Very interesting results. I was having problems with the parser but was able to eventually read the logs by manually removing them after each test. It seems dex does have an effect but nearly as much as atk does in equal ratio. I did 3 tests and Min/Max dmg as well as Avg. dmg all increased by ~6 when adding to my DEX or ATK stats. I did add an extra 2 DEX to my last test but that's because the atk gear had 2 dex on it so I couldn't really prevent that but 2 dex should have had a very minimal if any effect on the test. I'm currently lvl35 Arch fighting lvl30 bombs, I know it was mentioned that fighting normal mobs is useless but being so close to level I don't think I was hitting the damage cap by far since I've done over 732dmg on weaker mobs. I would ignore the dps as it's clearly off due to the way the experiment is carried out and total dmg may as well be off and I'm thinking this is due to the way FFXIV logs data, I.E. start, stop.
Results.
T1: DEX: 175
STR: 129
PIE: 136
ATK: 250
Atks: 49
DPS: 73.76
Crit: 10.20%
Min: 138 Max: 162
Avg: 150.97
CrtMi: 192 CrtMa: 207
DMG: 5728
T2: DEX: 145
STR: 129
PIE: 136
ATK: 250
Atks: 50
DPS: 53.83
Crit: 13%
Min: 132 Max: 155
Avg: 142.88
CrtMi: 182 CrtMa: 201
DMG: 5531
T3: DEX: 177
STR: 129
PIE: 136
ATK: 261
Atks: 50
DPS: 16.45
Crit: 20.90%
Min: 144 Max: 169
Avg: 156.37
CrtMi: 199 CrtMa: 224
DMG: 8194
Conclusion, while Dex does seem to have an effect on overall dmg, the ratio of DEX needed to Dmg increase is lower compared to Atk needed to increase dmg. It looks something around DEX5: DMG1 vs ATK2: DMG1. Not sure the cap on either though and I'm interested in doing or seeing someones results with STR, PIE and a lot more ATK. Would also like to see a more thorough experiment on normal mobs near same lvl since that is whats most important to me at this time as I'm solo lvling Arch.