Performance Differences In Sports Between Men And Women

During my years of Parkour training I’ve seen numerous times that women have trouble with the sport. Guys usually learn all the vaults quickly and proceed to learn flips too. On the other hand, girls tend to have trouble with most vaults, especially the monkey vault. It’s easily attributed to lower relative upper body strength. The issue is obvious when we look at climb-ups. When we look at flips, girls often stay with flips that take less strength and power to execute. 

The differences are expected and observed not only in parkour but in all other sports. On average, men tend to be stronger, faster and more powerful. Most of us know this and it’s easily observed in the world records of different sports. It’s interesting to observe by how much.

My hypothesis is that there will be larger differences between the sexes in areas where more strength or power is required to excel. Also, there should be smaller differences where endurance and non-strength skills are required. I looked at the data and this mostly seems to be the case. Powerlifting and weight-lifting are both very dependent on strength. Events like the long jump are still dependent on it but not as much. Then we have the running events which are even less dependent on it.

We’ll first look at powerlifting. I looked at the 148, 165 and 181 pound weight classes. These classes are somewhat in the middle. I picked records for drug-tested athletes whenever I could. I also picked records for squats with no wraps. All lifts are in-meet lifts. For squats the ratios are 1.56, 1.25 and 1.27. For the bench press, the ratios are 1.46, 1.35 and 1.42. For the deadlift the ratios are 1.30, 1.26 and 1.45. For the totals we have 1.30, 1.27 and 1.21. The average between the classes is about 1.32 for squat, 1.41 for bench, 1.34 for deadlift and 1.26 for the total. The errors in this data are probably somewhat high but it’s clear that there are large differences.

Next, we’ll look at weight-lifting. The men’s and women’s division do not have the same weight classes but by plotting the records it seems like they fit a quadratic relationship. 

 I took the men’s 56, 62, 69 and 77 kg weight classes and the women’s 58, 63, 69 and 75 kg classes. Using quadratic fits for the men’s records, I adjusted them to what they would be like if the classes were the same as the women’s. The the snatch’s ratios for the four classes are 1.29, 1.32, 1.29 and 1.27. For the clean and jerk’s they are 1.23, 1.29, 1.24 and 1.27. For the totals they are 1.26, 1.32, 1.27 and 1.28. I trust this data because the judging rules are stricter.

Now, let’s look at the running events. I picked the records for 50 m, 100 m, 200 m, 400 m, 800 m, 1 km, 1.5 km, 2 km, 3 km, 5 km, 10 km, 15 km, 21 km (half marathon), 25 km, 30 km, 42 km (marathon), 100 km. The ratios do not seem to follow any obvious relationship but the average is 1.11 and they are between 1.05 and 1.14. 

Interestingly enough, the 50, 100 and 200 meter races lead to ratios of 1.07, 1.09 and 1.11. I repeated the analysis for the 100 m, 200 m, 400 m and 800 m races with the top 5 best times in each category. The resulting ratios were quite close to the ones derived earlier. If I have more time I might look at more data to see what’s going on.

Finally, I looked at some other track and field events. For the broad jump the records are 3.71 m for men, 2.92 m for women, and the ratio is 1.27. For the long jump the numbers are 8.95 m, 7.52 m, 1.19. For high jump, 2.45 m, 2.09 m, 1.17. Triple jump: 18.29 m, 15.5 m, 1.18. Pole-vault: 6.14 m, 5.06 m, 1.21. Looking at the top 10 performers of each event, we have the following ratios 1.19, 1.16, 1.18, 1.18

To summarize, this is the data I’ve gathered on the differences between male and female elite athletes. As expected, the differences are greater when strength sports are concerned. The difference is smaller when endurance and technique are concerned, even in events like the 100 meters. I think there is some great analysis that can be made on this data but I have to save this for next time. For example, the numbers suggest that absolute strength plays a major role in weight-lifting and standing long jumps, a somewhat major role in field events, and only somewhat of a role in track events.

Homeostasis is the biggest enemy of personal change

I'm fed up with a lot of self-help material out there, so here is my take on it. Things don't have to be complicated to be effective. We all want to improve certain things about ourselves. We all try and most of us fail again and again. Those that succeed are deemed lucky and special. Those that fail are normal and acceptable.

Your worst enemy when you are trying to change yourself is ... you. We have to be careful with this common knowledge. On one hand, it's easy to externalize the blame. People are assholes, nobody cares about you, life's unfair. It's also easy to internalize. I'm not good enough, I'm a failure, I'm weak, I'm fat, I'm unealthy. Negative spirals of doom and gloom are common too. Been there, done all that.

What's really stopping you from becoming better is homeostasis. It's your body's property to restore your last stable state and maintain your identity. It's machinery, not a personal flaw. It's still a part of you, though, and it's not other people. Unless you are a dualist, your mind is part of the system and is greatly affected. Your mind has homeostasis on its own. It's why positive and negative feelings go away, you revert back to normal.

The desire to change is not enough. It's just a fleeting thought through your brain. You may feel like you are already different, but it's just an illusion because the moment the thought ceases, you revert back to your old self. This is a very crucial time. You need to record things and make a quick plan. No matter how long you maintain it, it ALWAYS ends. These pockets of inspiration should serve to restart your process of change if you fall off the tracks.

The reason that thoughts don't do much on their own is that our thoughts are fleeting. The structure and mechanics of the brain don't change. Once the fleeting thoughts are gone, you are left with the same old body, same old brain.  This is why it's so important to make a physical mark of change, so that you can start from there, for when your thoughts revert back to normal. So, the goal is to change physically.

Once you begin the change, it feels great for some time. You feel different and you are on your way. Its stability is an illusion. In a way, your brain is simulating what it would be like to be a changed brain. You need to maintain and regain this state as much as you can for your brain to actually remodel itself. But at first, you are only successful at temporarily changing your brain chemistry. When you pause, homeostasis is already doing its business to undo you. And oh boy, does your body try hard or what! This is where the real game begins. Your body aches and feel iffy, you feel like staying home and doing nothing, you feel sad, tired and lazy, you feel doubt, and so on. The gimmicks are many, the goal is one - stop you from doing this big overhaul. At this point it's very easy to give in to these desires of your body but you have to remember that it's just trying to bring things back to normal, and it's trying everything it can, but normal is not what you want.

It's part of the game and to some extent you cannot avoid this. I believe the best way to overcome the barrier is to achieve victory by attrition over your body. Don't give a chance for your body to work on getting you back to normal. It's really important to get the ball rolling and gain momentum. If you want to be strong, train every day. If you want to sing well, sing every day. If you want to be a nicer person, do it every day. But what about over-training? Screw over-training. It's over-training that you want. Your body needs a very good reason to change or it won't. Train once, twice, three times a day if you need to, just do it and stick to it. Obviously, there are precautions. Start small.It's better to do 10 minutes a day every day than 70 minutes once a week. But why is this better? Think about it, if you train 3 times a week, for example, you are giving your body 4 days to push back at you. If you are poor and you want to be rich, would you work 3 times a week, or every freaking day? If you are not working on your goals every day, you are fighting an uphill battle. Train every day and you'll exhaust your body's tricks. It'll just give up and play along for the time being. I personally felt amazing during my 9 weeks of intense weight training. It was somewhat scary to see my body recovering every time. As a concrete example, suppose you want to go to the gym every day. Everything that contributes towards your gym session is positive (brings you closer to your goal) and everything else is negative (brings you back). It can take 1/2 hour to plan your session, 1/2 to eat a proper meal that will help with the session. The session, including changing can be 1 or 2 hours, say. The post workout hormones can work their magic for a few hours (say 4?). Well, that's about 5-6 hours out of 24 that bring you closer to your target.

"Doing" is more important that "Doing well" at first. You have time to add the "well". Don't plan much yet. The window of opportunity is short-lived and you might miss the train. That's why in most cases, you should just start doing something every day. Form may be bad, exercises may be dumb, recovery might be bad, diet may be bad. That's ok, because once you start, you'll have plenty of opportunity to improve things as you go. Hit the ground running. Start easy and light and adjust as you go. It's a very organic approach.

So, once you've done this for a month, you are good, right? Wrong. You seriously think that a month is enough to undo years of habituation to another life? One thing is for sure, you need to keep it up for some time. After x number of days all you've achieved it is a working status - you've kept this lifestyle for x number of days and you can keep it up indefinitely, if you want to. But it still takes effort and a constant reminder. And then it takes much more time to physically ingrain it in you and make it a part of you. The moment you stop, your body will work against you, unless you've done it for long enough, so that the thing is part of you. Where's the end though? It's when you notice that you haven't thought about your newly-formed habit for so and so days/weeks/a month. Suddenly, it's a part of you and you cannot not do it.

In the end, people do change as result of life events and habits. It's usually a slow change. If you put on 100 pounds over 20 years, that's only 5 pounds a year. Your body weight probably fluctuates by more than 5 pounds every day. Big change happens slowly. However, it doesn't have to take long, until you perceive change. Getting twice as strong may be far into the future but getting 1.1 times stronger is easy. You only need to effectively only nudge your body and mind a little bit every day in the right direction, and over time the transformation will be significant. This is why people become fit/fat, relationships win/fail, careers become established. The stability of a transformation is proportional to the amount of time it prevails. In other words, getting fit is difficult, but staying fit is not.

Density of Math language

I'm sure this is a common problem among mathematicians reading mathematical text. If we read a bit too fast we glaze over. This also happens when reading novels and other text but with math, it happens much quicker. Of course, this has to be because math is denser than the English language. Unravel the meaning of any mathematical expression and it becomes clear. The solution to the problem is quite simple - we need to spend more time reading the mathematical expressions themselves. This post could easily end here but it’s more interesting to explore further.

There are a few ways to read mathematical formulas. The first and most obvious is to read them the way we would say them if we were dictating to someone or the way we would input them into a computer program, or $\LaTeX$. Then we might read them the way we would communicate them to a fellow mathematician, presumably short-handed somewhat. Finally, we might combine math words into more complicated objects which would themselves be treated as single words in our brain. Let’s try an example – the Fundamental Theorem of Calculus. We have

$$ \int_a^b f’(x) dx = f(b) – f(a). $$ 

Here are different ways we can read this 

- The integral of the derivative of f with respect to x from a to b equals f of b minus f of a (23 words)

- Integral, a to b, of f prime of ex dee ex equals f  of b minus f of a (19 words)

- Integral, a to b of f prime equals f b minus f a (13 words)

We can become very picky with the way we assign a number of words to a mathematical expression. One can easily just look at the way $\LaTeX$ math formulas works and assign a word for each symbol or operation. For example, the expression above goes by \int_a^b f’(x) dx = f(b) – f(a) in $\LaTeX$. Counting \int, _, a, ^,b, f,‘,(), x, dx, =, - as one word each, we arrive at 19 words, which is the same as the second interpretation. I will use a mix of this approach the the one with which we would communicate to others.

As an example, we will take the proof of a theorem from an analysis textbook and count the number of math words in it. It is the proof of theorem 17.1, page 148 of “Analysis with Introduction to Proof” by Steven R. Lay, $3^{rd}$ edition.

To get an idea for how my counting works, here are a few examples. 

"$ \lim (s_n + t_n) = s + t $" is read as "limit of s sub n plus t sub n equals s plus t", which is 13 words. 

"$|s_n t_n – s t| = |(s_n t_n – s_n t) +  (s_n t – s t)|$" is read as “absolute value of s sub n times t sub n minus s times t, equals, absolute value of bracket s sub n times t sub n minus s sub n times t plus bracket s sub n times t minus s times t”, which has 43 words. 

"$|t_n – t| < \epsilon/2M$" is read like “absolute value of t sub n minus t, is less than or equal to epsilon over two em”, which has 18 words. 

The process is tedious so that one has to go line by line tallying up all the numbers. The numbers are striking. There are only 160 English words in the entire proof that goes on for more than a page. There are about 350 in-line math words hidden inside mathematical symbols and about 450 such words off-line, for a total of 800. Coincidentally, this is exactly 5 times the number of English words in the proof. It's becoming clear why we can easily glaze over when we read a proof like this.

We can take this idea further. Each page seems to have about 45 lines of text. I scanned about 20 different lines of text to count the number of words in each line and estimated that the average number of words in this math textbook is about 13.92 per line. This comes to about 626.4 words per page of block text, on average. The proof takes about 1.2 pages of text. If we accumulate the English words into a block of text, we would have about 0.26 pages of text. If we do the same for the math words, we would get 1.27 pages. The total for the proof is about 1.53 pages of solid text, which is only somewhat larger than the current proof. It's only slightly longer than the actual proof, and this is due to formatting. Off-line formulas take up a lot of space area but the actual symbols are dense. I propose the following rule of thumb: "Treat the empty space around math expressions as text. Pay that much more attention to these areas."

Just for fun, define the mathematical word index of a proof to be the number $r$

$$ r = \frac{m}{m+w} $$

where $m$ is the total number of words hidden in mathematical expressions over the work we are looking at, and $w$ is the total number of actual English words in the same section. For this proof, $r \approx 0.83$. This proof is 83 percent math. Well, actually, the proof is 100 percent mathematics but this is beside the point.

Let’s look at a proof or two just for fun. Here is a proof of the density of the rationals inside the real numbers, using the Archimedian property of the natural numbers.

12.12 Theorem (Density of $\mathbb{Q}$ in $\mathbb{R}$) If $x$ and $y$ are real numbers with $x < y$, then there exists a rational number $r$ such that $x<r<y$.

Proof: We begin by supposing that $x > 0$. Using the Archimedian property, there exists an $n \in \mathbb{N}$ such that $n > 1/(y-x)$. That is, $nx+1 < ny$. Since $nx > 0$, it is not difficult to show (Exercise 12.9) that there exists $m \in \mathbb{N}$ such that $m - 1 \leq nx < m$. But then $m \leq nx + 1 < ny$, so that $nx < m < n$. It follows that the rational number $r = m/n$ satisfies $x < r < y$.

Finally, if $x \leq 0$, chose an integer $k$ such that $k > |x|$. Then apply the argument above to the positive numbers $x +k$ and $y+k$. If $q$ is a rational satisfying $x +k < q < y+k$, then the rational $r  = q - k $ satisfies $x  < r < y$. QED


Alright. Going line by line (of the blog entry) the number of english words is

$$ w = 10 + 9 +12 + 6 +5 + 11 + 11 + 4 = 68. $$

The number of mathematical words is

$$\begin{aligned} m &= 4 + (3 + 9 + 10 + 6) + 3 + (12 + 15 + 9) + (5 + 7) \\ &\quad\quad+ (6 + 6) + 6  + (11 + 5  +7) \\ &=  124\end{aligned}$$

where in the brackets I've counted the words in each expression on the line to make less mistakes. This gives us a ratio of $ r = 124 / ( 124 + 68) = 0.6458333$ or roughly 65 percent.