The Tower mostly excludes people who are too weak to test for the shawl. Not only do we know this for a fact, we actually know what percentage of women are excluded for this reason. 37.5% to be precise.

That means 37.5% of women are weaker than Dagian, by the way. The one indisputable fact we have in this debate.

Now, if you took a random sample, you would expect the average to be more or less equal to the average of the entire population. But this sample (the Aes Sedai)is not random, in fact, it excludes 37.5% of women who are weaker than Dagian. Thereofore, a sample consisting only of Aes Sedai would be skewed to the stronger side of the range. In other words, the average for this sample would fall above the average for the entire population.

Therefore, the average Aes Sedai is stronger than the average woman. Now from my recollection, women like Alanna and Verin are already above average for an Aes Sedai. That means, according to your theory, Alanna and Verin would each be more than 50% as strong as Rand.

But let's go back to the stats for a moment.

So we know now that the average strength woman must fall somewhere between Dagian and the average Aes Sedai. So what we are saying is that what we would consider a "weak" modern day Aes Sedai is actually comparable to the average strength woman from the general channeling population.

And according to you, such a weak Aes Sedai must be 50% as strong as Lanfear, and EFFECTIVELY, 50% as strong as Rand.

See the problem yet?

Now let's extrapolate a bit further.

So we see that out of our sample of 1000 women (the Tower), only 5 women (6 if you include Cadsuane) are of Moiraine strength or higher. I think they were given in New Spring as Moiriane, Siuan, Lelaine, Romanda and maybe Elaida. (Someone who has actually read New Spring might correct me on this).

Anyway, 5 out of 1000 gives you a percentage of 0.5, meaning that 99.5% of observations do not fall this far away from the mean. Let's double the number of people this far from the mean to 10 (to account for Cadsuane, and others who might not have been picked up, and let's also include those equi-distant from the mean on the lower end of the spectrum, like Dagian etc.), and make it a round 95%, for simplicity's sake. And what do you know. This means that Moiraine, Lelaine and Romanda's level is 2 standard deviations away from the White Tower mean (which in this case is the average Aes Sedai, not the average woman, by the way).

Let's say that again, just to hammer it down. In the 1000 woman sample from the Tower (which is already SKEWED TO THE STRONGER SIDE), someone of Moiriane's strength is 2 standard deviations above the mean. Wow.

That means you would be hard pressed to find a stronger channeler than her. That also means that Dagian is at least 2 standard deviations below the average Aes Sedai in strength.

So, since we know from Siuan that she is currently around one third of her previous strength (and only barely above Dagian in strength) we can now extrapolate the standard deviation from this.

Siuan strongly suggests that if she was two thirds her old strength, she would be of about average Aes Sedai strength.

Therefore, we can estimate that Moiraine is about half again as strong as the average woman. And since we have established that Moiraine is about 2 standard deviations from the mean, that means that the standard deviation - for Aes Sedai - equates to about a quarter of the average Aes Sedai's strength.

So, since we know that the average Aes Sedai is stronger than the average woman, it means that according to your model, we have to put the average Aes Sedai at more than 50. Let's put her at 60, just to have a round figure. It wouldn't be be materially different if it was 55 or 56 or whatever. Anyway, if we take the average Aes Sedai to be 60, then one standard deviation above that would take you to 75, and two standard deviations above that would take you to 90. So Moiraine's level would be 90!

And since we know that Egwene is as strong as Amys and Melaine combined, (Amys is equal to Moiraine by the way), that means that Egwene would have a strenth of approximately 90 + 75 = 165.

That gives us a tiny problem, seeing as Lanfear, the strongest woman ever, is only on 100 on the female scale.

Clearly, therefore, the average Aes Sedai strength has to be FAR below 60, and therefore the average woman's strength has to be far below 50.