A huge, HUGE rookie mistake.
Remember this drawing from the Homestuck Post?
It was more correct than I was about the probability of a female-female conversation occurring in a group of size N containing F females.
I said that in a cast of characters including equal numbers of males and females, the probability that a randomly selected conversation takes place between two women is 0.5 x 0.5, or 25%. I wanted to use Karkat's shipping grid to illustrate my findings. But you'll see that there are, in fact, six possible pairings depicted on the grid above, of which only one pairing has the female-female capacity to pass the Bechdel Test. It bothered me at the time, but I figured that it wasn't relevant.
Turns out it was pretty obviously relevant!
Here are all possible Pesterchum conversational combinations for the first four kids, color-coded for convenience:
ectoBiologist + turntechGodhead
turntechGodhead + gardenGnostic
gardenGnostic + tentacleTherapist
turntechGodhead + gardenGnostic
gardenGnostic + tentacleTherapist
tentacleTherapist + ectoBiologist
ectoBiologist + gardenGnostic
tentacleTherapist + turntechGodhead
It's pretty clear that in a group of two boys and two girls, there are six possible two-person conversations, and only one of them involves both girls. That's only about a 17% chance of a F-F conversation, much lower than my original estimate. What gives? Smaller-than-infinite numbers of characters, that's what gives.
See, when I said that you could calculate the probability of any two women having a conversation by multiplying 0.5 x 0.5, I was thinking about it in really oversimplified steps. Probability that the first participant is a woman = 50%, probability that the second participant is a woman = 50%, probability that both participants are women = 25%. Except that's not really true in small groups, like the group of four above.
The probability of picking one female participant out of a 50/50 group is indeed 50%, but the probability of picking a second female participant out of the same group is dependent upon the size of the group and the outcome of the first event. For example, if I picked Jade as the first participant, then my remaining pool of participants is no longer 50% women-- it's two men and one woman, and there's only a 33% chance that I'll pick Rose instead of John or Dave. 1/2 x 1/3 = 1/6, or about 17%. My original calculation operated on the assumption that Jade could have a conversation with herself, which is impossible.
So, what IS the generalized formula for the probability of a female-female conversation, given a cast of size N containing F female characters? I figured out the real formula with combinatorics, one of my favorite branches of mathematics, but since my laptop died a few weeks ago I can't tell you about it in pretty LaTeX-generated PDFs. I have to tell you about it in craptastic MS Paint.
If you don't know how to work with factorials (or that exclamation points in mathematical equations represent factorials, not excited mathematicians), check out Dirk's explanation of pairing math. It's a very simple explanation of how to calculate the number of two-person pairs you can select from a group of a given size, and why we calculate it that way. Of course, Dirk describes this sort of math as "shipping science," not "conversation science," which highlights the fact that all of my conversation-math is equally applicable as shipping-math. So, if you're more into shipping than Bechdel scores, this math is still useful to you! Just replace the words "female-female conversation percentage" with "likelihood of sloppy lesbian makeouts" and go to town.
In order to calculate a theoretical female-female conversation percentage for Homestuck using this formula, I need to figure out what numbers I'm using for F and N. Ideally, two women participating in a Bechdel-pass conversation should be major, recurring characters rather than extras. So, like, do the Exiles count as major characters? PM plays a much more important role in the story than, say, Meulin, but Meulin has more conversations in one flash game than PM does in the entire comic. And what about alternate realities? Do Aranea and Marquise Spinneret Mindfang count as two different characters, or just the one? Is Fefeta a unique character, distinct from both Feferi and Nepeta? When Fefeta talks to herself, does it pass the Bechdel test? Will nobody think of dear, sweet, precious Fefeta???
Yeah, quantifying the number of characters in Homestuck is hard, so I'm just gonna say, if you can buy their shirt in the Homestuck store, they're a major character. So there are 34 major characters, of which 17 are female. Therefore, the theoretical percentage of female-female conversations in Homestuck should be (17x16)/(34x33) = 24.2%. A little lower than my original 25% estimate, but that just makes the 21.6% Homestuck Bechdel-pass percentage look better in comparison.
The 25% figure isn't entirely without merit, though. As the size of a cast of characters approaches infinity (given that the cast of characters maintains a 1:1 female-to-male ratio), the probability of a randomly-chosen conversation being female-female approaches 25%. See, I made a graph and everything!
The graph makes it clear that for very small groups, like the four Beta kids, it's unlikely that you're going to have many female-female conversations. The chance of a female-female conversation occurring jumps up really quickly once you introduce more than two female characters into your story. You wind up with a lot more options for female-female pairs, whereas if you only have two women, there's just one.
The shape of the conversation-curve changes as the ratio of males to females changes, too. This graph shows three other female-to-male ratios, and how they affect the predicted percentage of female-female conversations as the cast of named characters grows larger. (I only calculated values for cast sizes that would split evenly into whole numbers of male and female cast members; hence, the 2:3 graph only stars at a cast of size 5, etc.)
Though it reflects real life, it's a rare work of fiction that employs the 1:1 female-to-male ratio in its cast of major characters. The Harry Potter books are closer to a 1:2 ratio (the green dots), and it's been demonstrated that the Lord of the Rings trilogy presents a 1:4 ratio of named female characters to named male characters (the red dots). In television, the five-person-cast-with-two-women is pretty popular (pink dots), but that means that there's only a 10% chance you'll see Robin and Lily talking together on How I Met Your Mother. And of course, not all works of fiction will maintain the same gender ratio over the course of the story, as the cast grows and shrinks.
ANYWAY. Point is, I schooled y'all wrong when it comes to estimating the probability of female-female conversations in a cast of a given size and gender ratio. This is the correct schooling. Disregard all previous schooling.
Also, I was messing around a little with that cool birthday heatmap over on the Daily Viz, and I think I found an explanation for why Nepeta is such a popular character. I mean, she's all right I guess, but... why so many Nepeta cosplays and fansongs and artwork and plushies? She barely does anything important!
As the heatmap makes clear, certain seasons are more or less popular for birthdays, at least in the United States. Late summer/early fall is super-popular. Logically, then, the distribution of astrological signs among U. S. citizens should not be uniform: July, August, and September signs should be more common than other signs. And each Homestuck troll corresponds to an astrological sign. Using the birthday-popularity data, I've ranked each astrological sign/Homestuck troll by number of U. S. citizens (probably) born under that sign, from most to fewest:
See, when I said that you could calculate the probability of any two women having a conversation by multiplying 0.5 x 0.5, I was thinking about it in really oversimplified steps. Probability that the first participant is a woman = 50%, probability that the second participant is a woman = 50%, probability that both participants are women = 25%. Except that's not really true in small groups, like the group of four above.
The probability of picking one female participant out of a 50/50 group is indeed 50%, but the probability of picking a second female participant out of the same group is dependent upon the size of the group and the outcome of the first event. For example, if I picked Jade as the first participant, then my remaining pool of participants is no longer 50% women-- it's two men and one woman, and there's only a 33% chance that I'll pick Rose instead of John or Dave. 1/2 x 1/3 = 1/6, or about 17%. My original calculation operated on the assumption that Jade could have a conversation with herself, which is impossible.
 |
| Mostly impossible. |
If you don't know how to work with factorials (or that exclamation points in mathematical equations represent factorials, not excited mathematicians), check out Dirk's explanation of pairing math. It's a very simple explanation of how to calculate the number of two-person pairs you can select from a group of a given size, and why we calculate it that way. Of course, Dirk describes this sort of math as "shipping science," not "conversation science," which highlights the fact that all of my conversation-math is equally applicable as shipping-math. So, if you're more into shipping than Bechdel scores, this math is still useful to you! Just replace the words "female-female conversation percentage" with "likelihood of sloppy lesbian makeouts" and go to town.
 |
| Source |
In order to calculate a theoretical female-female conversation percentage for Homestuck using this formula, I need to figure out what numbers I'm using for F and N. Ideally, two women participating in a Bechdel-pass conversation should be major, recurring characters rather than extras. So, like, do the Exiles count as major characters? PM plays a much more important role in the story than, say, Meulin, but Meulin has more conversations in one flash game than PM does in the entire comic. And what about alternate realities? Do Aranea and Marquise Spinneret Mindfang count as two different characters, or just the one? Is Fefeta a unique character, distinct from both Feferi and Nepeta? When Fefeta talks to herself, does it pass the Bechdel test? Will nobody think of dear, sweet, precious Fefeta???
Yeah, quantifying the number of characters in Homestuck is hard, so I'm just gonna say, if you can buy their shirt in the Homestuck store, they're a major character. So there are 34 major characters, of which 17 are female. Therefore, the theoretical percentage of female-female conversations in Homestuck should be (17x16)/(34x33) = 24.2%. A little lower than my original 25% estimate, but that just makes the 21.6% Homestuck Bechdel-pass percentage look better in comparison.
The 25% figure isn't entirely without merit, though. As the size of a cast of characters approaches infinity (given that the cast of characters maintains a 1:1 female-to-male ratio), the probability of a randomly-chosen conversation being female-female approaches 25%. See, I made a graph and everything!
The graph makes it clear that for very small groups, like the four Beta kids, it's unlikely that you're going to have many female-female conversations. The chance of a female-female conversation occurring jumps up really quickly once you introduce more than two female characters into your story. You wind up with a lot more options for female-female pairs, whereas if you only have two women, there's just one.
The shape of the conversation-curve changes as the ratio of males to females changes, too. This graph shows three other female-to-male ratios, and how they affect the predicted percentage of female-female conversations as the cast of named characters grows larger. (I only calculated values for cast sizes that would split evenly into whole numbers of male and female cast members; hence, the 2:3 graph only stars at a cast of size 5, etc.)
Though it reflects real life, it's a rare work of fiction that employs the 1:1 female-to-male ratio in its cast of major characters. The Harry Potter books are closer to a 1:2 ratio (the green dots), and it's been demonstrated that the Lord of the Rings trilogy presents a 1:4 ratio of named female characters to named male characters (the red dots). In television, the five-person-cast-with-two-women is pretty popular (pink dots), but that means that there's only a 10% chance you'll see Robin and Lily talking together on How I Met Your Mother. And of course, not all works of fiction will maintain the same gender ratio over the course of the story, as the cast grows and shrinks.
ANYWAY. Point is, I schooled y'all wrong when it comes to estimating the probability of female-female conversations in a cast of a given size and gender ratio. This is the correct schooling. Disregard all previous schooling.
Also, I was messing around a little with that cool birthday heatmap over on the Daily Viz, and I think I found an explanation for why Nepeta is such a popular character. I mean, she's all right I guess, but... why so many Nepeta cosplays and fansongs and artwork and plushies? She barely does anything important!
As the heatmap makes clear, certain seasons are more or less popular for birthdays, at least in the United States. Late summer/early fall is super-popular. Logically, then, the distribution of astrological signs among U. S. citizens should not be uniform: July, August, and September signs should be more common than other signs. And each Homestuck troll corresponds to an astrological sign. Using the birthday-popularity data, I've ranked each astrological sign/Homestuck troll by number of U. S. citizens (probably) born under that sign, from most to fewest:
1. Virgo (Aug. 23 - Sep. 23), Kanaya
2. Leo (July 23 - Aug. 22), Nepeta
3. Cancer (June 22 - July 22), Karkat
4. Libra (Sep. 24 - Oct. 23), Terezi
5. Gemini (May 22 - June 21), Sollux
6. Scorpio (Oct. 24 - Nov. 22), Vriska
7. Pisces (Feb. 20 - Mar. 20), Feferi
8. Sagittarius (Nov. 23 - Dec. 21), Equius
9. Taurus (Apr. 21 - May 21), Tavros
10. Aquarius (Jan. 21 - Feb. 19), Eridan
11. Aries (Mar. 21 - Apr. 20), Aradia
12. Capricorn (Dec. 22 - Jan. 20), Gamzee
The second-most common sign in America is Leo, Nepeta's sign. If Homestuck readers subconsciously (or consciously) esteem the troll who bears their astrological sign, the large number of Leos in the United States could account for the disproportionate Nepeta-love. By that logic, Kanaya should be the most popular troll. Which she is. Because, let's be honest here, she's the best one. I'm not just saying that because I'm a Virgo, nope, definitely not. When it comes to being a troll, Kanaya is simply the best there is, end of story.
In conclusion, thank GOD I'm not an Aquarius.
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