Sense of scale
February 26, 2021

1 billion gallons of water & remembering a sense of scale

By Nick Radcliffe

Why should data or analytics leaders be interested in such a volume of water or a sense of scale?

To answer that question, I am delighted to welcome a new guest blogger to our fold. Dr Nick Radcliffe is a very suitable new addition during a month when we are thinking about Statistics. Nick is a specialist in analytical customer retention, cross-selling & targeting using uplift modelling and other state-of-the-art data science.

Nick’s expertise straddles both academia & business. He is a visiting professor of Maths at the University of Edinburgh, as well as CEO of Stochastic Solutions & Data Lead for the Global Open Finance Centre of Excellence. In this first post, he addresses the mundane data-to-day challenge of retaining a sense of scale when we hear big numbers.

A big number grabs your attention (1 billion)

A friend mentioned that the diving events during the last (Glasgow) Commonwealth games were actually held at Edinburgh’s Commonwealth Pool. The image above is a screenshot from Glasgow 2014, which you can still find as a claim online.

But what’s this “1 billion gallons of water”? Surely that can’t be the amount of water in the pool, can it? I mean, that’s obviously wrong. Isn’t it? It is obviously wrong. But let’s just double-check.

A quick calculation to sense check

As we all know, an Olympic/Commonwealth pool is 50m by 25m. And the depth is something like 2m on average. So the volume is 50m x 25m x 2m = 2,500 cubic metres. That still doesn’t seem like it’s going to turn into a billion gallons, but let’s carry on. A cubic metre is 1,000 litres (a cube of side 10cm has a volume of 1 litre). So the pool has about 2.5 million litres of water. And a gallon is more than a litre. So QED.

Trying slightly harder (and still not looking anything up) a gallon is 8 pints and there are about 1.75 (imperial) pints in a litre (or about 2, if we only care about orders of magnitude). So we need to divide that 2.5 million litres by something between 4 and 5 (8/1.75 = 4.57) which will give us something a little over half a million gallons.

So there not only aren’t a billion litres of water in the pool: there aren’t even a million by my estimate. The claim is out by 3 orders of magnitude.

Now, of course, if you want to defend Glasgow 2014, they don’t actually say that the pool’s volume is a billion gallons. They just stick the phrase “1 billion gallons of water” under “Royal Commonwealth Pool”. Maybe it’s the amount of water the pool will use over its lifetime. Maybe it’s just an impressive amount of water. Who knows.

Why retaining a sense of scale matters for everyone

Why do I care? Only because I think it’s important people develop a good sense of scale and can spot when numbers are “obviously” wrong (even if sometimes “obviously” wrong numbers turn out to be correct). An excellent discipline that physicists are often taught is:

  • Guess
  • Estimate
  • Calculate

Meaning that whenever you’re calculating anything you should first guess the answer (literally, just guess).

Then make a crude estimate by approximating the key components of the formula; then calculate the precise answer. The idea is that this leads you to first to develop better intuition about orders-of-magnitude sizes (by seeing when your guess is significantly off), but also helps you to avoid believing calculations that are orders of magnitude off because the guessing and estimation stages lead you to be surprised by an answer that seems as if it can’t be right.

Do you have a feel for how big a billion really is?

Ordinary people, quite reasonably, have no feeling at all for what a billion is, because in ordinary life you rarely if ever encounter such numbers in contexts where they can really be appreciated. Sure, people might know that the population of the Earth is about 7 billion, but you can’t see 7 billion people.

Even more, people will hear about states and companies spending and earning quantities of currency measured in billions (or occasionally even trillions), but again, those are just abstract numbers. Even when there’s hyperinflation and prices begin to measured in millions or billions, the appreciation doesn’t really increase, because the million or billion just becomes a suffix.

Two ideas to remember a sense of scale (e.g. a feel for a billion)

If you don’t have a feel for a billion (and who really does?), my two favourite ways of getting some kind of handle on it are the following:

  1. I always remember the number of seconds in a year as “pi times ten to the seventh”, i.e. about 31 million. Since pi squared is ten (roughly), this means that a billion seconds is about 31 years.
  2. The other way I like to think about it is with respect to centimetre cubes and metre cubes. When I was at school, we had wooden 1cm cubes that you could assemble into larger volumes. Obviously, a 1-metre cube contains one million centimetre cubes (100 ⨉ 100 ⨉ 100 = 1,000,000). If you don’t immediately get a sense of just how big that means a million is, get hold of some of those 1cm cubes, and start arranging them to make the bottom layer of the 1-metre cube. (You’ll need 10,000 of them.)

A billion-centimetre cube is either one thousand of these metre cubes or a single cube of side 10 metres. Now a 10m by 10m room is a pretty decent size, but will typically only be 2–4 m tall. So a billion-centimetre cube would fill a room (say) 50m by 10m by 2m tall.

Back to that swimming pool & your daily big number challenges

And so we come full circle. Because that 50m x 10m x 2m is not so far off the size of an Olympic swimming pool (too narrow, at 10m, but pretty similar otherwise). And it only contains a billion centimetre cubes. So even if you have no real sense of how big a gallon is, you probably know it’s orders of magnitude bigger than a 1cm cube.

Originally published in Scientific Marketer (Nick’s own blog) here.

Many thanks to Nick for his first post & for challenging us to improve our ability to sense when numbers are orders of magnitude wrong. The kind of mental process that Nick describes reminds me of the statistical thinking encouraged by Prof David Spiegelhalter in his book. It is also a theme that will reoccur even more strongly in my pending review of Tim Harford’s latest book.

Until then, I’d love to hear what helps you retain a sense of scale for the numbers you hear each day. Do you remember the kind of examples that Nick suggests above, or have you learned to remember some key commercial numbers in your business? If you’ve got tips or tricks to share, please let us know in the comments below. Keep thinking!