The Three-Point Shot Delusion

The vast majority of NBA analysts claim today that the NBA has changed. It has become more fast-paced, and there is a significantly greater emphasis on teams attempting more three point shots. The evidence for this is the repeated recital of the fact that over the last number of years, the average three-point attempt rate has increased. An example of such an article can be found here. 

It is my hypothesis that this is all based on a very shallow analysis of what is actually going on. In particular, there are more than 60 variables on that classify each team’s play. It seems strange that analysts have picked out one statistic, noticed a trend, and have made conclusions ushering in the “modern-day” NBA. As I will demonstrate below, using concepts from statistical and machine learning, many things have been missed in their analyses. What is even more strange is that there have been an increasing number of articles claiming that, for example, if teams do not shoot more three point shots, they will probably not make the playoffs or win a championship. Examples of such articles can be found here, here, and here.

I will now demonstrate why all of these analyses are incomplete, and why their conclusions are wholly incorrect.

Using the great service provided by, I looked at the last 15 seasons of  every NBA team, looking at more than 60 predictor variables that classified each team’s performance in the season. Some of these included: MP FG FGA FG% 3P 3PA 3P% 2P 2PA 2P% FT FTA FT% ORB DRB TRB AST STL BLK TOV PF PTS PTS/G oG oMP oFG oFGA oFG% o3P o3PA o3P% o2P o2PA o2P% oFT oFTA oFT% oORB oDRB oTRB oAST oSTL oBLK oTOV oPF oPTS oPTS/G MOV SOS SRS ORtg DRtg Pace FTr 3PAr TOV% ORB% FT/FGA  TOV% DRB% FT/FGA, where a small “o” indicates a team’s opponent’s statistics.

What classifies a playoff team?

Building a classification tree, I wanted to analyze what factors specifically lead to a team making the playoffs in a given season. I found the following:


(For this classification tree, the misclassification error rate was 2.73% indicating a good fit to the data.)


At the top of the tree, we see that the distinguishing factor is the average MOV/”Margin of Victory” measured per game. Teams that on average beat their opponents by more than 2.695 points are predicted to make the playoffs, while teams that on average lose by more than 1.825 points are predicted to not make the playoffs. Further, the only factor relating to three-point shooting  in this entire classification tree is the o3PA, which is the number of opponent 3-point attempts per game. For example, suppose a team can has an average MOV of less than -0.54 but greater than -1.825. If that team’s opponent attempts more than 16.0732 3-point shots per game, the team is expected to make the playoffs. In this particular case, getting your opponent to take a lot of three point shots is indeed desirable, and leads to the expectation of a team making the playoffs.


What classifies a championship team?

The next question to analyze is what characteristics/features classify a championship team. Looking at the last 20 years of playoff data, we see that the following classification tree describes the championship criteria for a given NBA playoff team.


(The learning error rate was 1.172% indicating an excellent fit to the data). One sees that at the very top is a team opponent’s field goal percentage (OFG.). If the average per game OFG% is greater than 44.95%, that team is predicted to not win a championship. Further, there are apparently three predicted paths to a championship:

  1. OFG% < 44.95 –> ORtg (Opponent Team Points Scored per 100 possessions) < 108.55 –> FT% < 73.5% –> Opponent Offensive Rebounds per game (OORB) < 30.2405 –> Personal Fouls per game (PF) < 24.1467
  2. OFG% < 44.95 –> ORtg > 108.55 –> O3P% < 32.45%
  3. OFG% < 44.95 –> ORtg > 108.55 –> O3P% > 32.45% –> AST > 19.9076 –> OAST < 19.0938

This shows once again that the three point shot is not at all relevant in winning a championship amongst playoff teams, in that, shooting a lot of threes, or playing as a “modern” team, does not uniquely determine a team’s success. What is tremendously important is defense, and offensive efficiency, and there are multiple ways to achieve this. One does not need to be a prolific three-point shooting team to achieve these metrics. 



The increasing  trend of teams shooting more threes and playing at a higher pace still does not uniquely determine whether a team will make the playoffs or win a championship, which is why I have called it a “delusion”. Indeed, the common statement that “nowadays, teams that make the playoffs also have the highest number of three-point shot attempts” is a very shallow statement, and is not actually why teams make the playoffs as this analysis very clearly shows. Further, attempting more three-point shots is not at all uniquely indicative of a team’s success in winning a championship.

Misinformation on Sikhism

Even though this posting is a bit different than my usual ones and is outside the scope of this blog, I thought that as a Sikh myself, I have stayed too silent on several issues with regards to the Sikh community, and certain principles of the Sikh religion, that are seemingly unknown to those both inside and outside of the Sikh community. I will address here some common misconceptions and misinformation about Sikhs that are both being spread inside and outside of the Sikh community.

  1. Sikhism is NOT a hybrid of Islam and Hinduism: Sikhism is a unique religion with a unique origin beginning with the teachings of the first Sikh Guru, Guru Nanak Dev Ji. Guru Nanak formed the religion to be uniquely different from Hinduism and Islam, as he opposed many of the practises common in those religions.
  2. Sikhs are not to cut their hair or trim their beards. This is perhaps the most common fact that is blatantly missed by those members of the Sikh community that wish to cut their hair and propagate this misinformation to defend their actions as being accepted in Sikhism. This is a very wrong ideology for several reasons:      Guru Gobind Singh Jee, the 10th Guru of the Sikhs explicitly described the form of a Sikh in Persian:

Ik Onkaar Sri Waheguru Jee Kee Fateh || Sri Mukhvaak PaatShaahee Dasvee||

Nishaanay Sikhi Ee Haroof Panj Kaaf|| Hargiz Na Baashad Ee Panj Muaf ||

Karra Kaardo Kachh Kanghaa Bida || Bila Kesh Haych Asat Jumleh Nishaa ||

Haraf Haae Kaat Asat Ee Panjkaaf || Bi Daanand Baavar Na Goyam Khilaaf ||

HukaaJaamat Halaalo Haraam || Baachishay Hinaa Kardaroo Sayaam Faam||

Note that the usual argument that this is for the Khalsa, and not Sikhs, is patently false, as Guru Sahib explicitly says “Nishaanay Sikhi”, and not “Nishaanay Khalsa”. This point is therefore a moot point.

Guru Gobind Singh Sahib also in his Hukam to the Afghanistan Sikh sangat said:

Tusi Khande da Amrit Panja to lena

Kes rakhne…ih asadee mohur hai;

Kachh, Kirpan da visah nahee karna

Sarb Loh da kara hath rakhna

Dono vakat kesa dee palna karna

Sarbat sangat abhakhia da kutha

Khave naheen, Tamakoo na vartana

Bhadni tatha kanya-maran-vale so mel na rakhe

Meene, Massandei, Ramraiye ki sangat na baiso

Gurbani parhni…Waheguru, Waheguru japna

Guru kee rahat rakhnee

Sarbat sangat oopar meri khushi hai.
Patshahi Dasvi

Jeth 26, Samat 1756

These are not my opinions. This is explicitly the command of Guru Gobind Singh Jee. Further, the aforementioned passages also explicitly state that Sikhs are not to drink alcohol or smoke. It is truly puzzling and embarrassing why drinking alcohol has become somewhat synonymous with Sikhs, particularly, in the Punjab region.

Further, Bhai Desa Singh Jee explicitly writes the following on trimming beards:

dhaarrhaa mushh sir kaes banaaee || hai eih dhrirrh jih prabhoo razaaee || maett razaaee j sees mu(n)ddaavai || kahu thae jag kaisae har paavai

It could not be more clear, that Sikhs are to emphatically not cut their hair or trim their beards.

3. Sikhs do not celebrate Diwali. There are many Sikhs around the world that insist on celebrating Diwali, the Hindu festival of lamps. Some have even justified this action by conflating the Bandhi Chorr divas related to Sri Guru Hargobind Sahib with the Diwali day. This is also wrong for the following reasons:

For years now, I have heard this constant story of how it is acceptable for Sikhs to celebrate Diwali as per the Hindu traditions of lighting lamps, etc… Further, Raagis and Bhai Sahibs in Gurdwaras have conflated Bandhi Chorr Diwas with lighting lamps as per Hindu Diwali traditions. They further support these ideas with the supposed Vaar from Bhai Gurdas Jee, in which they ironically only mention and repeat the first line! : “deewaalee dee raath dheevae baaleean”. Of course, just by reading this line, it would suggest that the aforementioned actions are justified. But taking one line completely out of context leads one to these conclusions. A full reading of Bhai Gurdas Jee’s Vaar on the Diwali matter which given the timeframe is also a historical first-hand account suggests that Sikhs are to practice completely the opposite and in fact, lighting lamps is contrary to Gurmat. The full Vaar’s transliteration is below:
Vaars Bhai Gurdaas 19-6

diwali dee raath dheevae baaleeani

thaarae jaath sanaath a(n)bar bhaaleean

fulaa(n) dhee baagaath chun chun chaaleean

theerathh jaathee jaath nain nihaaleean

har cha(n)dhuree jhaath vasaae ouchaaleean

guramukh sukhafal dhaath shabadh samhaaleean

The essence of this Vaar is in every line after the first. Namely, in the third, fourth, and fifth lines, Bhai Gurdas Jee compares those that celebrate Diwali by lighting lamps akin to those who go on long pilgrimages to find God, and to those who search for God by worshipping the stars, or things in nature, etc… All contrary to Gurmat by a simple reading of Japjee Sahib! Indeed, Bhai Sahib Jee in the last line clearly states that a person of Gurmat does not practice any of these things, which he declares to be temporary and pointless.

So, there you have it. A simple reading of the full Vaar changes the entire context of the “importance” of Diwali in Sikhism. I doubt many Sikhs will read this posting with sincerity, but someone has to speak the truth.

4. Sikhs Do Not Eat Meat: Sikhs most certainly do not eat meat. Despite this, many Sikhs continue to insist that eating meat is permissible as long as it is not Halal, etc. This is also wrong for the following reasons:

The Sikh Gurus including the Guru Granth Sahib Jee (the present living Guru of the Sikhs) very explicitly discuss how eating meat is not for Sikhs, some examples below:

1. Guru Granth Sahib – Page 1374 – “Kabeer Khoob Khaanaa Keecharee Jaa Mai Amrit Lon, Heraa Roti Kaaranay Galaa Kataavai Kaun”.

2. Guru Granth Sahib – Page 140 – “Jay Rat Lagay Kaparay Jaamaa Hoay Paleet, Jo Ray Peeveh Maansaa Tin Kio Nirmal Cheet”.

3. The essence of why a Sikh cannot be satisfied with “Jhatka” and simply opposed to Halal is due to Guru Naanak Dev Ji, Page 468 on SGGSJ: “Daaiaa Jaanay Jee Kee Kichh Pun Daan Karay”

Several other key points are as follows:

4. “Jee Badhoh So Dharam Kar Thaapoh, Adharam Kaho Kat Bhai.

Anpas Ko Munwar Kar Thaapoh, Kaa Ko Kaho Kasaaee. (SGGS 1103)

5. “Bed Kateb Kaho Mat Jhoothhay, Jhoothhaa Jo Na Bichaarey.

Jo Sabh Meh Ek Khudai Kahat Ho,To Kio Murghi Maarey” (SGGS 1350)

6. “Rojaa Dharey, Manaavey Mlah, Svaadat Jee Sanghaarey.

Aapaa Deldi Avar Nahin Dekhey,Kaahey Kow Jhakh Maarey” (SGGS 1375)

7. “Kabir Jee Jo Maareh Jor Kar,Kaahtey Heh Ju Halaal.

Daftar Daee Jab Kaadh Hai, Hoegaa Kaun Havaal” (SGGS 1375)

8. “Kabir Bhaang, Machli, Surapaan Jo Jo Praanee Khahey.

Tirath, Barat, Nem Kiaye Te Sabhay Rasaatal Jahey” (SGGS 1376)

9. “Kabir Khoob Khaana Khichri, Ja Meh Amrit Lon

Heraa Rotee Kaarney Galaa Kataavey Kon” (SGGS 1374)

These are examples/Hukams explicitly from Sri Guru Granth Sahib Jee going back to the days of Bhagat Kabeer Jee, to Guru Naanak Sahib, all the way to Guru Gobind Singh Sahib. This is factual evidence that your claims are incorrect, and any such claims made by Sikhs that only “Halal” is forbidden, is simply wrong, as these Sikhs do not have an answer/simply ignored such aforementioned verses from Guru Granth Sahib Jee, because of a desire to not leave meat.

5. Women in Sikhism

In Sikhism, women are completely equal to men, there is zero tolerance for those men that would treat women poorly or lower than them. Indeed, according to Sri Guru Granth Sahib Jee, women are to be treated higher than men:

From woman, (man) is born; within woman, (man) is conceived; to woman he is engaged and married. With woman

(man) establishes friendship; through woman, the future generations come. When woman dies, (man) seeks another

woman; because of woman, (man) becomes related (to other people – ਲੈਣ-ਦੇਣ ਦੇ ਸਾਰੇ ਸੰਸਾਰਕ ਬੰਧਾਨੁ, etc.). From her,

(even) kings are born; so why call her bad? From woman, woman is born; without woman, there would be no one at all.  (sggs 473).

6. Sikhs and Rakhdi/Raksha Bandhan Ceremony

There are many Sikhs around the world that insist on Rakhdi/Raksha Bandhan ceremony. Sikhs absolutely are not supposed to participate in this festival. On Raksha Bandhan, sisters tie a rakhi (sacred thread) on her brother’s wrist. This symbolizes the sister’s love and prayers for her brother’s well-being, and the brother’s lifelong vow to protect her.

This very concept implies that women are weaker than men and therefore need their protection. This is in direct conflict with Guru Nanak Dev Ji’s philosophy regarding women as described above, and very much against the Khalsa Rehat, where women and men take amrit equally! Therefore, the idea that women should tie a thread to their brother’s wrist for “protection” is very much against the concept of equality, and ignores the roles played by Sikh women in history that didn’t need any man for protection. Some examples are:

  • Mata Gujri Jee
  • Mata Ganga Dev Jee
  • Mai Bhago
  • Rani Sada Kaur
  • Mata Jito Jee
  • Bibi Rajni
  • Mata Kishan Kaur

And many, many others!


The purpose of this piece was simply to stop the misconceptions related to the aforementioned points from spreading.

Further, the majority of us live in free societies, where freedom of religion is the expectation and the norm. I am a big believer in this concept, and fully appreciate having such a law in existence. This of course means that people are free to practice their religion as they see fit. However, while this principle must absolutely be firm, I question what it means for someone to call themself a Sikh on one hand and disagree with the tenets that the Sikh gurus established. This is more ironic because Sikh means “disciple”. So, you can call yourself a Sikh, but if you don’t follow the principles of the Sikh religion, what “religion” are you following?

On The Acausality of Heat Propagation

In many physics and chemistry courses, one is typically taught that heat propagates according to the heat equation, which is a parabolic partial differential equation:

\boxed{u_t = \alpha u_{xx}},

where \alpha is the thermal diffusivity and is material dependent. Note also, we are considering the one-dimensional case for simplicity.

Now, let f(x-at) be a solution to this problem, which represents a wave travelling at speed a. We get that

\boxed{-a f' = \alpha f''}.

This implies that

\boxed{u(x,t) = -\frac{\alpha c_1}{a} \exp\left[-\frac{a (x-at)}{\alpha}\right] + c_{2}},

where c_{1}, c_{2} are constants determined by appropriate boundary conditions. We can see that as a \to \infty, u(x,t) < \infty! That is, that even under an infinite propagation speed (greater than the speed of light), the solution to the heat equation remains bounded. PDE folks will also say that solutions to the heat equation have characteristics that propagate at an infinite speed. Thus, the heat equation is fundamentally acausal, indeed, all such distribution propagations from Brownian motions to simple diffusions are fundamentally acausal, and violate relativity theory.

Some efforts have been made, and it is still an active area of mathematical physics research to form a relativistic heat conduction theory, see here, for more information.

What we really need are hyperbolic partial differential equations to maintain causality. That is why, Einstein’s field equations, Maxwell’s equations, and the Schrodinger equation are hyperbolic partial differential equations, to maintain causality. This can be seen by considering an analogous methodology to the wave equation in 1-D:

\boxed{u_{tt} = c^2 u_{xx}}.

Now, consider a travelling wave solution as before f(x-at). Substituting this into this wave equation, we obtain that

\boxed{a^2 f'' = c^2 f'' \Rightarrow a^2 = c^2 \Rightarrow a = \pm c}.

That is, all solutions to the wave equation travel at the speed of light, i.e., a = c! Therefore, wave equations are fundamentally causal, and all dynamical laws of nature, must be given in terms of hyperbolic partial differential equations, as to be consistent with Relativity theory.