Updating your numerical intuition

For my whole life, I have been very quantitatively curious.

I always wanted to know the numbers behind things. As a kid, I loved to read maps, and would always imagine the large distances and compare them to the mile which we regularly had to run in school. Later on as a teenager, I remember working in a skateboarding store and trying to understand how the economics of the store worked.

Recently, living in a country that has undergone massive inflation and a currency crisis (Turkey), I have had to continuously recalibrate my numeric sense (stay tuned for a whole post on that!). Even more interestingly, everyone else is simultaneously doing this, so it is quite a dynamic (and difficult for many) process.

I wanted to write about some heuristics that help me anchor my thoughts

How Big are Billions?

One way to try to intuit big numbers is to compare them to quantities that we know.

One hundred thousand may seem like a huge number. For many young people, reaching a six-figure salary is a big milestone. Our first houses are typically. in the low hundred low hundred thousands. But did you know that 100,000 seconds is just a little over a day? Correspondingly, we pass a million seconds in under two weeks.

When it comes to infrastructure or other large projects, it is useful to be able to divide large numbers.

Approximating the US population at 330 million people, a project that costs 1 billion dollars costs around $3.00.

Thus, a project like California’s High Speed Rail, which was estimated to cost around 100 billion dollars, would cost each person in the United States about $300. That might sound very expensive - but if we are restrict the cost of the California High Speed Rail to California’s population of nearly 40 million people, then that figure jumps to about $2,500 dollars per person. For my family of 7, that is one person short of $20,000.

Fortunately, infrastructure projects are paid over time. So those $2,500 dollars per person might come out of a $250-$500 property tax each year for a few years.

I am very interested in statistics and demography. It helps to have a few anchor points I can quickly reference mentally.

Here are some population baselines:

1,000 people - The Vatican. Many performance halls hold this many people.

10,000 people - Nauru and Tuvalu are small island nations with around 10,000 people.

50,000 people - Many Caribbean island nations have less than 50,000 people. Many American suburbs have around this number of people. For Europeans, Liechtenstein has around 40,000 people, and Greenland has a little over 50,000 people. Many sport stadiums easily hold 50,000 people.

100,000 people - The bigger Caribbean islands (like the US Virgin Islands, Antigua & Barbuda) have around 100,000 people. The small European country of Andorra has around 80,000 people, and the Seychelles are a little over 100,000. The largest soccer and football stadiums can fit around this many people.

500,000 people - Iceland’s population is just a little under 400,000 people. They make up just 0.005% of the world’s population, but a much larger fraction of the world’s powerlifters. The Maldives, Malta, and Brunei all have around 500,000 people. In the US, cities like Tucson, Fresno, Kansas City, Mesa, Omaha or Milwaukee have around 500,000 people (though most of these cities have vast suburban areas which add to the effective number of people in the area). The total number of Qatari nationals is estimated to be around 320,000 people.

1,000,000 people - Djibouti’s population is right around 1 million people. Abu Dhabi has around 1.5 million people. Seattle has about 1 million people within the city limits, and another million within an hour drive from it.

5,000,000 people - Countries like Oman, Lebanon, Ireland, New Zealand. Many people are connected with just a few degrees of separation, especially professionals and specialists. Cities like Guadalajara, Washington DC, Alexandria (Egypt), and Saint Petersburg have around 5 million people.

10,000,000 people - Megacities are often defined as having more than 10 million people. Places like London (9 million), Seoul (10 million), Bangkok (10 million), Paris (11 million), Hyderabad (9.5 million) and Jakarta (10.5 million) are in this range. Countries like the United Arab Emirates (10 million), Portugal (10.4 million), and Jordan (11.4 million) have around 10 million people.

50,000,000 people - This number is beyond any individual cities. The closest one to it, Tokyo, has around 37 million people. Countries like Spain (47 million), Colombia (50.8 million), and South Korea (51.3 million) are in this range. Indian states like Andhra Pradesh (50 million), Gujarat (60 million), and Karnataka (61 million) are in this range. There are about 45 million people in all of Oceania.

100,000,000 people - There are only a few countries with more than 100 million people. The Democratic Republic of Congo (99 million), Vietnam (99.5 million), Egypt (105 million), Ethiopia (105 million), and the Philippines (110.5 million) are in this range. The Indian states of Bihar (105 million), Maharashtra (112 million), and Uttar Pradesh (200 million) are in this range.

500,000,000 people - Only China (1.41 billion) and India (1.39 billion) have more than this amount of people. There are about 441 million people in all of South America and about 610 million people in all of North America.

1,000,000,000 people - The Americas (including North and South America, plus islands) have about 1.05 billion people. The entire African continent has about 1.5 billion people.

8,000,000,000 people - The world population is nearly at 8 billion.

Crowds

For the more visual among us, this set of images can also be a helpful anchor.

Areas

It takes a bit more practice for most of us to develop intuition for areas.

This picture from Wikimedia is 1,000 by 1,000 pixels, so it covers 1 million pixels.

Humans have been measuring areas for as long as we have written history, so there are many units that remain in common use.

Dunam - This is a unit that has a long history, and it remains in use across the Middle East. Today, it has been standardized to 1,000 square meters. So when I visualize a dunam, I decompose this to 10 x 100, 20 x 50, or 32 x 32 components. When I visit farms or orchards here in Turkey, I like to challenge myself to see how close I can estimate the size of their fields as I am walking around.

Hectare - Many Americans are not used to hearing about hectares, so this is an important graphic to understand for those of use who work overseas.

From Wikipedia’s page for Hectare.

Acre - Only used in the US. I generally calculate in hectares and multiply by 40%.

Square kilometers and Square miles - I highly recommend looking at Hans Hack’s How Big tool, which allows you to interactively compare these units to areas you know. You can see that 1 hectare is approximately the size of the city blocks of Seattle.

Volumetric and Yield

Coming soon in a future post!

Practicing

Programmers are used to being asked to make numerical estimates and describe how we come up with them. I am so used to doing this that I do it without even thinking about it. It is both fun and useful, so I suggest at least practicing from time to time. You may be surprised at how quickly you get better at honing your intuition.