An interesting discussion came up at work the other day, when a co-worker mentioned that a friend of hers (a guy) had opted to undergo an expensive and painful surgery to gain a few inches in height. While the consensus seemed to be that such an operation would ultimately not be worth it, I must admit that I entertained the idea briefly…or at least long enough to run a few numbers for my own curiosity.

Most people have probably heard of the phrase “tall, dark, and handsome” being used to describe the physical attributes of an attractive male. While it’s certainly a generalization, there’s little doubt that when it comes to traditional heterosexual relationships, women tend to prefer taller men. Although attraction is often the result of a variety of factors, I wanted to focus on just one attribute (height) and quantify the impact of being two inches taller on a man’s romantic prospects.

Assumptions

I started with a sample population of 200 million people (100 million men and 100 million women). I then assumed that the heights for both men and women were normally distributed: “adult male heights are on average 70 inches  (5’10”) with a standard deviation of 4 inches. Adult women are on average a bit shorter and less variable in height with a mean height of 65  inches (5’5”) and standard deviation of 3.5 inches.” Although relationship matching is a two-sided problem, I assumed for the purposes of this exercise that only women’s preferences counted (and that every woman shared the same preferences), while men were assumed to be indifferent.

For example, if a 5’5” woman preferred men who were at least her height but not more than 12 inches taller than her, then her potential prospects would include all men between 5’5” and 6’5” (85,429,107 men, or 85.4% of the male population). Conversely, a 5’10” man’s prospects would include all women between 4’10” and 5’10” (90,068,614 women, or 90.1% of the female population).

As always, my data is available here.

From a Woman’s Perspective…

I first examined the impact of women’s preferences on the number of potential men that they would be attracted to. The bold black and red lines represent the number of males and females (left y-axis) distributed by height (x-axis). The dotted blue, green, and purple lines represent the number of potential male prospects (right y-axis) for a given female’s height. The three colors represent three different preference cases:

• Case A: All females prefer between her height (0” lower bound) and 12 inches taller (+12” upper bound). This is our base case
• Case B: All females prefer between 3 inches shorter (-3”) and 12 inches taller (+12”)
• Case C: All females prefer between 6 inches shorter (-6”) and 12 inches taller (+12”)

Of course, there are a multitude of different preferences that we could test, but I mainly wanted to see what would happen if women became more receptive to dating shorter men.

The total number of potential men increases from Case A to Case B to Case C. This should surprise no one, as the total preference range is expanding in each case. What should also be intuitive is that while every woman is better off (which we will define as having a greater number of potential male prospects), how much each woman benefits is dependent on her height. Because we are pushing out the lower bound of women’s preferences from 0” to -3” to -6” in the three cases, it makes sense that doing so disproportionately benefits taller women. This is represented graphically by the right tails of the distributions shifting out (from blue to green to purple) while the left tails remain anchored.

It’s worth noting that even a woman of average height (5’5”) would experience notable benefits from the expansion of lower bound preferences. In Case A, her potential prospects would include 85.4% of the male population, but that figure jumps to 93.7% in Case B and 95.7% in Case C. However, the same shifts in lower bound preferences would be a complete game-changer for taller women. A 5’8.5” woman (1 SD above the mean) would see her potential prospects increase from 64.2% in Case A to 86.5% in Case B and 96.5% in Case C. Lastly, the “ideal height” (i.e. the height at which a woman would have the most potential prospects) in each of the 3 scenarios would be 5’4” in Case A (86.6% of male population), 5’5.5” in Case B (93.9%), and 5’7” in Case C (97.6%).

From a Man’s Perspective…

The bold black and red lines remain unchanged, but the dotted blue, green, and purple lines now represent the number of potential females prospects for a given male’s height in Case A, Case B, and Case C, respectively.

The dotted lines are a mirror image of the ones in the first chart. The total number of potential women increases from Case A to Case B to Case C. Everyone man is better off, but shorter men are benefiting disproportionately. This is represented graphically by the left tails of the distributions shifting out (from blue to green to purple) while the right tails remain anchored.

A man of average height (5’10”) would experience benefits from the women expanding their lower bound preferences, but these benefits are not as significant as the gains that the average woman would experience. This is due to the fact that men are on average taller than women and that women tend to prefer taller men. In Case A, the average man’s potential prospects would include 90.1% of the female population (already very high), and that figure increases to 96.6% in Case B and 97.6% in Case C. The relative gains from Case A to Case B to Case C (6.5% and 1.0%) are indeed less than the gains of the average woman (8.3% and 2.0%).

For the same reasons, women changing their lower bound preferences would be an even greater game-changer for shorter men than it would be for taller women. A 5’6” man (1 SD below the mean) would see his potential prospects increase from 61.2% in Case A to 87.3% in Case B and 97.6% in Case C (gains of 26.1% and 10.3%, versus 22.3% and 10.0% for the tall women 1 SD above the mean). Lastly, the “ideal height” (i.e. the height at which a man would have the most potential prospects) in each of the 3 scenarios would be 5’11” in Case A (91.4% of female population), 5’9.5” in Case B (96.8%), and 5’8” in Case C (99.0%). Just to hammer the point home, notice how these peak percentages are higher than those of their female counterparts (86.6%, 93.9%, and 97.6%). Lastly, I’d like to point out that in our base case preference scenario (Case A), the ideal heights are 5’4” for women and 5’11” for men, which is very close to their average heights of 5’5” and 5’10”.

The Impact of Two Inches of Height

All this brings us to our main question: as a man, how much is it worth to be two inches taller? At this point, it should surprise no one that the answer is “it depends” (doesn’t it always?). Given everything that we’ve learned so far, we should expect shorter men to benefit more from a potential height increase than taller men.

We see that this is in fact the case in the chart below.

The dashed lines measure the percentage increase (right y-axis) in the number of potential female prospects for a man of given height (x-axis). These dashed distributions resemble exponential decay, with extremely short men benefiting significantly from being two inches taller, but with those benefits rapidly diminishing as men approach 5’6” and taller. The shapes of these distributions are also impacted by the female preference case. Case A shows the steepest decline while Case C shows the shallowest. This should make intuitive sense, because Case A is the least tolerant preference range while Case C is the most tolerant.

Lastly, if we zoom in on the impact of being two inches taller on only men who are 5′ and taller, then we can get a clearer sense of where the breakeven height is for each preference case. Here, we define breakeven as the height where a man receives no benefit from being two inches taller. Any man taller than the breakeven height would in fact be worse off if he received the height increase of two inches.

In the base case preference case (Case A), the breakeven male height is 5’10”, which is exactly the average male height! In other words, a 5’10” man and a 6′ man have the same number of female prospects, which ties in beautifully to our earlier conclusion that 5’11” is the ideal male height in Case A (5’10” and 6′ are simply symmetric points on a distribution where 5’11” is the peak). By the same analysis, the breakeven male height is 5’8.5” in Case B and 5’7” in Case C.

The fact that the breakeven male height decreases as the female preference range increases (from Case A to Case B to Case C) is significant because it very clearly shows that there are two ways to solve the attraction disadvantage that shorter men face: (1) make them taller or (2) get women to change their preferences. The former is a solution that comes at a steep physical and financial cost (to an individual), while the latter is more of a mental and cultural challenge (that may in fact be more difficult to achieve as a society).

As a 5’7” man, I find it amusing that I would boost my potential prospects by 21% if I were two inches taller in Case A, while that figure would shrink to only 5% in Case B and 0% in Case C.

So ladies, please show some love for shorter men! It will make our decision to not get height enhancement surgery that much easier.

As some of you may recall, I kicked off the Fun with Excel series with a post on attraction, where I hoped to explore the mechanics of physical attraction from a statistical perspective. Due to the amount of feedback I have received (positive, skeptical, or otherwise), I have decided to write a follow-up post.

In the first part of The Laws of Attraction, I focused solely on physical attraction, and the impact of bias in our perception of attractiveness on seeking a compatible partner. In part two, I focus on the bigger picture: given a set of personal traits, what is the probability that you will find someone with those traits at the specific level that you desire?

Background: In the song One In A Million, Ne-Yo sings about a girl who he calls “one in a million.” Of course, not content with just enjoying the music, I wondered to myself what it actually meant to be “one in a million.” One way of measuring this is by breaking down attraction into a larger set of personality traits and trying quantify our desires, which is essentially what online dating services do with their “matching formulas.” For purposes of our exercise, let’s say you have a list of 10 distinct characteristics that you believe to be important and that you actively look for when searching for a partner. You might be more picky on some traits than others, but it isn’t too hard to quantify your objectives. Similar to my previous project, I quantify these objectives in terms of percentile, which, at least from a guy’s perspective, is pretty straightforward. For example, I might say, “I’m only interested in a girl who’s in the 80% percentile for Trait 1, 90th percentile for Trait 2, 50th percentile for Trait 3…” and so on and so forth. Now, the question is “what are the chances that such a girl exists?” A closely related question is “how many such girls are out there?”, followed by the not-so-fun reality-check of “what are the odds that I’ll actually find such a girl?”

The Model: While we won’t tackle the last question in this post, the first two are pretty straightforward to simulate from a mathematical standpoint. For each trait, the probability of finding someone who is at the X-percentile or higher of that trait is (100-X)%. For multiple traits, all we have to do is multiply these probabilities together, but the key assumption here is that all the traits are independent. Obviously, this isn’t true in real life, but we’ll revisit this point in a little while.

Assuming we start with a set of 10 traits, I will define a person having N “Perfect” Traits as someone who ranks at the 90th percentile or higher in N traits, and at the 50th percentile or higher in the remaining 10 minus N traits. Thus, assuming a world population of 7.12 billion, a male/female split of 50/50, and that you are heterosexual, the number of potential partners with 0 “Perfect” Traits  walking on the planet is 3,476,563, or 1 in 1,024 (the mathematically inclined should immediately realize that 1,024 = 2^10). On the other hand of the spectrum, there are theoretically only 18 people with 9 “Perfect” Traits, or 1 in 200 million. Note that a person with 10 “Perfect” Traits technically doesn’t exist, as probability indicates a 1 in 10 billion chance. At this point, the astute reader will note one possible answer to Ne-Yo’s earlier problem: if you consider a smaller set of 6 traits rather than 10, a “one in a million” girl would simply be a girl who has 6 “Perfect” Traits (all 6 traits at the 90th percentile or higher) in that scenario.

The Results: I plotted the entire spectrum of N “Perfect” Traits in the scenario of 10 traits, to arrive at the following graph:

It should be no surprise that our graph strongly resembles a normal curve, as we are working with a binomial distribution.

I suppose the lesson here is that it doesn’t pay to be picky, but recall the very important (and incorrect) assumption we made earlier that all traits occur independently of one another. In the real world, however, this couldn’t be further from the truth. Creativity may be correlated with Curiosity, Honor may be correlated with Kindness, and Intelligence may be highly correlated with (or the cause of) all the other traits. Accounting for the dependencies between and among all 10 traits would require us to estimate both marginal and conditional probabilities, which would not only be difficult, but also complicate our model very quickly. Statistical mumbo jumbo aside, what this means is that the probabilities estimated by a simple binomial model are far too conservative (too low). This should be great news for all the picky daters out there.

An alternative way of tackling the dependent traits problem is to simply consider a smaller set of traits. For example, if we created a list of 10 traits, and then realized that two of them were very highly correlated with each other, then we could eliminate one of them and simply consider a 9 trait model, which in turn would be a more accurate simulation of what the actual probabilities might look like in real life. To that point, I also plotted out graphs for scenarios involving 7 traits and 5 traits:

Note that as we decrease the number of traits, the number of potential candidates increases exponentially. So if you only considered 5 main traits, and furthermore were only picky about 3 of them (3 “Perfect” Traits in the graph above), then you would only be looking at a probability of  1 in 400. Not bad.

Conclusion

At the end of the day, it is perhaps silly to attempt to model real life human dynamics with 50 lines in Excel. But that would also be missing the point of the exercise. Thinking about real world problems from a different perspective (whether it is psychologically, statistically, or otherwise) can shed new light on the issue, or simply affirm something we already knew or suspected. Even if it is only the latter, there is still value derived from being able to connect the dots between a variety of different frameworks.

As for me, my dream girl in the 10 Trait model is about 1 in 5,925,926, and about 1 in 53,333 in the 5 Trait model. I’m not sure if I’ll ever find her, but it’s satisfying to know that she’s out there.

-J