases of failing over 10 attempts on a 45% success chance...
I hate materia melding...

[StouterTaru]

The HQ rate feels like it's different from what it says, but it would take a gigantic sample size to get an accurate reading.

For melding, the ~35% success rate on the first meld always seems to be closer to 50% for me; it seems like more than half succeed first attempt, the worst I've had is 6 tries, and at least 90% happen in first 3 tries (should be about 70% chance). The 2nd-4th overmeld have a much worse rate than listed from what I've seen.