The Bell Curve revisited - Edit 2
Before modification by Shannow at 29/10/2012 09:45:29 AM
Given the discussion Sidious’s Androl/Pevara thread has generated on the Bell Curve, I feel it is necessary to once again raise this issue as a separate topic of discussion. (I know, I know, the forum is overly clogged up with new threads, but I'm daring to post another one anyway.) Smiley face.
In any case, for those who aren’t aware, this thread is based on RJ’s statement that channeler strength in WoT can be represented by a Bell Curve distribution. For those unfamiliar with statistics, a Bell Curve is a perfectly normal distribution, peaking at the average and petering off to either side in a perfectly symmetrical manner.
A Bell Curve by definition means that the distance from the weakest to the strongest channeler is intersected at exactly the 50% mark by the mean (the average channeler). Any skewing of the distribution would mean that the term “Bell Curve” cannot be applied to the distribution. Instead, it would then be either a positively or negatively skewed distribution. But not a Bell Curve.
So the basic rule is that the average channeler has to be exactly half as strong as the strongest channeler. Or to put it differently, a channeler x standard deviations away from the mean on the weak side, must be exactly as far from the mean as a channeler x standard deviations away on the strong side.
RJ has also said that 62.5% of channelers are strong enough to become Aes Sedai. This means that Daigian – who is the weakest possible Aes Sedai – lies exactly on this margin. And it then means that 12.5% of all female channelers lie between Daigian and the average strength woman.
Since it has been strongly suggested that Lanfear is the strongest possible woman, we therefore know that 12.5% of all female channelers lie between Daigian and the channeler who has 50% of Lanfear’s strength.
By applying this logic, the key issue then is to determine – based on the evidence in the books – which channeler lies closest to 50% of Lanfear’s strength – in order to determine how strong all modern Aes Sedai are compared to the Forsaken.
We do know that:
Siuan was previously as strong as Moiraine. We are further told that “even if Siuan was healed to half her former strength, most Aes Sedai would still be stronger than her. This means that Siuan is now less than half her former strength. Yet she is still slightly stronger than Daigian.
A reasonable estimate is that Siuan is therefore about one third as strong as her former strength. We also know that she is not far above Daigian at this stage.
It is therefore very reasonable to suggest that the weakest Aes Sedai is around one third as strong as Moiraine and Siuan’s former strength. The question then is, was Moiraine’s previous strength above or below the average strength female (who has 50% of Lanfear’s strength).
So the question can be broken down to: Can two channelers of Moiraine’s strength match a Forsaken strength channeler?
This has been a topic of some discussion, especially because there are posters who fervently want Moiraine – and by extension all modern Aes Sedai - to be stronger than they actually are.
There are numerous pieces of evidence showing that it is impossible for Moiraine to be half as strong as Lanfear. Without providing an exhaustive list, here are a few examples:
Moiraine, who has felt the strength of a Forsaken – says that Egwene and Aviendha combined might come close to matching Lanfear. This at a time when Aviendha has stated that Egwene is stronger than Amys and Melaine combined. Amys is very close to Moiraine’s level.
The above heavily implies that Lanfear is roughly as strong as Egwene and Aviendha combined, at the end of Fires of Heaven. This would put Lanfear at at least 3 times Moiraine’s strength.
This ties in with Egwene’s statement in one of the later books that She, Romanda and Lelaine combined would be necessary to match one of the male Forsaken, which perfectly corroborates Moiraine’s earlier estimate.
Looking at the Cleansing, we see more corroboration of this ratio, given that circles of at least 3 mixed channelers (mixed circles are stronger than single gender circles) and often 4 modern channelers were necessary to match each of the Forsaken.
Demandred tells us that the circle of Flinn plus two average Aes Sedai EXACTLY matches the strength of his own weaving. Given that Flinn is the most powerful of the Ashaman after Taim and given that a mixed circle adds strength to the weave, this tells us that one Forsaken at the very least matches about 4 average modern Aes Sedai in pure strength, possibly even 5.
This is important, because we know that even if Siuan was healed to two thirds her former strength, she would still only be of average strength for a modern Aes Sedai. This means that Siuan (and Moiraine’s) former strength is about 50% greater than the average Aes Sedai.
So, if a top level Forsaken is as strong as 4 (possibly 5) modern Aes Sedai, then it means that Moiraine is just over a third as strong as a Forsaken.
Therefore, she cannot be half as strong as Lanfear. Therefore, she cannot be as strong as the average channeler on the Bell Curve.
Therefore, all modern Aes Sedai fall below the mean of the Bell Curve. And they all come from the tiny part of the channelling population that falls between Daigian and the average woman. In fact, they come from maybe just the bottom half of that 12.5%.
Hence, the Bell Curve cannot represent the known channelling population.
So we are left with a problem. The pure Bell Curve description does not fit the evidence.
The solution, in my view, is that RJ used the term Bell Curve loosely, and in fact meant a skewed distribution, with the majority of channelers falling on the weak side. Thus, the “Bell” would be skewed towards the left (leaning towards the left) with a long tail stretching towards the upper half.
Certainly, based on the One Power section in the Guide to the Wheel of Time, the vast majority of channelers during the Age of Legends were weak, with only a small percentage being of great strength. This refutes a symmetrical Bell Curve distribution, and in fact points to a skewed distribution that is dominated by a huge portion of weak channelers, and a small percentage of very powerful outliers on the upper side.
The Bell Curve is therefore a misnomer, casually used by RJ, when in fact a skewed distribution represents the evidence as presented in the books and in the Guide.
In any case, for those who aren’t aware, this thread is based on RJ’s statement that channeler strength in WoT can be represented by a Bell Curve distribution. For those unfamiliar with statistics, a Bell Curve is a perfectly normal distribution, peaking at the average and petering off to either side in a perfectly symmetrical manner.
A Bell Curve by definition means that the distance from the weakest to the strongest channeler is intersected at exactly the 50% mark by the mean (the average channeler). Any skewing of the distribution would mean that the term “Bell Curve” cannot be applied to the distribution. Instead, it would then be either a positively or negatively skewed distribution. But not a Bell Curve.
So the basic rule is that the average channeler has to be exactly half as strong as the strongest channeler. Or to put it differently, a channeler x standard deviations away from the mean on the weak side, must be exactly as far from the mean as a channeler x standard deviations away on the strong side.
RJ has also said that 62.5% of channelers are strong enough to become Aes Sedai. This means that Daigian – who is the weakest possible Aes Sedai – lies exactly on this margin. And it then means that 12.5% of all female channelers lie between Daigian and the average strength woman.
Since it has been strongly suggested that Lanfear is the strongest possible woman, we therefore know that 12.5% of all female channelers lie between Daigian and the channeler who has 50% of Lanfear’s strength.
By applying this logic, the key issue then is to determine – based on the evidence in the books – which channeler lies closest to 50% of Lanfear’s strength – in order to determine how strong all modern Aes Sedai are compared to the Forsaken.
We do know that:
Siuan was previously as strong as Moiraine. We are further told that “even if Siuan was healed to half her former strength, most Aes Sedai would still be stronger than her. This means that Siuan is now less than half her former strength. Yet she is still slightly stronger than Daigian.
A reasonable estimate is that Siuan is therefore about one third as strong as her former strength. We also know that she is not far above Daigian at this stage.
It is therefore very reasonable to suggest that the weakest Aes Sedai is around one third as strong as Moiraine and Siuan’s former strength. The question then is, was Moiraine’s previous strength above or below the average strength female (who has 50% of Lanfear’s strength).
So the question can be broken down to: Can two channelers of Moiraine’s strength match a Forsaken strength channeler?
This has been a topic of some discussion, especially because there are posters who fervently want Moiraine – and by extension all modern Aes Sedai - to be stronger than they actually are.
There are numerous pieces of evidence showing that it is impossible for Moiraine to be half as strong as Lanfear. Without providing an exhaustive list, here are a few examples:
Moiraine, who has felt the strength of a Forsaken – says that Egwene and Aviendha combined might come close to matching Lanfear. This at a time when Aviendha has stated that Egwene is stronger than Amys and Melaine combined. Amys is very close to Moiraine’s level.
The above heavily implies that Lanfear is roughly as strong as Egwene and Aviendha combined, at the end of Fires of Heaven. This would put Lanfear at at least 3 times Moiraine’s strength.
This ties in with Egwene’s statement in one of the later books that She, Romanda and Lelaine combined would be necessary to match one of the male Forsaken, which perfectly corroborates Moiraine’s earlier estimate.
Looking at the Cleansing, we see more corroboration of this ratio, given that circles of at least 3 mixed channelers (mixed circles are stronger than single gender circles) and often 4 modern channelers were necessary to match each of the Forsaken.
Demandred tells us that the circle of Flinn plus two average Aes Sedai EXACTLY matches the strength of his own weaving. Given that Flinn is the most powerful of the Ashaman after Taim and given that a mixed circle adds strength to the weave, this tells us that one Forsaken at the very least matches about 4 average modern Aes Sedai in pure strength, possibly even 5.
This is important, because we know that even if Siuan was healed to two thirds her former strength, she would still only be of average strength for a modern Aes Sedai. This means that Siuan (and Moiraine’s) former strength is about 50% greater than the average Aes Sedai.
So, if a top level Forsaken is as strong as 4 (possibly 5) modern Aes Sedai, then it means that Moiraine is just over a third as strong as a Forsaken.
Therefore, she cannot be half as strong as Lanfear. Therefore, she cannot be as strong as the average channeler on the Bell Curve.
Therefore, all modern Aes Sedai fall below the mean of the Bell Curve. And they all come from the tiny part of the channelling population that falls between Daigian and the average woman. In fact, they come from maybe just the bottom half of that 12.5%.
Hence, the Bell Curve cannot represent the known channelling population.
So we are left with a problem. The pure Bell Curve description does not fit the evidence.
The solution, in my view, is that RJ used the term Bell Curve loosely, and in fact meant a skewed distribution, with the majority of channelers falling on the weak side. Thus, the “Bell” would be skewed towards the left (leaning towards the left) with a long tail stretching towards the upper half.
Certainly, based on the One Power section in the Guide to the Wheel of Time, the vast majority of channelers during the Age of Legends were weak, with only a small percentage being of great strength. This refutes a symmetrical Bell Curve distribution, and in fact points to a skewed distribution that is dominated by a huge portion of weak channelers, and a small percentage of very powerful outliers on the upper side.
The Bell Curve is therefore a misnomer, casually used by RJ, when in fact a skewed distribution represents the evidence as presented in the books and in the Guide.