Keeping in line with ZOS and normal and veteran versions of everything. I present a normal and veteran version of this post.

Normal
Julianos is slightly preferred over Twice-Born Star but the difference is rather small. In game single trial test may lead to an incorrect conclusion. Instead, I suggest simply calculating the average damage with the following

where Crit is your spell critical chance and Modifier is the extra damage done on a critical strike.

Veteran
I was recently watching Deltia’s BIS Magic DPS Gear in ESO Kagrenacs vs Law of Julianos vs Twice Born Star. On his Molten whip test, he found that Twice-Born Star slightly outperforms Julianos but on his full rotation test he concluded that Julianos is preferred over Twice-Born Star. Now I am already convinced that Julianos is better but I was curious as to how likely is it for someone who performs a similar whip test to reach the incorrect conclusion. I’ll be using a lot of data from @Deltia’s video just because it is readily available data. I would like to acknowledge Deltia in creating the video and providing a resource for the community.

To answer the question of how likely is it for someone who performs a similar whip test to incorrectly conclude that Twice-Born Star outperforms Julianos, I will create the normal distribution of total damage for whip damage for Law of Julianos and Twice-Born Star. Damage calculations in ESO are rather straightforward, you either crit (and you do your base damage plus an additional critical modifier damage) or you don’t crit (and just do your base damage). This leads to a binomial distribution and if we do it sufficiently long we can approximate the total damage distribution with a normal distribution with the following statistics

where n is the number of whips, Crit is (spell) critical chance and modifier is the extra damage done on a crit. In Deltia’s case this is 0.69 (67 points in Elfborn, as an aside on Live 66 and 67 points in Elfborn result in the same modifier) Unfortunately, Deltia was not using a combat log so we cannot get accurate Base Damage numbers. We can however guess that it is 12684 (max Molten whip) / 1.69 = 7505. Some readers might suggest solving the equation for total damage

Unfortunately, combat logs occasionally incorrectly record the number of crits because sometimes non-crits are counted as crits and crits are counted as non-crits. Also if you tried this out your results won’t match Deltia’s scrolling combat text. So the first way is preferable since it matches the scrolling combat text shown in the video where non-crits are 7.5k and 12.7k for a crit. Similarly, the Base Damage for Twice-Born Star is most likely 7102. Now the stats Deltia’s listed in his video are a bit perplexing. Here is a comparison of stats in his video for Julianos and Twice-Born Star

I’ve no idea why max magicka and spell critical is lower for Julianos than Twice-Born Star when the max magicka should be the same and the spell critical should be higher for Julianos. I’ll just use his Twice-Born Star spell critical and add an additional 6.2% for Julianos. We can easily plot both normal distributions and come to the conclusion that Julianos is better than Twice-Born Star.

But let us suppose that you didn’t do all this and just looked at the total damage done by molten whip when using Law of Julianos and compared that to Twice-Born Star. This situation is just the difference between the two normal distributions

Deltia observed a difference of 14045 total damage for molten whip (448 DPS). If were to set a damage threshold of 10000 that is if the difference is less than -1000, Twice-Born Star is better and if the difference is greater than 10000 Julianos is better and anything in between means we are not sure, we arrive at the following.

So about 25% chance to reach the incorrect conclusion that Twice-Born Star is better than Julianos, 35% chance that the test is inconclusive and 40% chance that Julianos is correctly identified as being preferrred over Twice-Born Star. Of course if you did multiple trials and then created a confidence interval you are more likely to get the right answer but why not just calculate the mean of both and be done with it in a significantly shorter time.

Note: If we used the stats provided by Deltia's video, the Twice-Born Star is clearly superior to Julianos because it has a higher critical chance