How to sense check numbers

Six quick rules of thumb for growth and proportions

When working on questions involving data and dynamics, I often need to get a quick handle on growth rates or proportions. Here are six particularly useful rules of thumb I’ve come across for these types of problems. The first three relate to growth, and the second three to proportions:

Rule of 72

If something grows at x% per month (whether an investment or epidemic), it will take around 72/x months to double. E.g. something growing at 6% per month will double in around 12 months. Likewise, if something shrinks at x% per month, it will take around 72/x months to halve. More here.

Doubling up

2^10 is just over a thousand, i.e. if you double something ten times, it will be about a thousand times larger. And 2^20 is just over a million.

Exponential calculations

If x is small, then exp(x) is approximately equal to 1+x. For example, if a population (or investment) is continously growing at a small rate r per unit time, then after a short period of time t, its size will be approximately 1+rt.

This also leads to a cautionary tale: if have a curve that is growing exponentially and you zoom in too much, it will look linear.

Swapping percentages

Calculating x% of y is the same as calculating y% of x. E.g. It’s easier to work out 25% of 16 than 16% of 25, and they both give the same answer = 4.

Rule of 3

If we sample N people or things at random and none of them have a particular characteristic (e.g. all test negative for a disease, or all respond ‘no’ to a survey), then we can be 95% confident that less than 3/N of the overall population have the characteristic. E.g. if 50 people are randomly surveyed and all say ‘no’, then we can be 95% confident that the true proportion of people who would say yes is between 0 and 0.06 (i.e. 0–6%). More here.

Margins of error: 10-3-1

If we have a random sample of 100 people, and a reasonable number of them have a certain characteristic (e.g. between 20 and 80 of the 100), then the 95% confidence interval for the true percentage will be approximately the percentage with that characteristic ±10%. E.g. if 40 out of 100 people answer ‘yes’ in a survey, we can be 95% confident the true percentage is between 30% and 50%.

If we have 1000 people, then the 95% confidence interval will be the percentage with the characteristic ±3%. And if we have 10,000 people, it will be the percentage ±1%.