1. Churn is more threatening than you think.

2. Use churn analysis as an opportunity to fix things, rather than getting dismayed.

## It is 5 to 25 times more expensive to acquire a new customer than to retain an existing one.

## Only 4% of unhappy customers complain.

## 90% of unhappy customers will leave.

## 100% of unhappy customers will talk about your business to their friends.

## Increasing customer retention rate by 1% increases profits by 5% to 20%.

**Bottom line**, take churn seriously. The most effective way to understand why customers are churning is by listening to them. Looking at historical data (cohort retention curves, core actions, etc) will help you understand what **retained users** are doing. This leads to survivorship bias. To truly sustain growth, you must understand non-survivors!

Wald (1943) recommended that the Navy reinforce the plane where there were no bullets.

- Segment customers and measure churn for the most valuable segments.
- Conduct periodic churn surveys to listen & understand “the why?”
- Build plans to fix problems which are in your control.
- Reward teams who successfully manage churn.
- If possible, build machine learning models to automate churn prediction and be proactive.

1. This clay tablet is one of the earliest record of a formal customer complaint. Click here to read the full translation

2. There is no universal definition of churn rate. This was the basis for the judge dismissing a case against Netflix back in 2004!

If you enjoyed reading so far, you will like these references.

- Customer Churn Management: The Key To Sustainable Margin Growth
- The Value Of Keeping The Right Customers
- Predicting Customer Churn with Amazon Machine Learning
- Prescription for cutting costs
- Turning SaaS Churn Into A Growth Strategy: Changing Our Perspective
- 75 Customer Service Facts, Quotes & Statistics

You will never forget how to solve quadratic equations once you see how Babylonians did it back in 2000 B.C. It’s striking that these methods date back 4000 years!

Quadratic equations are everywhere. You see some manifestation of it, often as parabolas, every single day. For example:

Babylonians, who lived in the fertile region between Euphrates River and Tigris River were also interested in quadratic equations. Consider this cute puzzle from 2000 B.C.

Babylon puzzle from 2000 B.C.

I have added the area and two-thirdsof [ the side of ] my square and it is35. What is the side of my square?

In modern notation, this puzzle corresponds to solving for in

Babylonians had a geometric way of solving such equations. Let me explain. To solve a general quadratic equations of the form

Babylonians considered the following square.

The area of the larger square with side length is the sum of the areas of the individual squares and rectangles shaded in various colors above. Therefore,

Now, . This gives

Therefore,

Here is a trivia if you enjoyed reading this far.

Why did the Babylonians represent a general quadratic equation in the form , as opposed to the contemporary form ?

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]]>This is one of the most common questions I get asked when I consult with companies. The practical answer here is that size does matter. I would love to hear your opinions and experiences in the comments. Here is the short summary.

**TL;DR version:**

For a medium scale company, the formula for growth attributable to A/B tests is

Annual growth % =

where is the number of growth engineers and is the average probability of success.

For example, if and , expect an annual growth of

Therefore, a well-oiled growth team needs to have 20 engineers and supporting product managers, designers, statisticians, and marketers to achieve annual growth driven purely by experimentation. Read on to figure out how I arrived at this formula.

**Extended version:**

In this post, I will provide you with a mental model and a Fermi calculator to quantify growth team size. Before jumping into details, we will assume the following.

- Your product has hit product-market fit and you are measuring a meaningful north star metric. And not some vanity metric like open rate for a clickbait email
- You have buy-in from the CEO and other execs, in principle and more importantly in action to build a growth team.
- You have an A/B testing infrastructure in place.

Here are some numbers to start with.

- A well-oiled growth team should strive to achieve an experimentation rate of
*one A/B test per engineer per week* - A new growth team typically starts with about a success rate. In other words,
*nine out of ten ideas fail*! - As the team matures, the success rate goes up. I have come across teams where the success rate is around , i.e.,
*three out of ten ideas succeed*.

These numbers are already good enough to estimate the team size. The only other number we need is how many experimentation focused weeks we have in a year.

- Letâ€™s assume the team spends two weeks per quarter for planning, feedback reviews, etc.

That means we have weeks to work on growth experiments. - Letâ€™s discount another to take into account vacations, sick days, removing technical debt, resolving bugs, etc.

This leaves us with 40 weeks in a year to run A/B tests.

Next, assume that there is an impact of an average of on the north star metric with each successful A/B test. We now get the following table.

Success rate | 10% | 20% | 30% |

Weekly A/B tests | (success, %-impact) | (success, %-impact) | (success, %-impact) |

1 | ( 4, 0.4% ) | ( 8, 0.8% ) | ( 12, 1% ) |

3 | ( 12, 1.2% ) | ( 24, 2.4% ) | ( 36, 3.7% ) |

6 | ( 24, 2.4% ) | ( 48, 4.9% ) | ( 72, 7.5% ) |

10 | ( 40, 4.1% ) | ( 80, 8.3% ) | ( 120, 12.7% ) |

20 | ( 80, 8.3% ) | ( 160, 17.3% ) | ( 240, 27.1% ) |

Click here to make a copy of this sheet.

What the table above suggests is that a YoY growth rate of about attributable to A/B tests needs 20 engineers delivering one experiment per week! You can now work backwards and estimate the total team size based on the PM to engineer ratio and other dynamics within your company.

**An approximate formula to compute YoY growth from A/B tests.**

If N is the number of experiments per week and p is the average probability of success, then for a *typical growth team*, the year over year metric growth attributable to A/B tests as a percentage is

For example, if and , this formula gives , which is approximately the last row in the column in the table above!

**Can this go on forever?**

Note that if you target to run 200 A/B tests per week, you get 2400 successful A/B tests in a year and a YoY growth of . So, is this achievable? The simple answer is no. Things donâ€™t always go to the top right. You will hit diminishing returns as the team scales up. Some underlying causes include

- More managerial challenges, more meetings, more communication overhead as the team scales up.
- Market saturation
- Concerns over user product experience.
- Worrying about multivariate tests rather than A/B.

Diminishing returns are just a fact of life. The mathematical way to model them is with the logistic differential equation. Check out the wiki link if you are curious about it.

So, the answer to the original question is the following: Aim to get to a stage where you can run one A/B test per engineer per week. Figure out the average movement in the north star metric per successful experiment. And then plug it into the calculator!

**Fun fact: can you estimate the number of engineers in Facebookâ€™s growth team?**

Yes, here is one approach. I was able to get the following numbers from various public sources for Facebook MAU over the years.

Reporting Date | MAU (millions) | YoY MAU Growth |

03/31/2009 | 242 | |

3/31/2010 | 482 | 99% |

3/31/2011 | 739 | 53% |

3/31/2012 | 955 | 29% |

3/31/2013 | 1155 | 21% |

3/31/2014 | 1317 | 14% |

3/31/2015 | 1490 | 13% |

3/31/2016 | 1712 | 15% |

3/31/2017 | 2006 | 17% |

At the scale at which FB operates, I would assume an impact rate of +0.01% in MAU and a success rate of 10%. The choice of these numbers reflects diminishing returns and not the quality of Facebook growth team. Letâ€™s be generous and assume 50% of the YoY growth is coming from A/B tests and the remaining 50% from the organic linear trend. That already gives us that FB needs to run 200 A/B tests per week to get to 8.3% YoY growth. Remember, Fermi calculations like these only give you an order of magnitude estimate. So the number of growth engineers at FB is definitely in the early to mid 100s!

Thanks for reading so far. Let me know your thoughts in the comments!

Acknowledgement: Thanks Julia Gitis for giving it a first pass, and helping out with the presentation.

]]>Thomas Babington Macaulay, Minute on Indian Education, 1835.

India was usually known to outsiders until very recently as the land of naked sadhus, snake charmers, overtly religious mystical land which taught the world how to make love. Such a perspective definitely helped the colonizers in assigning an inferior status for the colonized, and justify their acts of colonization and “civilization”. Macaulay’s remarks in his Minute on Indian Education is a good read to understand the British perspective then about India.

Gladly enough, thanks to the IT age, outsourcing and the recent economic boom, this image is rapidly changing. Such a categorization, not only misrepresented India, but also ignored completely many other aspects like agnosticism (Buddhism), atheism (Carvaka), science and mathematics which were being developed in ancient India. Buddhism, which originated in India and spread as a rebel against caste system and rituals, was in full glory for about a millenium, and was one of the prime exports of India during those days to China and other eastern countries. So much that China refered to India as the Kingdom of Buddhism. Rationalism and skepticism too, was a school of thought which cannot be ignored during these times. Hinduism, Buddhism, Jainism, Atheism and many other schools were competing at a time for acceptance as a mainstream philosophy.Â Amartya Sen‘s book The Argumentatvie India forms an excellent read on this subject.

As far as ancient Indian sciences are concerned, it was developed and transmitted to regions outside India by Chinese and Arabs. Many mathematics and science texts were translated by Al-Beruni^{[2]}, Al-Khwarizmi (who is responsible for the terms algebra and algorithm) into Arabic and found its way to Europe there on. One of the prime motivations for doing mathematics those days was astronomy. Approximations for trigonometric values, , circumference of circle, etc were developed. Bhaskara‘s approximation for as a rational function of , Brahmagupta‘s calculations of eclipses, Aryabhatta I’s approximation of , Aryabhatta I‘s method to solve linear Diophantine equations are just some examples amongst a huge literature.

Broadly, ancient Indian mathematics can be categorized into the following based on their period of development.

- Pre Vedic Indus mathematics dating back to as early as 3000BC. Development of mathematics were highly influenced by practical applications like measuring scales, calculating brick ratios etc.
- Vedic or Sulbasutras, which contained rules to construct altars for various rites and rituals. Various constructions based on Pythagoras theorem are listed, approximation to square root of 2, approximation to are some of the achievements. This period probably lasted till around 500BC.
- Jaina mathematics, from 600BC to 500AD. Prime achievements are various notions of infinities, Pascal’s triangle, form of set theory, operations with roots of order larger than 2 etc
- A set of manuscripts were found around 1880s, called the Bakhshali manuscript. The dates of these manuscripts are assumed to be around 400AD. This book contains sets of problems and solutions in linear equations, fractions, square roots etc
- Golden age of Indian mathematics set off by Aryabhatta I. Numbers become more abstract, and makes it possible to consider zero and negative numbers. Brahmagupta, Mahavira and Bhaskara lists down rules for multiplying, subtracting and (wrongly) dividing by zero. All of them missed that dividing by zero does not make sense It was during this period that Al-Khwarizmi, who lived from 790 to 840, wrote
*Al’Khwarizmi on the Hindu Art of Reckoning*and passed on Indian achievements to Islamic and Arabic countries. Ibn Ezra, who lived from 1092 to 1167 in Spain also helped transmit Indian number system to Europe. - Keralese mathematics: The golden age started to decline after Bhaskara’s works around 1200. There is some consensus that political instabilities are partly to be blamed for this decline. Despite all these, mathematics continued to develop in Kerala. The time period between 1400 and 1600 is considered to be the peak period of this development. In addition to building upon and extending previous works by Bhaskara, Brahmagupta, Aryabhatta etc, one of the significant achievements of this period was mathematical inductive proof in the works of Nilakantha, Jyeshtadeva. The prominent figure during this time period is definitely Madhava of Sangamagramma. He built tools essential for modern analysis. Though not fully certain, he is supposed to have independently derived taylor series expansion for arctan, sin, cos and many other function. A list of 13 different expansions attributed to Madhava is listed here. Madhava also managed to calculate the value of to 17 decimal places (3.14155265358979324), much ahead of his contemporaries.

People have started recognizing and acknowledging role of Indians, but awareness still seems to be a big issue. I think it will take some time before terms like Madhava-Gregory series, Leibniz-Gregory-Madhava constant, Hemachandra-Fibonacci series, Bhaskara-Brouncker Algorithm, Aryabhatta algorithm, Aryabhatta Remainder Theorem, Aryabhatta Kuttaka etc become commonplace.You might want to check the following links to dig deeper into the history of Indian mathematics.

C. T. Rajagopal and M. S. Rangachari. ‘On an untapped source of medieval Keralese mathematics’, *Archive for History of Exact Sciences***18** (pages 89-102). 1978.Keralese MathematicsPossible transmission of Keralese mathematics to Europe.

Here is a list of prominent Indian mathematicians in chronological order. The names and brief description are obtained from various websites including but not limited to University of St. Andrews.

Baudhayana 800BC – 740BC **Baudhayana** was the author of one of the earliest Sulbasutras: documents containing some of the earliest Indian mathematics.

Apastamba 600BC – 540BC **Apastamba** was the author of one of the most interesting of the Indian Sulbasutras from a mathematical point of view.

Panini 520BC – 460BC**Panini** was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology.

Katyayana 200BC – 140BC**Katyayana** was the author of one of the Sulbasutras: documents containing some of the earliest Indian mathematics.

Yavanesvara 120 – 180**Yavanesvara** was an Indian astrologer who translated an important Greek text on astrology.

Aryabhata the Elder 476 – 550**Aryabhata I** was an Indian mathematician who wrote the *Aryabhatiya* which summarises Hindu mathematics up to that 6th Century.

Yativrsabha 500 – 570**Yativrsabha** was a Jaina mathematician who gave a description of the universe which is of historical importance in understanding Jaina science and mathematics.

Varahamihira 505 – 587**Varahamihira** was an Indian astrologer whose main work was a treatise on mathematical astronomy which summarised earlier astronomical treatises. He discovered a version of Pascal’s triangle and worked on magic squares.

Brahmagupta 598 – 670**Brahmagupta** was the foremost Indian mathematician of his time. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations.

Bhaskara I 600 – 680**Bhaskara I** was an Indian mathematicians who wrote commentaries on the work of Aryabhata I.

Lalla 720 – 790**Lalla** was an Indian mathematician who wrote mainly on the application of mathematics to astronomy.

Govindasvami 800 – 860**Govindasvami** was an Indian mathematical astronomer whose most famous treatise was a

commentary on work of Bhaskara I.

Mahavira 800 – 870**Mahavira** Mahavira was an Indian mathematician who extended the mathematics of Brahmagupta.

Prthudakasvami 830 – 890**Prthudakasvami** was an Indian mathematician best known for his work on solving equations.

Sankara Narayana 840 – 900**Sankara Narayana** was an Indian astronomer and mathematician. He wrote a commentary on the work of Bhaskara I.

Sridhara 870 – 930**Sridhara** was an Indian mathematician who wrote on practical applications of algebra and was one of the first to give a formula for solving quadratic equations.

Aryabhata II 920 – 1000**Aryabhata II** was an Indian mathematician who wrote about astronomy as well as geometry. He constructed tables of sines accurate up to about 5 figures.

Vijayanandi 940 – 1010**Vijayanandi** was an Indian mathematician and astronomer who made some contributions to trigonometry.

Sripati 1019 – 1066**Sripati** was an Indian who wrote works on astronomy and arithmetic.

Brahmadeva 1060 – 1130**Brahmadeva** was an Indian mathematician who wrote a commentary on the work of Aryabhata I.

Acharya Hemchandra 1089 – 1173**Hemachandra** was a Jaina scholar who presented what is now called the Fibonacci sequence around 1150, about 50 years before Fibonacci (1202). He was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length (nâˆ’1), or a long syllable to one of (nâˆ’2). This recursion relation F(n) = F(nâˆ’1) + F(nâˆ’2) is what defines the fibonacci sequence.

Bhaskara 1114 – 1185**Bhaskara II** or **Bhaskaracharya** was an Indian mathematician and astronomer who extended Brahmagupta’s work on number systems.

Narayana Pandit 1340-1400**Narayana Pandit** is the author of an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II’s Lilavati, titled Karmapradipika (or Karma-Paddhati).

Madhava of Sangamagrama 1340-1425**Madhava of Sangamagrama** was the founder of the Kerala School and considered to be one of the greatest mathematician-astronomers of the Middle Ages. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475 but all that is known of Madhava comes from works of later scholars.

Parameshvara 1370-1460**Parameshvara,** the founder of the Drgganita system of astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha. He is stated to have made direct astronomical observations for fifty-five years before writing his famous work, Drgganita. He also wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II’s Lilavati, contains one of his most important discoveries:

Nilakantha Somayaji 1444 – 1544**Nilakantha** was a mathematician and astronomer from South India who wrote texts on both astronomy and infinite series.

Jyesthadeva 1500 – 1575**Jyesthadeva** was a mathematician from South India who wrote an important work on mathematics and astronomy which summarises the work of the Kerala school.

Citrabhanu c. 1530**Citrabhanu** was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:

Kamalakara 1616 – 1700**Kamalakara** was an Indian astronomer and mathematician who combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists.

Jagannatha Samrat 1690 – 1750**Jagannatha** was an Indian mathematician who is important as a translator of important Greek works into Sanskrit.

Sankara Varman 1800-1838**Sankara Varman**There remains a final Kerala work worthy of a brief mention, Sadratnamala an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands as the last notable name in Keralese mathematics. A notable contribution was his compution of Ï€ correct to 17 decimal places.

^{[1]} The Mahatma and the Poet : Letters and Debates between Gandhi and Tagore 1915-1941, Compiled and Edited by Sabyasachi Bhattacharya, National Book Trust, India

^{[2]} Alberuni’s India, Edward C. Sachau, Trobner & Co., London, 1888, Rupa & Co., 2002. You can read a review of this book at hindu.com or obtain a pdf file of his book from infinityfoundation.co