A/B and Rh Antigens in Blood Types: A Statistical Test of Independence among IISER Kolkata Students

A couple of days ago one of my juniors in college (Indian Institute of Science Education and Research, Kolkata) found unguarded in the guest account of our computing system a spreadsheet.

This spreadsheet contained the blood types of the Masters students across five batches (‘07-‘13). With this information he made a nice little bar diagram of the frequency distribution of blood types, which I lift here:


When I looked at this graph, what I noticed first was that this contains the distribution across both the antigen type (A/B/AB/O) and the Rh factor (+/-). It struck me that it may be possible by constructing a contingency table to check whether these two properties are independent of each other.

For this I first constructed from the graph the two-way contingency table of the joint distribution of antigen and Rh factor:


Rh\Ag→ A B AB O Marginal totals
+ 76 116 28 122 342
1 4 1 6 12
Marginal totals 77 120 29 128 354 (total)


Consider now the marginal distributions, i.e. the subtotals along the last row and column. If we divide these numbers through by the total number of data points (354), we obtain the relative marginal frequencies. For a large sample, this may be identified with the probabilities of occurrence of the individual properties irrespective of that of the other.

Now comes the important part. If the occurrence of these two properties are independent of each other, then the joint probability in any cell shall be the product of the marginal probabilities for that row and column (P(A∩B)=P(A)P(B)). In terms of frequencies, this means that the number in any cell should be the product of the subtotals for that row and column, divided by the total (354). You can think of it this way: the frequency in any cell is the product of one of its marginal frequencies with the other relative marginal frequency (signifying the independent conditional probability). Example:

n(A+) = n(A).P(+) = n(A).n(+)/N.

Taking this assumption of independence then, it is possible to construct a contingency table inwards starting from just the outer marginal frequencies. This has been constructed below with the marginal frequencies of the actual data, and rounded to the nearest integers.

Rh\Ag→ A B AB O Marginal totals
+ 74 116 28 124 342
3 4 1 4 12
Marginal totals 77 120 29 128 354

If you compare the joint frequencies in the two tables, you can see immediately that they are very close, lending support to the assumption that these factors are independent.

For a more graphical idea, I decided to plot cluster bars of the actual and the computed frequencies assuming independence.


The average error between the actual and computed frequencies was 0 over the eight blood types. That doesn’t say much because errors of opposite signs reduce the net effect. The root mean squared error was 1.16, which too is tiny in comparison to the frequencies themselves.

Thus, it is pretty safe to suppose that the antigen and the Rh factor are uncorrelated properties.

There are stronger, more explicit tests of association than what has been done here. The usual correlation coefficient, though, cannot be calculated here because the variable (blood type) is not a quantitative one. However, there are others that you can read about here.

If you want to test this method or any of the other methods on other datasets of blood types, this page provides quite a bit of data for various countries (although last I checked, they seemed to not be completely factually correct).

A Statistical Problem on Laptop Uptimes

Suppose you are in a large university campus. Most students here use laptops, and if you look around, you’d see most of them either working, listening to music or doing something else on their laptops. Suppose now you think of a quick project, of listing the uptimes of the laptops (how long they’ve been running). In Windows this is quite simple. Under the ‘Performance’ tab in Task Manager, you’ll notice that ‘Uptime’ gives the duration for which the laptop has been running. This timer, however, keeps counting from where it left off if you resume from hibernation, as it should, but we shall assume that no such cases happen in our campus.

Now, the campus is huge, and many students are using their laptops. You figure out some methodical way of visiting each student so that there are no over- or under-counting errors. But it takes you quite a while to visit all of them and note down their uptimes. So that if, for example, it took you six hours to collect all the data, then you took the last uptime reading six hours after the first.

If we assume a very large campus where the uptimes of the laptops of different students are completely independent of each other, the questions are the following:

1. On an average, is it going to make any difference to the statistics you collected, if instead of taking a long time to go around the campus, you could somehow acquire all of the uptimes at one instant of time?

2. Is the data you collected going to be distributed differently from that of the maximum uptimes of student laptops (duration before they shut it down)?

Unless I find myself without the time and effort, I plan to return and solve it in this blog post. (I haven’t solved it yet.)

What is Beauty? Case in Point: the ‘China Campus Model’ Controversy

With this post, I am about to take somewhat of a bold first leap into issues that are socially and culturally controversial. I have not spoken directly about such topics ever before on my blog, but I have thought about it and see no reason why I cannot write unequivocally on issues that I feel so strongly and clearly about.

From September to October of 2012, a beauty contest was organized in China by the name of ‘Campus Model’, by the China Campus Model Network, which aimed to find the ten most beautiful college students in the Hubei province of China. So far, so good. But what irked netizens in China and consequently worldwide soon after is the particular well-defined criteria they had enforced as qualification to enter the competition. This included the following, now much-talked-about and controversial, diagrams, of allowed physical proportions:


and of allowed breast configurations for entrants:


From what I read and watched, it appears that in order to qualify for the competition, a participant’s nipples could be no more than 20 cms apart, among several other rigidly defined criteria. Now, there was a lot of hue and cry over this matter, to which it is not my intention to contribute. I just want to draw your attention to several common forms of reaction that followed this piece of news. Standard, civilized, sane reactions and outbursts by the literate, thinking, progressive world. I need to do this to be able to present the points that I finally aim to make. So here’s some of them:

  1. “Why more than 20 centimetres? I honestly don’t know who came up with these figures,” said a user on Weibo, China’s version of Twitter.
  2. “How can beauty standards include breast distance? Do they take women as toys?” Judging women by such rigid criteria is so ‘out’!” said TV personality Yang Lan on the microblog.
  3. ‘In addition to considering traditional measurements like chest, waist and hips, the contest also said the space between candidates’ pupils should be 46 percent of the distance between their pupil and their ear.’
  4. @无语_2012: “Why is a university selecting a beauty queen?”

source: Daily News, Beijing Cream

Take a look now at the following YouTube video from which I learnt about this whole affair in the first place. This is a popular video channel which voiced their criticism and ridicule for the competition. Listen especially to the points that the female host makes:

My point, through this blog post, is not so much to criticize the original competition, as to criticize this kind of response and in general the whole deal that has been made of this incident by ‘progressive, forward-thinking’ society. Yes, you heard that right.

As I make my points, I shall always be talking about female beauty contestants, but only because of this particular context. My points will mostly extend easily to male beauty competitions, so I encourage you not interpret some of my statements as being sexist.

So, there are several aspects to the reasons behind my stand, and I shall start with the current top comment below this video, because it  hits the nail very near the heart of what I wish to convey:


I do not know exactly in which sense the OP meant this comment. Perhaps he/she meant that there goes on unjust discrimination even for American beauty contests, which is never exposed or publicized by the organizing body. However, I choose to believe that they meant something deeper: that every time there is a beauty pageant, anywhere in the world, in any spirit, with any sort of rule or criteria, there is, unavoidably, as a direct necessity of such a program, an evaluation of the bodies of several contestants against some fixed ideals that are either decided in advance or left to a panel of judges. I seriously doubt whether in such cases TV personality Yang Lan (comment 2 in the numbered list) expresses outrage over treating women as toys. Where does she choose to draw the line? Does it even matter where she chooses to draw it? I don’t see anything about hosting beauty competitions to gauge contestants against some pre-decided, inflexible ideals of beauty in our heads as anything short of a nonsensical treatment of humans on the pure basis of their bodies minus the persons they are: exactly what Yang Lan hastens to call a ‘toy’.

Oh wait, let me correct something about beauty contests being only about physique. Nowadays they also have a personality, knowledge and intellect assessment round.

Cute, huh?

I regard that as a modern, hastily politically corrected version, in which the intellectual assessment is still insignificant in importance compared to the core objective of choosing the best-looking person. In fact, I find this whole concept of introducing such a round in a beauty contest absolutely hilarious. Because if you’re at all interested in intellect to any degree, then why is it a beauty competition in the first place? Do you expect intelligence and personality to naturally follow beauty? Well, sorry about this, but reality check: that doesn’t happen. Take a look at an example here:

Beauty and brains are independent parameters. Worse than independent, actually. There is often a negative correlation between the two. I don’t mean to sound prejudiced or biased when I say that. I am not saying that beautiful girls cannot be smart. However, if I look at the average contestant in a beauty pageant, who is generally even otherwise obsessed with how she looks (you know that’s very true, don’t become all proper suddenly), and spends great lengths of time amending and chiseling various facets of her face and body to approach the unwritten general standard of beauty, that doesn’t leave a lot of time to do real smart things. And to do smart things, to be smart, to think about, discuss and make an impact on matters of intellectual weightage, requires effort, dedication, and years and years of time. A human being, in general, has to choose what they wish to do with their life, at least with their youth. Without speaking from narrow-minded prejudice, even from raw statistics and common sense, I can bet you that most entrants in serious beauty competitions do little else of impact with their time. Before you retort to this, please present me evidence of the contrary. And if you have followed my argument so far, you will realize that a couple of non-examples will not help your case. It is your task to conclude from evidence that at the time that most women enter themselves into serious beauty contests, these entrants are also engaged in something of serious intellectual impact.

So anyway, at this juncture, when you are throwing in this round at the end, you are making a laughingstock of the whole idea and agenda of a beauty contest. If you decide to look for one thing, and be honest about exactly what you are looking for, I am prepared to let you be (well, not entirely, but at least for the time being, for the sake of the argument). And if you are somehow ashamed of being progressive humans and holding beauty contests, that is in no way remedied, my friend, by slapping a personality and intellect assessment round at the end. You have found the inherent problem, which I must laud you for, but that is a ridiculous and bird-brained fix to it. It does not fix the core problem that you are here essentially to judge bodies. It only serves to aggravate the stupidity of the whole idea.

Another point that the female presenter tries to make in the video is that the rules for this competition seem to be laid down too precisely for her (or most others’) comfort. I mean, she’s okay with the very common ‘hourglass shape, no fat, considerably sized and pleasantly shaped boobies and booties, symmetric face, smooth complexion, thin eyebrows, beautiful hair, adequate height’ criteria, but not okay with ‘nipples must be 20 cm apart or under’. This is what comment 3 in the numbered list also mentions. ‘Traditional’ measurements are okay. Not these new, exactly-defined ones. What exactly is the point here, sorry? That vague, but nevertheless very-much-there criteria are fine, but you are not allowed to be precise about it, for you run the risk of degrading the idea of civilized humans?

Let’s talk about American beauty pageants. I don’t know if their rules are written down explicitly anywhere. (I’m assuming they’re not, otherwise I don’t see what’s so outrageous about this Chinese incident.) If these are not, then the competition must be left up to the subjective evaluation of a panel of judges or of a larger population via public vote. How do you think that evaluation happens? Don’t we, in our heads, match those bodies to the perfect ideal that has been fed to us by various social and media channels through our lives, and choose the bodies and faces that best fit this standardized idea? In other words, the evaluation is not really as subjective as you might think. If every person had very different opinions of what good-looking is, including beauty contest judges, I find it surprising that these contest applicants, and the winners, are usually so gorgeous. Gorgeous according to me, according to my friend, according to a random guy on the street. According to most viewers of that show (otherwise the program will be laughed off and won’t run after a while). The success of such programs hinges crucially on the fact that our idea of beauty is much more standardized than we might like to believe. It is a separate debate as to where this shared idea of beauty comes from, and I have often wondered about it, but the point of relevance at the moment is that the facial and bodily features that rock my goat is not hugely different from what rock yours, barring of course some fringe tribes and micro-civilizations that anyway do not participate in this commerce of modern media. And it is this same, very standard, ideal, to which bodies and faces are pitted against by the so-called ‘subjective’ evaluation, in what these shocked presenters in the video might regard as a more correct and sensible beauty contest. Just because the parameters are not written down does not make them vague; they are deeply and clearly etched in everyone’s mind. They have developed over years and generations and are reiterated and reinforced every passing day that we are exposed to modern media. The message does not change with every iteration. The beauty sold by the face cream ad is the beauty in the hair removal ad five minutes later, which is the beauty in the tummy-trimming ad in the newspaper the following day. It’s the same image of beauty, over and over again. There is no variation allowed, because that will be a commercial failure. That is, of course, in the long run logical and convenient from the commercial point of view, if different arms of industry reinforce each others’ campaign. Then there’s a single goal to motivate people towards and everything is greatly simplified and business has a happy day. A new product then has only to uphold that single image of beauty. And the more that different products and brands uphold that image, the more this single image strengthens. A vicious cycle. And we conveniently choose to not speak about the long-lasting scars that this leaves with us.

Look now at that woman who’s presenting this video. She looks pretty to me, and I suppose to most others. And it is, I hope, a no-brainer that one of the topmost parameters for the selection of hosts or presenters on TV or online channels must be the pleasantness of their faces. Her job depended on fitting this standard ideal. She has make-up on. She did her hair. What is the function of this, I ask. It is to correct and enhance her already existent prettiness in order to approach even closer to that ideal. And she knew that that’s the reason she put on make-up or shampooed her hair. If she looked like the following, she would not be here at the presenter’s desk today expressing her shock at definitions of beauty that degrade what it means to be civilized humans:


She sat there in the studio of her progressive, forward-thinking channel, of the modern enlightened civilization, with her standard pretty face with additional make-up helping her approach closer to the standard, and she said: ‘You’re not only discriminating against and degrading all the people in this pageant, you’re also discriminating against all women because you’re saying that if you do not fit these specific stereotypes, you’re not beautiful, and you do not fit the typical image of what beauty really is.’

I was in fact taken aback by the awkwardness of the whole situation as she said this. You are this close, I thought. A few more words down that line and this would have been hilarious. For me, though, the irony was complete at that point, and I decided to write this post.

So I was talking about this ideal of beauty that’s embedded in our collective psyche. So entrenched is this idea in fact, that, for example, I can whip up a quick list of what we’re supposed to find beautiful (I already listed some). So can you, no matter whether you’re a girl or a guy. And yes, those are still the parameters that these contestants in more ‘correct’ contests are judged against. There is nothing vague about this. In fact, as far as this incident is concerned, I even deem it an improvement over the otherwise senseless bs, to lay down, once and for all, concrete markers on what is supposed to be beautiful. As the top commenter there rightly said, that is just honesty. That is not brushing the issue under the carpet. And in this particular context, I actually feel a sense of admiration for the organizers of Campus Model.

I believe that in their zeal to voice their shock, most of the responders, and particularly this video duo, have got some of their reasons mixed up. They are not entirely clear on the list of reasons they find this idea so outrageous, or on the separation between their reasons, and so lightly spring from reason to reason with every sentence. One of these reasons, as I gleaned from the commentary, is probably that the criteria imposed by China Campus Model Network are arbitrary. They are disturbed, for example, because they cannot figure out where that 20 cm came from, or that, according to them, such criteria have no bearing on the final perceptible beauty that is registered by the statistically averaged human mind. Very disturbing indeed, yes? These two presenters even spend some time deliberating upon this particular issue (mostly cracking jokes). They do discuss some sane points regarding this arbitrariness issue, and she does mention at some point the very valid argument that even without such arbitrarily set criteria, a general beauty pageant is a degrading idea in itself. But I wish to dwell on this matter a bit more deeply than that.

I think you can already see where I might want to go with this arbitrariness issue. Yes, I wish to know how the currently accepted conditions of beauty are not exactly as arbitrary as the ones these people are so shocked and angered about imposing. I understand that a raw number such as 20 cm between the nipples might come as a sudden shock and may sound very arbitrary at first glance, so let’s ease you into it with an example that’s easier to digest. If I told you that applicants for the beauty contest need to be at least 5 feet tall in order to enter, would you be as enraged as in this case? Aren’t such conditions actually found in many beauty contests? And even if I turn very politically correct and host a beauty contest with no entry restrictions, it shall still be ensured, by the very nature and purpose of the competition, that the bodies and faces that do not conform to that outrageous imposition, will anyway be weeded out in the process of the competition, upon evaluation against that implicit mental beauty standard. A mere advance assertion of the rule does not appear very shocking or outrageous to me at all. Again, all it is, is honesty.

Now you may still contest that ‘not more than 20 cm between the nipples’ is not a standard criterion of beauty. Not as standard anyway, according to you, as ‘no less than 5 feet between the head and feet’. And I ask you, and I ask the presenters in the video so jubilantly waving their flag of progressive, rational thinking, as to where exactly they found the logic and comfort they have in adhering to the more standard criteria of beauty. How do they argue, I ask, that the second criterion above is not as morally and rationally illogical, as the first? Just because we are going with a certain configuration at this point in the cultural history of the human species? Because she herself has been able to subconsciously imbibe this particular notion of beauty over years and is now familiar and comfortable with carrying it? Is it just a well-dug zone of comfort and familiarity that is arduous and disturbing to climb out of? Are they trying to say that what has been repeatedly fed to our heads by money-mongering businesses is the only possible criterion, and therefore the only ethical demand, and that which has not, therefore invades upon human dignity and womanhood, and hence deserves to be butt-thrashed publicly? Is that the whole point of their agenda?

And once again, as they were joking about the arbitrariness of that ‘20 cm’, I winced inwardly and thought, ‘You were this close. This close to the ridiculousness of it all. And you stopped just short. Oh, the irony.’

Anyway, the only comment in the numbered list that I found myself wholeheartedly agreeing with was number 4, presumably by a Chinese student on twitter. He’s right. The function of a university is to promote thoughts and ideas, and hopefully aid in breaking out of the shackles of standard, nevertheless illogical, thoughts and ideas imposed through previous generations and fed through various channels on a daily basis. Also, like I said before, intellect and ‘beauty’ are independent parameters at best. Colleges are for the first. Not the second. This is my final, unequivocal statement. The whole idea of a ‘Campus Model’ tries to mix these up in a way that I greatly dislike.

Anyway, I think I have exhausted all that I wanted to say on this topic. But yes, I am really looking forward to your comments and feedback on this issue. So please take the time to leave a little comment, especially if you find yourself disagreeing with my viewpoint in any aspect.

What is this writing?

Something funny happened today. Two of my college friends and I were lounging around in one of the classrooms of the college after hours. Because it has this sexy AC. There was this big clean inviting blackboard in front, and I just went ahead and scribbled some stuff on it in a way I had taught myself when I was in school:


Here it is from up close:


We were taking these photos and transferring them on a laptop and doing related useless stuff when suddenly one of my junior students, A, entered the room, followed by Professor P, Head of Department of Physics and another professor, probably, who I don’t know. And I thought, shit.

They ambled into the room. Their gaze fell upon the board. Now that looks like some serious Arabic or Klingon or aboriginal script, so they kept staring at it. And I thought again, shit.

Professor P asked, ‘what is written here?’ My heart accelerated mildly. I thought at the back of my head that it’s going to take some time for them to figure this out. This can be salvaged comfortably.

A explained immediately how it was written. Shit came to my mind. He had seen me do this before some time.

Okay, we don’t know what it means, or who wrote it. We were just in the room, I thought.

P asked, ‘who wrote this?’ A pointed to me and said it was me; it couldn’t have been anyone else.

You know what came to my mind? Right.

I had to intervene before it all went out of control. I raced up towards the board from behind as the professors stood reading it. The second professor was already saying, ‘I can tell what the first word is…’

I grabbed a duster and started wiping it off, saying, ‘Sir, I’ll show you. I’ll show you right now how it’s written.’

Professor P started saying he wanted to know what was written. I kept  erasing until I had wiped the whole thing clean.

Then I took a chalk and said, ‘Look, Sir, I’ll write your name.’ And I wrote his name. He was impressed.

I came back to my seat. The professors took their seats as A started explaining some physics stuff on the board.

My friends and I stayed for a while in the room, then got the hell out of there when we couldn’t stop sniggering.