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. SmartSassyBBWinLA says:

    I loved this. Thanks for taking the time to analyze the numbers. As a plus sized woman I have found that there are men who find me to be their dream girl and others who find my body type totally unappealling. Thankfully most of the unattracted men don’t bother to send me a message. Rarely over the years of online dating sites have I encountered rude men who go out of their way to chastise me for my weight issue. The comment about men messaging the sexy pic is pretty true. I used to put up that type of pic and had fun with all the responses. As I have matured a bit I realize that I am not interested in someone drooling over me…unless it is because they also think that I am funny and smart. The men who like the sexy me quite often didn’t even want to know the rest of me. Getting laid would have been easy…a quality relationship was not. I love your site because I can read their answers to know if they are balanced in their approach to women in general or if they are really just here for the sexual payoff.

  2. Anthony says:

    I guess there is some truth in everything said here by the blog and the reactions to it.

    I could say I’m equally ignored by all the above. Most really attractive women aren’t interest, most fairly attractive women aren’t interest and even some you would think are in my “league” (as to whom determines that?) ignore me.

    I just don’t have the stomach for American sites anymore. In my age “group” there are far too many single parents and that’s more of a turn-off than “looks” so blog post like this don’t take these things into account as one commenter said.

    As for getting more responses or whatever, being aggressive (not with your messages) with how many messages you send out is the best advice I can give to those having problems.

    Ha, I can’t even find 50 profiles I like….

  3. SigmaEyes says:

    I was pleased to read this blog. I wonder though, if you should equally stratify the males on some criteria segmented into a 1 to 5 scale also, could you do a bi-variate analysis? The Z-axis dependent variable would be the probability of a message based on variable x and variable y. I would like to see the 3-D graph that would produce, if you think it an appropriate step in your analysis. I would think that men who consider their options to be limited (based on their past experience with approaching women) make very different decisions than taller, or more attractive, or more confident, or more successful men. I wonder if this would explain the “middle” portion of your analysis.

    What criteria would you think appropriate for rating the men on that 5 tier scale? I cannot post a graphic here, but I would imagine the resulting surface plot would have a “mountain” that is not centered, surrounded by a dip, which in turn would be surrounded by a ridge that forms a ring around the mountain and dip. – A somewhat common form. I’d enjoy the chance to look at the methods you chose to use for your data and analysis.

  4. Chris says:

    You know we have all of formulas now that let us control for Gender, sexual orientation and a whole bunch of other metrics.

    The one thing I see people complaining about the most is that it’s all about guys trying to see who’s most likely to put out. I’d really like the next Article, or one following it to be a complete statistical breakdown of what people state they are after and what they’re actually after in a potential mate, and what are statistically significant factors contribute towards more successes.

    And there is one thing I find rather disingenuous about these reports. Success is always termed as messages received for girls or responses for guys. Shouldn’t the only acceptable metric for success on a dating site be the number of people you ACTUALLY went on a date with, good or bad? Because that’s what the entire purpose of the site was, and messages in no way indicate increased success at finding a date.

  5. charleneearl says:

    I’ve been basing my ratings on the basis of the overall profile, not just the photo. Simply a way for me to keep up with the ones I’ve checked out.

  6. gil111 says:

    cute :)

    but i disagree with the interpretation. i have a different view:
    two types of girls:
    - cute girls – these get attention from guys looking for a serious relationship
    - provocative/fuckedup girls – these get attention from guys looking for sex, and also get low ratting from nice guys

    i clam experienced guys can consciously detect ‘easy’ girls. others can unconsciously do the same. ‘cute’ girls r those judged as good looking conventional girls

  7. Keovar says:

    All the users listed are quite attractive. ☺

    I prefer “cute” beauty over “glam” beauty, myself. I tend to rate most women 4, reserving 5 for ones that have really cool profiles. If someone actually writes back, I’ll usually upgrade her a star. Both 4 and 5 have the potential of a mutual 4-5 match, which can be really helpful because she’s already shown some interest before I message. I tend to rate 3 for women that I am unlikely to message, but might consider meeting if they make first contact. Having guts or an unconventional view of gender roles does count for something. I rarely rate 1 or 2, as many times I’ll just skip rating if I’m really turned off. Regardless of rating, I do make an effort to write back to anyone that messages me, even if all I have to say is “no thanks”.

    Personality and intelligence end up having a very strong influence over how I perceive a woman after I’ve gotten to know her a bit, so looks aren’t the most important thing up front, because I know she’ll grow on me if I like her mind.

    I do admit a bit of a shallow weakness for redheads. I blame my Celt ancestry for that. Dark hair or curls of any colour can be great too. I guess that comes from my dating history. Bleach-blonde is generally a turn-off, but that goes back to perceptions about her personality. if someone buys into such a generic idea of “glam” beauty, she tends to be less interesting mentally.

  8. Mari says:

    So now apply this to the “my best face” and maybe it will be a more useful gizmo!

  9. Moe Faux says:

    It’s good to see at least a little bit of info on how you chose your regression model in the “if you’re into algebra” box, but I’m still curious what the p values were for the coefficients you actually included. For instance, it would be interesting to know the significance of that -.1 term for m4. If the p value is high for that too, you might not really be able to say that being rated a 4 detracts from your message count (not with any confidence, anyway).

    Since your main argument here is that a woman receives more messages if she receives both high and low ratings, the way to test this would be to include an interaction term in your model. Just create a new variable that is m1*m5 and include that in your regression. If your hypothesis is correct, that could be a stronger predictor than either m1 or m5 alone.

    One of the common criticisms people seem to be making in the comments here is that ratings and message decisions aren’t just based on attractiveness. This analysis could be a perfect starting point for evaluating the effect of other factors (like, say, match percentages or personality traits) on messaging behavior. I’ve often been surprised at how consistent the “Dating Persona” test is with what is written in people’s profiles. What if you were to include dummy variables for each component of the persona test (random vs. deliberate, sex vs. love, etc) and see whether they had a significant impact on messaging? If guys are, as some people in the comments here seem to think, preferentially messaging the women they think will “put out,” then maybe you could test this idea using the dating persona (or your question-derived personality traits) as a proxy.

    Having some understanding of who is doing the messaging (and what they might have in common with the recipient) seems crucial to making any recommendations (like emphasizing your weird snaggletooth) based on these statistics. I can only imagine that the women receiving tons and tons of messages are mostly getting lots of unwanted attention. A really useful question to ask would be: how do you draw messages from people you might actually like? If you were able to determine which factors drew messages from “good” matches rather than just more-more-MORE, would you still be able to say that appearing “attainable” is a good idea? Maybe you would, but then maybe you wouldn’t. It’s entirely possible that the women receiving fewer messages are getting “better” messages and are happier with their experience on the site as a result (either because they end up with better dates or just because they have to wade through less crap every time they log on). The trick, I guess, would be in deciding how to measure which messages are “good.” Maybe you could weight all messages received by the match percentage between sender and recipient, or only count messages from users over a certain match compatibility threshold?

  10. Squirrelloid says:

    Your analysis is fundamentally flawed because your equation is dictated by one fact: you’re holding average rating constant. Since average rating is constant, more high ratings automatically implies more low ratings, which means the positive term for m1 doesn’t provide new information relative to the positive term for m5.

    So, here’s what’s really going on:

    You have a population of males looking at women. Each man is more likely to message a woman he thinks is more attractive. Each man has some threshhold below which he is unlikely to message a woman. (Or probably more accurately, each man has a gradient which determines the frequency with which he sends messages to women he rates at a given attractiveness.)

    So we have two women, both rated the same on average, one with higher variance of ratings than the other. But we’re not actually interested in the overall distribution, we’re interested in the truncated distribution which describes the men who send messages (on average). Ie, we could draw a line at the average threshhold and everyone above that line was likely to have sent a message. Thus, whichever women have more ratings above that line get more messages.

    There is something interesting in your data here, however. If the average threshhold were low, you wouldn’t notice a large effect of variance for women who averaged high scores. Since you consider to see a rather large effect on women who rate in the 80th percentile for beauty, that means that the average threshhold for men, on the basis of looks alone, is higher than the 80th percentile, based on mens’ personal ratings of a woman’s beauty.

    That given a particular average score, there are more men who dislike a woman’s look if there are more who really like her look is non-data, and predicated on the fact that we’re looking at a set average score.

  11. brent says:

    Wait … you use the stars strictly to weigh attractiveness?

    Then why show a chunk of their profile?

    I’m more likely to rate someone 1 star for an incomprehensible profile than for an ugly picture.

    I am more forgiving over things people have less control over … and that reflects in my ratings.

  12. Matt says:

    Are we ever going to get a male version of this? I’d like to see how women’s views of male attractiveness stack up against this data.

  13. Lola says:

    Just wondering if you considered the body language of the pose as part of the phenomenon you are studying. For example if 2 women have the same attractiveness rating does the way in which they are posing influence the number of messages. For example does a seductive pose change the results?

  14. Rory says:

    Dear Chris,
    Marry me.
    Sincerely,
    Rory

  15. Karl says:

    So…with all that analysis…why don’t you show us the “variation” of voting that people have given to us? Both so I can see what sort of voting proportions people have given to me, and so that I can see the proportions of the person I might be thinking of sending a message too?

  16. duckie says:

    wow! this analysis was really insightful! i think the phenomena is probably true on some extent. men and women alike do sense competition and go for what they believe they can get, rather than who is conventionally perceived to be in demand.

    however i think the “cute” girls are the ones who men prefer to be in a serious relationship with whereas the tarted up types look easy for a quick lay. those tarted up types can either really fool a dumb guy and make him believe she is really hot or be extremely offputting/turn off for a guy. obviously those ones who are fooled would go for her for the easy lay. some men can sniff “easy” from a mile away.

    but the question is, would you rather be conventionally pretty or something else? I know this guy who was convinced his ex was a 10 when literally everybody else thought she was hideous, now compared to me, as someone who is “conventionally” pretty, he was always ready to voice his opinions on me being second to her….it’s when this happens that this phenomena becomes problematic. probably a slightly bad example since she is pretty conventionally ugly/plain-looking, he was just very deluded. but id imagine there may be a few ones like him in the world.

  17. vasha says:

    There is another variable: the picture itself. The verrer photograph won in the polls shown.

  18. ComtriS says:

    Where’s the article for men?

  19. Nana says:

    I think it’s because the women who get messaged the most look the most interesting. “Interesting” looking women can be beautiful, but they will not attract a mainstream audience. Some guys may think a woman is unattractive simply because of her style (ex: piercings, tattoos, weird hair) However, the people who they do attract will probably be curious enough to write them a message.

  20. Chris says:

    The results are very intriguing and counter-intuitive. It may be informative apply regression to each individual so that you can examine the heterogeneity of regression coefficients across individuals and relevant subgroups. If there is considerable heterogeneity, the aggregate regression will be misleading. Another problem regarding heterogeneity is called Simpson’s paradox, a phenomenon in which correlations can reverse direction due to averaging. In either case, there might be interesting factors that account for the heterogeneity.

  21. insonmiacgrl says:

    Thanks for the info. I’d appreciate another article explaining the “sup” message? Cuz whatever it means to men (which might be volumes), is I think lost on me.

  22. MadRooster says:

    Hi, I do think that there’s an important factor that’s left out: Novelty is attractive. I, myself, like several attributes that the mainstream would call unattractive, and don’t find myself all that attracted to women who are excessively Barbie-like. I think that being polarizing is an important factor in creating a following, in whatever sphere you care to consider.

  23. wendy says:

    RE: THE WOMEN WITH CONSISTANT RATINGS GETTING LESS MESSAGES..

    I THINK YOUR HYPOTHESIS IS MOSTLY INCORRECT THOUGH TRUE FOR LESS THAN A MAJORITY OF MEN. I THINK THE LACK OF COMPETITION THEORY THOUGH WOULD CORRELATE MORE WHEN WOMEN ARE DOING THE MESSAGING.

    UNFORTUNATELY, MEN OFTEN DON’T EVEN MAKE SUCH THOUGHTFUL CONSIDERATIONS OR THOUGHT EVEN TO THAT LEVEL (LESS COMPETITION)-
    WORST OF ALL-THEY ARE COMPLETELY AND MALADAPTIVELY PROGRAMMED FROM EVOLUTION (GREATLY ENCOURAGED OR AGGRAVATED BY THE MEDIA AND CULTURE) TO BE DISTRACTED BY THE VERY TOP OF THE SCALE. THAT IS 90% OF THE MEN ATTEMPT TO ATTAIN 10% OF THE WOMEN.

    THIS PHENOMENON IS EASILY OBSERVABLE AT FOR EXAMPLE A POOL PARTY FOR SINGLE WITH MIXED AGES. SAY THE PARTY IS MOSTLY FOR THE OLDER CROWD BUT OPEN TO ALL AGES. THERE ARE PLENTY OF MEN AND WOMEN AGED 45-62. AND THEN A COUPLE OF VERY YOUNG ATTRACTIVE WOMEN AGED 23 HAPPENED TO SHOW ALONG WITH A GUY OR TWO ALSO ABOUT THAT AGE. 70% OF THE MEN MAYBE MORE WILL BE FOCUSED ON THE 2 YOUNG WOMEN.. EVEN SOME WHO ARE FAIRLY LOW ON THE ATTRACTIVENESS RATING THEMSELVES WHO WOULD BE LUCKY TO GET THE ATTENTION OF SOME OF THE WOMEN EVEN MORE REMOTELY THEIR AGE..

    BEST WAY TO HANDLE THIS IS AGE RANGE EVENTS TO AVOID THE DISTRACTION FOR THE MEN..

  24. nowcinterested says:

    Hello,

    I find this very interesting and thank you all for your input. SInce I am Heterosexual I am writing this that way.
    I have wondered about this off and on through my life. Men have always labeled me a “cute” woman. Even now at this age.
    And this is what I have found here.

    Some things to note: personality is a part of the equation that is not mentioned just looks. My experience tells me several things
    1 I initially thought the more you answer all of the Q&A (including sex) honestly the more it would help find a good man to date for a ltr… But that was not the case due to men mostly using it for their own misguided sexual reasons about me.
    And that when I answer too many Q&A about my sexuality and sensuality on OkCupid (and else where) I get quite a bit more contacts and more often then not in undesirable ways, far to forward, rude and obnoxious.
    2 When I let my cuteness also show in my personality I get more attention and a better quality of contacts.
    3 I also have observed over the years that being very attractive can be and is often a hinderance as this article states. Many men do not approach you because they think you are going to turn them down. Or so they have said at different times in my life. And whether I smile, have eye contact longer then average… gives them a signal if I will welcome their approach. Bottom line find the nerve to contact the other person. It’s well worth the risk in the long run.

    And there are still something to the old fashioned points about men. If you want to attract nice men, good men, men who are worth a whole lot more then just sex think first and be more careful with the Q&A. There is a lot of science to all of this. Once you start talking to individual men you can share more about yourself on a personal level.
    The bottom line is your heart and health are the most important things.

    Please, feel free to contact me with comments on anything I write.

    nowcinterested

  25. Regina29 says:

    Hi, As always I find your research absolutely fascinating! This is the first article I’ve read out here in a long time and I’m certainly glad to discover that you are continuing to do such fascinating mathematical magic!

    Several of my friends have told me in the past that they think I’m just slightly too pretty for most men. I don’t think of myself as very attractive, so I didn’t take them very seriously. Now that I read your theories, maybe my friends were on to something.

    I’m attracted to powerful, confident men…so if your theories are correct, they are blessings as far as I’m concerned. What you imply then is that men who are fearful of competition naturally self select to opt out of contacting women they perceive as being potentially too popular. In effect, this screens out men who aren’t strong enough to be with a woman like me. I definitely recommend that women continue to post their nicest pictures out here, for why would any woman want to be with an insecure man? Men in their power and comfortable with their Warrior energy are the ones that turn me on! Those men aren’t going to be afraid of the competition. This natural self selection process is great news as far as I’m concerned!

    Thanks for doing what you do! It never fails to be thought provoking!

    Peace, Regina

  26. NO MATH NEEDED says:

    What’s up with the whole analysis thing?

    The answer is SIMPLE. You don’t need mathematical interpretation/s.

    Men prefer the ones who they think they’d have a better chance of dating with, rather than the more attractive but ‘almost impossible’ ones. Men decide to hit (more) on the ‘real ones’.. the less attractive but hot enough.. you know.. ;)

    In other words, don’t WASTE your time on something you knew you can’t have.

  27. chartguy says:

    It’s actually a lot simpler than you think it is. You’ve used a sledge hammer (standard deviation and statistical analysis) to kill an ant.

    Compare the number of “5-star” votes a woman gets to the number of emails that she receives. I’ll bet that you find that they correlate very highly.

    It’s simple. If a guy does not think she’s a five, he’s not going to write her.