The Mathematics Of Beauty

January 10th, 2011 by Christian Rudder

This post investigates female attractiveness, but without the usual photo analysis stuff. Instead, we look past a woman's picture, into the reaction she creates in the reptile mind of the human male.

Among the remarkable things we'll show:

Fair warning: we're about to objectify women, big-time. The whole purpose of this blog is to analyze OkCupid's data, and without a little bit of objectification that's impossible. Men will get their turn under the microscope soon enough. As usual, none of this (with the exception of the celebrity examples) is my opinion. All data is collected from actual user activity.

Let's start at the beginning.

All people, but especially guys, spend a disproportionate amount of energy searching for, browsing, and messaging our hottest users. As I've noted before, a hot woman receives roughly the messages an average-looking woman gets, and 25× as many as an ugly one. Getting swamped with messages drives users, especially women, away. So we have to analyze and redirect this tendency, lest OkCupid become sausageparty.com.

Every so often we run diagnostic plots like the one below, showing how many messages a sampling of 5,000 women, sorted by attractiveness, received over the last month.

These graphs are adjusted for race, location, age, profile completeness, login activity, and so on—the only meaningful difference between the people plotted is their looks. After running a bunch of these, we began to ask ourselves: what else accounts for the wide spread of the x's, particularly on the "above-average" half of the graph? Is it just randomness?

What is it about her:

that gets more attention than her:

...even though according to our users, they're both good-looking?

Not all 7s are the same

It turns out that the first step to understanding this phenomenon is to go deeper into the mathematically different ways you can be attractive.

For example, using the classic 10-point 'looks' scale, let's say a person's a 7. It could be that everyone who sees her thinks exactly that: she's pretty cute.

But something extreme like this could just as easily be going on:

If all we know is that she is a 7, there's no way to tell. Maybe for some guys our hypothetical woman is the cat's pajamas and for the rest she's the cat Garfield. Who knows?

As it turns out, this distribution of opinions is very important.

Celebrity photos: to titillate and inform

Let's look at what the ratings distribution might be for a couple famous people. I imagine that for, say, the actress Kristen Bell it would be roughly like this:

Ms. Bell is universally considered good-looking, but it's not like she's a supermodel or anything. She would probably get a few votes in the 'super hot' range, lots around 'very attractive', and almost none at the 'unattractive' end of the graph.

Compare her to Megan Fox, who might rate like this:

On the far right, you have the many dudes who think she's the sexiest thing ever. On the far left, you have the small number of people who have seen her movies.

Unlike Ms. Bell, Ms. Fox produces a strong reaction, even if it's sometimes negative.

Real People

Now let's look back at the two real users from before, this time with their own graphs. OkCupid uses a 1 to 5 star system for rating people, so the rest of our discussion will be in those terms. All the users pictured were generous and confident enough to allow us to dissect their experience on our site, and we appreciate it. Okay, so we have:

As you can see, though the average attractiveness for the two women above is very close, their vote patterns differ. On the left you have consensus, and on the right you have split opinion.

To put a fine point on it:

  • Ms. Left is, in an absolute sense, considered slightly more attractive
  • Ms. Right was also given the lowest rating 142% more often
  • yet Ms. Right gets as many messages

When we began pairing other people of similar looks and profiles, but different message outcomes, this pattern presented itself again and again. The less-messaged woman was usually considered consistently attractive, while the more-messaged woman often created variation in male opinion. Here are a couple more examples:

We felt like were on to something, so, being math nerds, we put on sweatpants. Then we did some work.

Our first result was to compare the standard deviation of a woman's votes to the messages she gets. The more men disagree about a woman's looks, the more they like her. We found that the more men disagree about a woman's looks, the more they like her. I've plotted the deviation vs. messages curve below, again including some examples.

The women along the graph are near the 80th percentile in overall attractiveness. You can click the tiny thumbnails to expand them.

As you can see, a woman gets a better response from men as men become less consistent in their opinions of her.

Our next step was to analyze a woman's actual vote pattern of 1s, 2s, 3s, 4s, and 5s:

If You're Into Algebra

We did a regression on the votes for and messages to a sample of 43,000 women. To keep everything consistent, all the women were straight, between the ages of 20 and 27, and lived in the same city. The formula given in the body of the post was the best-fit we found on our second regression, after dropping the m3 term because its p-value was very near 1.

msgs are the number of messages the woman received during the observation period. The constant k reflects her overall level of site activity. For this equation, R2 = .28, which isn't great in a lab or on a problem set, but is actually very good in a real-world environment.

This required a bit more math and is harder to explain with a simple line-chart. Basically, we derived a formula to predict the amount of attention a woman gets, based on the curve of her votes. With this we can translate what guys think of a woman's looks into how much attention she actually gets.

The equation we arrived at might look opaque, but when we get into it, we'll see it says some funny things about guys and how they decide which women to hit on.

The most important thing to understand is that the ms are the men voting on her looks, making up her graph, like so:

And those ms with positive numbers in front contribute to messaging; the ones with negative numbers subtract from it. Here's what this formula is telling us:

The more men who say you're hot, the more messages you get.

How we know this—the .9 in front of m5 is the biggest positive number, meaning that the guys who think you're amazing (voting you a perfect '5') are the strongest contributors to your messaging income. This is certainly an expected result and gives us some indication our formula is making sense.

Men who think you're cute actually subtract from your message count.

How we know this—because the .1 coefficient in front of m4 is negative. This tells us that guys giving you a '4', who are actually rating you above average-looking, are taking away from the messages you get. Very surprising. In fact, when you combine this with the positive number in front of the m1 term, our formula says that, statistically speaking:

If someone doesn't think you're hot, the next best thing for them to think is that you're ugly.

This is a pretty crazy result, but every time we ran the numbers—changing the constraints, trying different data samples, and so on—it came back to stare us in the face.

What We Think Is Going On

So this is our paradox: when some men think you're ugly, other men are more likely to message you. And when some men think you're cute, other men become less interested. Why would this happen? Perhaps a little game theory can explain:

Suppose you're a man who's really into someone. If you suspect other men are uninterested, it means less competition. You therefore have an added incentive to send a message. You might start thinking: maybe she's lonely. . . maybe she's just waiting to find a guy who appreciates her. . . at least I won't get lost in the crowd. . . maybe these small thoughts, plus the fact that you really think she's hot, prod you to action. You send her the perfectly crafted opening message.

"sup"

On the other hand, a woman with a preponderance of '4' votes, someone conventionally cute, but not totally hot, might appear to be more in-demand than she actually is. To the typical man considering her, she's obviously attractive enough to create the impression that other guys are into her, too. But maybe she's not hot enough for him to throw caution (and grammar) to the wind and send her a message. It's the curse of being cute.

The overall picture looks something like this:

Finally: What This Could Mean To You

I don't assume every woman cares if guys notice her or not, but if you do, what does all the above analysis mean in practical terms?

Well, fundamentally, it's hard to change your overall attractiveness (the big single number we were talking about at the beginning). However, the variance you create is under your control, and it's simple to maximize:

Take whatever you think some guys don't like—and play it up.

As you've probably already noticed, women with tattoos and piercings seem to have an intuitive grasp of this principle. They show off what makes them different, and who cares if some people don't like it. And they get lots of attention from men.

But our advice can apply to anyone. Browsing OkCupid, I see so many photos that are clearly designed to minimize some supposedly unattractive trait—the close-cropped picture of a person who's probably overweight is the classic example. We now have mathematical evidence that minimizing your "flaws" is the opposite of what you should do. If you're a little chubby, play it up. If you have a big nose, play it up. If you have a weird snaggletooth, play it up: statistically, the guys who don't like it can only help you, and the ones who do like it will be all the more excited.

427 Responses to “The Mathematics Of Beauty”

  1. Renée Yoxon says:

    By gosh you’ve done it again!

  2. Waq says:

    Another amazing post. I loved it. So, so, interesting. It’s amazing what the data tells you, regardless of how paradoxical it might seem.

  3. Maya says:

    What about those of us who give star-ratings based on both pictures and profile content? Maybe someone with a bi-modal rating distribution says something like “I hate reading” or “God is super important in my life,” in their profile. These statements will obviously attract a certain type of person while simultaneously turning others off.

  4. Andrew says:

    While I generally love the analysis on the topics posted on OKT, this one seems to have a flaw. The one problem I notice with this analysis falls under your “What We Think Is Going On” post.

    It tends to assume that other guys in the OKC community have access to the data of the attractiveness data of a given companion. Obviously you guys can do a database dump and grab that, but the only way this is really available to everyone is with My Best Face – something that not everyone participates in no less analyses when contemplating sending a message.

    Do you have any more insight on how these “votes of attractiveness” might influence “subconcious messageship,” if you will?

  5. Joshua says:

    This is kind of a stupid “trend” since each person has their own comprehension of what “beautiful” is. Opinions vary person-to-person. Psychology will state that extremely beautiful women are less likely to be messaged, approached since this falls under the realm of the guy thinking she’s too good for him. Again, opinions vary. Some people like skinny women, others like bigger. Some like tall, some like short. ETC!!!

  6. I_am_quipster says:

    Very Cool, do you guys use R?

  7. AllForTheRush says:

    You guys do good stuff here. That being said It seems that we are missing a critical variable about what differentiates a cute girl who gets messages and a cute girl who does not (I believe okc trends covered this variable before). The missing bit is (wait for it)… which girl is most likely to have sex with you quickest. A man will see it easily in the pictures above, note that this also accounts for why tattooed girls get more messages as well. I don’t think it accounts for Megan Fox as the permanent ‘bitch face’ she sports is universally accepted as “I won’t have sex with you”.

    Its very easy to notice once you stop basing everything on one variable (looks) and start looking at men like complicated human beings (who shockingly enjoy sex) instead of lizard brained hominids.

  8. bleep bloop says:

    This is hideous pua bullshit, and you should be ashamed of even pretending this is scientific. I’m ashamed to have a profile on a site that honestly believes you can quantify a person’s attractiveness. Grow the hell up.

  9. Brendan says:

    As always, these articles are fascinating–thanks for continuing to post them.

    But the China-drowning-babies joke was unfunny and beneath you.

  10. Hughman says:

    Interesting analysis, I wondered why alt girls on here always have loads of awards and rarely reply, when you would expect them to be overlooked.

    I’d come to this conclusion naturally: alt girls are more interesting to the average guy on here: fashion sense, kinkier, geekier.

    So I go for the classically cute girls. Much easier to get responses.

  11. Matt says:

    Hey, what about those of us who message women because they sound like interesting people? Can’t we get in on this shallow bullshit too?

  12. David S. says:

    Guys- wouldn’t it seem that you’re making a pretty basic correlation/causation mistake here? Just because women with a higher deviation of attractiveness ratings tend to get more messages does not mean that the former is the cause of the latter. Thus, your advice to “play up what some guys don’t like” seems unproven, aside from not making a whole lot of sense to begin with.

  13. Ephemily says:

    It is a rare treat to be treated to insightful science and humor in the same post. Thank you.

    Now if you’ll excuse me, I need to go book a photographer for the next week when I’m retaining water…

  14. K says:

    Pretty crazy, but then I am always astonished at how much the slightly chubby girls try to play that down, despite that a ton of guys actually dig their curves. I wonder what the data says for males.

  15. shy_at_first625 says:

    A possible explanataion for this is that hiding your flaws (especially if you fail at it) communicates a lack of confidence. And as many of us know, confidence is the most attractive thing a woman can wear.
    So, the women who let their pictures say “I don’t need you to like me, random Joe” are getting more attention than the women who are forcing their pictures to say “Please don’t anyone see my flaws.”

    Great post, and another example of “Everything We /Know/ About Profiles Is Wrong.”

  16. bornyesterday says:

    I think this is actually one of the better done analyses that they’ve posted so far, mathematically speaking. I’m not sure that their conclusion is necessarily accurate that women should show their “unattractive” features, though (at least, not using that terminology).

    OKC did no real analysis of what in particular made the variance dependent upon certain “unattractive” features. Which is understandable, as defining such things can be rather difficult, especially when there is no way of insuring that people are rating solely on physical appearance (though it is probably safe to assume that the portion of votes limited to that evaluation is the most statistically significant.)

    From the sample pictures in the article, it would seem to me that the women with the higher levels of variance have certain characteristics, that aren’t necessarily “unattractive”, but which mark them as belonging to certain groups within the population. The “alternative”, heavily tattooed and pierced and stylistically made up girls, the geeky girls, the “ethnic” girls, etc. are more likely to get high votes from people who are interested in/associated with those population groups and more likely to get low votes from people who aren’t. Whereas the more “all American” girls are generally appreciated across the population.

    The way I’m interpreting the information, if women want to increase their messages, they shouldn’t post pictures which emphasize unattractive features, they should post pictures which emphasize their association with certain subcultures and population groups.

  17. Alex says:

    Couldn’t this just mean that men particularly like to message chicks they consider really hot? For example, if men only message 8s and up, and everyone rates you a 7, you get no messages. If men only message 8s and up and you get some 6s and some 9s averaging out to a 7, you get messages from the guys who rated you 9, thus more messages than if you got all 7s.

  18. John says:

    you’ve nailed it. before you even mentioned “game theory” i had it in my head. i pretty much do this consciously. this applies to profiles, too, though. when i give someone a star rating, it’s not just for looks. bland profiles, even if they sound awesome, never get an email from me. but if someone plays up their uniqueness, and they haven’t turned me off by the end, i’ll inevitably message them.

  19. Perkelix says:

    I’m really looking forward to your next part about how this applies to men.

  20. jtl says:

    Don’t forget to account for the role QuickMatch may play in rating other users. I often rate someone a four hoping that they have/would also rate me a four, but I would only message them if we got QuickMatched.

  21. George says:

    It would be very interesting if ok decided to give away some anonymized data. Kind like netflix did some time ago.

  22. Jaared says:

    Cool post, but I’d like to suggest an alternate explanation for why “ugliness” seems to help that’s not based on the game theory…

    A woman who has a lot of ugly votes and few or no superhot votes is probably outside of the range of what most people think of as reasonably ok. But a person who’s got a lot of hot votes is probably getting the ugly votes not because of some “objective ugliness” like obesity or awful skin or a deformity, but because of a particular look. That is, she’s *interesting*. My guess is that the right kind of interesting will boost a woman’s overall attractiveness, regardless of whether I think she’s lonely or popular.
    You could test this hypothesis by asking viewers to rate not just how hot a person was or wasn’t, but how much of her personality they thought showed in the picture.

  23. Ryan says:

    okay, the on in the sidebar isn’t so much a typo as an ambiguous wording. Got it.

  24. Gregor says:

    Fantastic data, as always. Unfortunately I believe that there are several unaddressed variables which effect the causations addressed in your theory.

  25. Aaron says:

    I’m a complete stats nerd and love these OKC trends, but I have a problem with this one when it comes to the ratings that you are using. Because ratings of 4 or 5 carry different implications than a 1,2, or 3 (the implication being if both parties are rated at 4 or 5, a messaged is sent on your behalf), this skews your data.

  26. Craig says:

    Can’t wait for the one for males. A lot of the data presented here and the studies really bring a lot of insight otherwise ignored out in the real world. Its the best dating advice, that its real statistics and can only be learned by observation. Great work.

  27. Mayson says:

    Great start but your not taking into account pair bonding at all. Guys with VERY little self worth embrace the “take what you can get” attitude as opposed to guys who have options for women in their lives.

    This is an awesome start but these variable falls VERY short when we ask WHY are they messaging the “ugly” girls.

  28. Aaron says:

    I’m a complete stats nerd and love these OKC trends, but I have a problem with this one when it comes to the ratings that you are using. Because ratings of 4 or 5 carry different implications than a 1,2, or 3 (the implication being if both parties are rated at 4 or 5, a messaged is sent on your behalf), this skews your data.

    Also, ratings are given for reasons quite different than attractiveness. Cool data, but I’m not sure about your validity

  29. Jacob says:

    Straight-A post! I can’t wait until you give guys the same treatment.

  30. Olive says:

    It is really crucial here to replot “measured attractiveness” vs. “messages per month” with your new formula, to see if that collapses the cloud to a finer line. Also messages vs. purely the number of “5” votes to convince us it’s not just that men only message women they consider tens.

  31. MrMr001 says:

    Have you overlooked that giving someone a 4-Rating IS sending them a message? I will often give a woman 4-Stars and SAVE her as a FAVORITE because I know the system will inform her of my choices. To me that constitutes sending her a message. It is a form of noninvasive signaling. This way, SHE can review my profile and signal if she has an interest.

    Why shouldn’t this form of messaging be counted? I also agree with Maya in that the difference between a 3, 4 and 5 might be due to the combination of her pictures, profile, quizzes, questions and question comments

  32. jerome cukier says:

    does your model work for boys too?

  33. Jimbo says:

    I think you’ve missed something that distorts everything and that is that giving someone 4 stars might be a means of trying to attract attention rather than a genuine rating. So you could have people giving everyone a 4* just to see if they will get a 4* back.

  34. Shuwen says:

    OMG, it’s really an amazing analysis,haha~How could you use game theory and formula~really hilarious~~

  35. Statist says:

    This was a pretty fun read. I do wonder if you made autocorrelation tests for the dependant variables and all that jazz, because all these terms are going to have relationships that could’ve colored your results. The fact that the number of people voting for a girl 5 is going to follow the trend of people voting for a person 4 could’ve contributed to your negative for that coeficient. Also could explain why the 3 was not significant, as it’s going to follow 3 and 4.

    I’d love it if you guys ran a regression with all the votes as a single categorical variable and added other aspects like your personality traits, profile completeness, etc. as additional variables to see how much hotness is really a factor.

  36. Dreidl says:

    Perhaps this begins to explain why straight men peep at/message lesbians – the guys are assuming a lack of male competition for women with more unusual (in straight people’s opinion) looks? (This is not to say that many gay females, in OKC’s category language, aren’t also conventionally attractive.)

  37. Marianne says:

    Nice post, very interesting.
    Only I feel like mentioning that this “it was like a baby we were trying to drown had somehow grown gills. (This happens all the time in China.)” seems in extremely bad taste, at least to me. Just sayin’.

  38. ablondecouple says:

    This ENTIRELY ignores people who have a habit of basing star ratings on the whole profile, not just pics.

    If a super hot woman has a 50% enemy rating with me, or thinks the earth is bigger than the sun, or is a bigot, she gets 1 starred.

    “Shiny on outside, like donkey dropping.”

  39. Brianhd71-4u says:

    Interesting..

  40. jamie says:

    you guys must have so much fun doing this
    congratulations

  41. doc_manhattan says:

    I completely agree with Maya: I think there’s a heck of a lot of male motivation not being discussed here. I wasn’t sure if it was just me, and then I saw the bit about tattoos –

    Now, any woman I know who has tattoos says that loads of guys approach her simply cos certain conditioning factors in the culture make them assume she has low self esteem and displays her tattoos because she’s desperate for a conversation piece with any man available.

    I think there’s a hell of a lot not being talked about in regard to what men think of as a potential “easy target” or even someone who they’d call a “skank” in front of their friends: after all, women’s beauty becomes abstract when you consider that 70% of mankind are lounging about with their stained T shirts, beer bellies and X Boxes complaining because every woman doesn’t look like a supermodel…

    Don’t get me wrong, I love OKstats and I think it’s a great piece of writing: but I’m just not 100% sure on the levels of honest assessment being done on why people message people.

    Oh and btw: when I saw this article was called “the mathematics of beauty” I knew *immediately* it would be about women’s profiles. Now, how would i know that…?

    ;-)

  42. Andy says:

    I’m not sure the star ratings are a direct measure of the girl’s attractiveness to the ranker. First, as someone else all ready pointed out, the profile can also factor in. And second, the system is designed so that users are informed of reciprocal rankings of four or five stars. It’s possible that male users are ranking girls four stars, when they really think there are a three, or a two, or even a one, just so they find out whether the other person thinks they are a four or a five star. And they might not have any intention of messaging that person unless they receive a high ranking themselves.

    Thus, because your conclusions are based on the assumption that the star rankings are only influenced by attractiveness, and because I think the star rankings are influenced by other factors, I think your conclusions are flawed.

  43. Ty says:

    ….and if you don’t look like ANY of there women ? the obvious…..your “scores” or “attractiveness” is significantly lower. Beauty must be redefined.

  44. JFontheR says:

    As a man, I don’t know, this study just makes me lose my faith in men more. Because the bottom line is, whether men objectify women and seek out only those that are ‘hot’ or the hottest, or objectify them and then do a kind of unconscious mental calculus to see which women will be most likely to be available (i.e., not the most attractive so maybe they feel they have a chance), it’s still objectification.

    Not that I pretend appearance doesn’t matter to me at all. And if anything appearance and fashion does say a lot about how a person chooses to be in the world. I consider myself a feminist and I firmly believe men don’t have “reptile” brains and base uncontrollable impulses (otherwise they wouldn’t be to blame for being chauvanistic and violent); I believe men are fully human, and are largely a product of social conditioning, as is masculinity in general. This study does support that to some degree I guess, it does show that men are more complex than just chasing the hottest women they see.

    Regardless, solid analysis and obvious effort put in, no doubt.

  45. Ty says:

    ….and if you don’t look like ANY of these women ? the obvious…..your “scores” or “attractiveness” is significantly lower. Beauty must be redefined.

  46. Remy Tennant says:

    Interesting, but there may be a methodology flaw here. Although you control for a range of variables (race, location, age, profile completeness, login activity), you do not account for women’s body types, even though you have that data. Did you try? You should control for that, and control for whether of not full-body pics are posted, if there is any way you have the time for that. For better or for worse, we wen care a lot about women’s bodies. Without teasing out this variable, I would call the results inconclusive. Good exploratory research nonetheless.

  47. Scott says:

    I’d agree with Maya’s point as well.

    I’d also like to know what the quantity vs quality breakdown of the messages were. Whether or not the message that was sent was a close copy of a message that the writer had written before (dramatically lowering the marginal cost of a message) or if it was something along the lines of “sup, how you.”

  48. RACHID says:

    Great…

  49. mynameisbrian says:

    I wonder how that coincides with the fact okcupid also displays how selectively someone is with responding to e-mail. As someone who looks average and has lower self-esteem, I rarely bother e-mailing someone who doesn’t “respond often,” as I figure there is little hope I will stand out, regardless of all the other things I can offer in a relationship.

    Would this variable affect your statistical outcome?

  50. winkingchef says:

    This blog is so nerdy, yet so interesting.
    I love you guys.