How many Pianists are there in New York city?

My answer to this Google estimation question.

1. Clarify - Can you describe a bit more on what you mean by pianists? Are you referring to professionals who play at concerts, students who are learning to play, piano tutors? There may be other categories but these are the ones I can think of. If you are fine, I'd like to pick a category - piano tutors and start there. So I will find the number of piano tutors in NYC

2. Equation - I will try to estimate this by determing how many piano lessons are required per week in NYC divided by how many lessons can single tutor deliver in a week. There will be some assumptions but it should lead to an upper bound - I will continue if this looks fine to you 

3. Calculation 

# of lessons that a tutor can deliver in a week

Average length of a lesson - 45 mins (based on my experience with my kids and their friends)

Break time after each lesson - 15 mins

Total time per lesson = 60 mins or 1 hour

Lessons usually happen after school on weekdays, between 5-8, so that 3 hours every weekday, and between 10-4 on Sat which is 6 hours on weekends (most tutors don't take lessons on Sunday from what I know - so I will go with that assumption)

If one lesson takes an hour, then a tutor can take 3 lessons per weekday (that is 15 lessons per weekday per tutor), and  5 lessons at the maximum on a Saturday (assuming an hour break for lunch), that's about 20 lessons per week per tutor on Saturday. 

# of lessons required

Students in the age group of 5 to 16 are most likely to take lessons compared to adults

Let's say the population of NYC is ~8M, if we assume life expectancy of 80 and a uniform distribution, the population by each group is

0-5 : 500K, 5-16: 1M, 16-55 : ~4M, 55-80 : ~2.5M

Likelihood of people taking lessons in each age group

0-5 : 0%, 5-16: 30%, 16-55 : 10%, 55-80 : 0%

Total # of people who take piano lessons in NYC = 700K

Average # of lessons per week = 1 

# of lessons required per week = 700K

# of tutors required = 700K / 20 = 35,000 tutors in NYC

4. Extrapolation

Assuming all professional pianists are also tutors (100% overlap), that means there are 35K pianists in NYC who are either professionals or tutors or both. Plus there are 700K students who are learning how to play, if we count them as pianists as well, that brings us to 735K pianists in NYC

5. Sanity check

735K is about 10% of the total population in NYC, which means 1 in every 10 people is learning, teaches or plays piano professionally. This looks very high. So I will revisit my assumptions of 30% students in 5-16 age group who are learning how to play, and 10% of adults in 16-55 age group who are learning how to play and evaluate if they are too high. Other area that I will evaluate is the demographics by age - that seems to be fine (~20% of the population is 16 or under) but I would subdivide it by income level. Piano lessons are quite expensive (~$50 per hour on average). Making an assumption that of across age groups, about 50% can afford piano lessons, then total number of tutors required would be about 17K, plus 350K students which is ~5% of total population in NYC.