Calculate any Population with a calculator

I wanted an answer to one simple question. What is the population of England? I set this as a challenge and the rule was paper, pen and calculator. No mainstream sources and my thoughts where to be methodical.

The initial struggle was finding a decent starting place. I posed another question, how do populations grow from generation to generation?

Two parents have two children. The two children, one girl and one boy go separate ways, forming couples with members of other families. I attempted drawing a diagram that visualised this.

Diagram A: Population splitting into two. Each generation the figure doubles.

Diagram B: Population as a series. Each generation remains the same.

Diagram A shows what I originally thought. But this representation only makes sense if every couple has 4 children. Diagram B is my revised version, 2 great-grandparents, 2 grandparents, 2 parents and 2 children. The vertical bars represent the coupling. Coincidently this makes my job simpler, if I calculate one generational bar then multiply by 4 generations then I get population size. So which one do I use?

Great-grandparents aren't at all useful for a measure. They are missing, few in numbers and the same problem I get with grandparents. Some are in retirement homes, bungalows, houses and I don't have the patience or resources to visit every elderly man or woman in the country for a half-assed census.

Parents are somewhat useful but they work in different towns, travel a lot and always move house and divorce. They have no place in my equation.

Kids provide just the data I'm looking for. They are more static because they live with their parents and go to local schools. Wikipedia provides a list of schools and average number of students. This is perfect.

When choosing between primary schools and high-schools was tricky to understand, you can't just use both. What if the same parents had two children one who's 8 and another 15, I might miscount and don't want family trees to overlap. Might I mention that primary school children might even share the same great-grandparents from high-school students, my results would therefore be inconsistent. So I went for the larger grouping.

Lets get started with my hometown Norwich.

15 schools x 1000 average students x 4 generations = 60,000 people

Real figure is 213,166. Damnit it's 72% off

Okay how about Leeds.

42 schools x 1000 average students x 4 generations = 168,000 people

Real figure is 784,800. Dammit it's 78% off

Surely Edinburgh is close.

27 schools x 1000 average students x 4 generations = 108,000 people

Real figure is 512,150. Dammit it's 78% off

How about Greater London.

463 schools x 1000 average students x 4 generations = 1.852 million people

Real figure is 8.825 million. Dammit it's 79% off

I noticed a pattern suddenly. I was consistently off by about 75%. If I multiply my original number by 4 then I'm within 95% accuracy for population. I might even argue my results are more accurate.

What's more this is a unique system. I can't find direct population equations on YouTube, Google or scientific journals. My half hazard attempt proved successful, this challenge only needed some out of the box thinking.

In England there are 4,168 high schools x 1000 average students x 4 generations = 16.67 million people
Alright so 16.67 million x 4 (as we found out) = 66.6 million people
The census is actually 54 million, so roughly 82% accuracy. Hooray.

Sources of error:

I assume all 4 generations are living

I assume both children go to school

I assume people have 2 children on average
I assume everyone forms as couple
I assume 4 generations
I assume a strict 20 year generational gap