The Social Network, the classic David Fincher film about the founding of Facebook, opens with the following exchange. It’s a Tuesday night in autumn 2003. Mark Zuckerberg, a computer science sophomore at Harvard, is at a tavern on a date with the fictional Erica Albright:

Zuckerberg: Did you know there are more people with genius IQs living in China than there are people of any kind living in the United States?

Albright: That can’t possibly be true.

Zuckerberg: It is.

Albright: What would account for that?

Albright’s intuition is right, though Zuckerberg derails the conversation before she can challenge him further. Some calculations show that Zuckerberg’s claim is ridiculous.

Establishing Numbers

According to the World Bank, in 2003 China had 1.29 billion people and the United States had 290,108,000 (rounding to the nearest hundred, not that it matters).

While there are many ways to define genius, I’ll use intelligence quotient, a well-known measure that is easy to work with. IQ is typically described as normally distributed, with mean 100 and standard deviation about 15. There is no universally accepted cutoff for a “genius” IQ. To be charitable, I’ll define it as 130 or greater, a mere two standard deviations above the mean.

If we assume each country has the same theoretical IQ distribution, how many geniuses would we expect in each?

Calculations

us_pop <- 290108000
china_pop <- 1290000000
mu <- 100
sigma <- 15
genius <- 130
cutoff <- pnorm(genius, mu, sigma, lower.tail = FALSE)
us_geniuses <- floor(cutoff * us_pop)
china_geniuses <- floor(cutoff * china_pop)

The China-US ratio is 4.45, so under these assumptions China has that many times more geniuses. But the ratio of U.S. population to Chinese geniuses is 9.89. There would need to be almost 10 times more Chinese geniuses than there are under these assumptions for Zuckerberg to be correct. If we use a higher IQ score as the genius cutoff, the ratio gets even worse.

There are infinitely many implausible IQ distributions that would justify Zuckerberg’s claim. For simplicity, let us pose an optimization problem. Given China’s actual population and a U.S. mean IQ of 100, how high must China’s mean IQ be for Zuckerberg’s claim to be correct? That is, what is the lowest mean IQ value for which the right tail area of the normal CDF is greater than the ratio of US to Chinese population?

validate_mean <- function(mu) {
  floor(pnorm(genius, mu, sigma, lower.tail = FALSE) * china_pop) > us_pop
}
grid <- seq(100, 150, 0.01)
minimum <- min(grid[validate_mean(grid)])
minimum

-- [1] 118.67

Zuckerberg’s claim would require the average Chinese to have an IQ more than one standard deviation above the U.S. average. I couldn’t find reliable IQ data for China, and intelligence is notoriously hard to measure. But for the claim to be true, the average Chinese would be more intelligent than about 89.3% of Americans, which is unbelievable.

Supposing instead China has the same IQ distribution as the United States, how many Chinese would there have to be for Zuckerberg’s claim to be correct? This can be solved algebraically:

floor(us_pop / cutoff + 1)

-- [1] 12751926040

That is about 10 times the actual 2003 population of China.

Conclusion

In fairness to the real Zuckerberg, this conversation is fictional, and I’m skeptical he would really have made the claim this post dissects. As a computer science sophomore at an elite university, he likely understood the properties of normal distributions well enough to know it was implausible. Or maybe the script means to imply that he’s trying to impress a listener he thinks is naive with nonsense, as I’ve always interpreted Han Solo’s nonsensical boast in Star Wars that the Millennium Falcon can make the Kessel Run in less than twelve parsecs. That would also explain why Zuckerberg keeps changing the subject whenever Albright presses him on the point. Either way, Albright is clearly right.

One other note: During the hacking montage that follows the tavern scene, Zuckerberg says (in voiceover), “So it’s definitely necessary to break out Emacs and modify that Perl script.” As a Neovim user, I’m required to remind you it is never necessary to break out Emacs. Editing bliss is always just a which vim away.

--- title: "Erica Albright was Right" author: "Ryan Heslin" date: "2023-06-13" categories: ["Miscellany"] urlcolor: "blue" --- [_The Social Network_](https://en.wikipedia.org/wiki/The_Social_Network), the classic David Fincher film about the founding of Facebook, opens with the [following exchange](https://www.youtube.com/watch?v=uvap4gEOC5I). It's a Tuesday night in autumn 2003. Mark Zuckerberg, a computer science sophomore at Harvard, is at a tavern on a date with the fictional Erica Albright: ::: {.callout} Zuckerberg: Did you know there are more people with genius IQs living in China than there are people of any kind living in the United States? Albright: That can't possibly be true. Zuckerberg: It is. Albright: What would account for that? ::: Albright's intuition is right, though Zuckerberg derails the conversation before she can challenge him further. Some calculations show that Zuckerberg's claim is ridiculous. # Establishing Numbers According to the World Bank, in 2003 China had [1.29 billion](https://data.worldbank.org/country/china?view=chart) people and the United States had [290,108,000](https://data.worldbank.org/country/united-states?view=chart) (rounding to the nearest hundred, not that it matters). While there are many ways to define genius, I'll use [intelligence quotient](https://en.wikipedia.org/wiki/Intelligence_quotient), a well-known measure that is easy to work with. IQ is typically described as normally distributed, with mean 100 and standard deviation about 15. There is no universally accepted cutoff for a "genius" IQ. To be charitable, I'll define it as 130 or greater, a mere two standard deviations above the mean. If we assume each country has the same theoretical IQ distribution, how many geniuses would we expect in each? # Calculations ```{r} us_pop <- 290108000 china_pop <- 1290000000 mu <- 100 sigma <- 15 genius <- 130 cutoff <- pnorm(genius, mu, sigma , lower.tail = FALSE) us_geniuses <- floor( cutoff * us_pop) china_geniuses <- floor(cutoff * china_pop) ``` The China-US ratio is `r round(china_pop / us_pop, 2)`, so under these assumptions China has that many times more geniuses. But the ratio of U.S. population to Chinese geniuses is `r round(us_pop /china_geniuses, 2)`. There would need to be almost 10 times more Chinese geniuses than there are under these assumptions for Zuckerberg to be correct. If we use a higher IQ score as the genius cutoff, the ratio gets even worse. There are infinitely many implausible IQ distributions that would justify Zuckerberg's claim. For simplicity, let us pose an optimization problem. Given China's actual population and a U.S. mean IQ of 100, how high must China's mean IQ be for Zuckerberg's claim to be correct? That is, what is the lowest mean IQ value for which the right tail area of the normal CDF is greater than the ratio of US to Chinese population? ```{r} validate_mean <- function(mu){ floor(pnorm(genius, mu, sigma, lower.tail = FALSE) * china_pop) > us_pop } grid <- seq(100, 150, 0.01) minimum <- min(grid[validate_mean(grid)]) minimum ``` Zuckerberg's claim would require the average Chinese to have an IQ more than one standard deviation above the U.S. average. I couldn't find reliable IQ data for China, and intelligence is notoriously hard to measure. But for the claim to be true, the average Chinese would be more intelligent than about `r round(pnorm(minimum, mu, sigma) * 100, 1)`% of Americans, which is unbelievable. Supposing instead China has the same IQ distribution as the United States, how many Chinese would there have to be for Zuckerberg's claim to be correct? This can be solved algebraically: ```{r} floor(us_pop/ cutoff + 1) ``` That is about 10 times the actual 2003 population of China. # Conclusion In fairness to the real Zuckerberg, this conversation is fictional, and I'm skeptical he would really have made the claim this post dissects. As a computer science sophomore at an elite university, he likely understood the properties of normal distributions well enough to know it was implausible. Or maybe the script means to imply that he's trying to impress a listener he thinks is naive with nonsense, as I've always interpreted Han Solo's nonsensical boast in _Star Wars_ that the Millennium Falcon can make the Kessel Run in less than twelve [parsecs](https://en.wikipedia.org/wiki/Parsec). That would also explain why Zuckerberg keeps changing the subject whenever Albright presses him on the point. Either way, Albright is clearly right. One other note: During the hacking montage that follows the tavern scene, Zuckerberg says (in voiceover), "So it's definitely necessary to break out Emacs and modify that Perl script." As a Neovim user, I'm required to remind you it is _never_ necessary to break out Emacs. Editing bliss is always just a `which vim` away.