Glossary / Exponential growth
According to an old legend, the inventor of chess asked the ruler for a modest reward: one grain on the first square of the board, two on the second, four on the third, and so on, twice as many each time. The ruler laughed at such a trifle. It seems it would come to a few sacks. Exponential growth is when a quantity is multiplied by the same number at every step, and it is the kind of growth intuition underestimates most of all.
“In plain words” is hidden on all puzzles.
Move the slider square by square. The brightness of each square shows how much rice it holds compared with the current one. Watch what happens to the earlier squares as you move on.
From which square a single square holds more rice than:
A grain of rice weighs about 20 mg: there are roughly 50 thousand in a kilogram.
The 64th square holds 263 ≈ 9.2 · 1018 grains, about 184 billion tonnes. Today the world harvests a little over half a billion tonnes of milled rice a year. So the last square is the whole planet's harvest for about 350 years. Humanity has not grown that much rice in all its history: since 1961, when the FAO started keeping records, about 33 billion tonnes of paddy rice, still in the husk, have been harvested, and harvests before that were much smaller.
“Rigorous” is hidden on all puzzles.
Exponential growth is when a quantity grows by the same factor, that is by the same percentage, over equal stretches of time. Ordinary, linear growth adds the same number each time. Exponential growth multiplies by the same number each time.
Put 10,000 dollars in at 10% a year and leave the interest in the account every year. Each year the amount is multiplied by 1.1. How long until it doubles?
1.1T = 2 → T = ln 2 / ln 1.1 ≈ 7.27 years
And it is the same for any amount: 10,000 becomes 20,000 in 7.27 years, 20,000 becomes 40,000 in the next 7.27 years, and so on. A constant percentage means a constant doubling time. After 30 years you get 10,000 · 1.130 ≈ 174,494 dollars. If the interest were taken out each year instead of added to the amount, there would be only 10,000 + 30 · 1,000 = 40,000.
You are offered a choice: get 1,000 dollars every day, or get 1 cent today, 2 tomorrow, 4 the day after and so on. Move the days slider and switch the scale. What a logarithmic scale is and why an exponential becomes a straight line on it is explained in the article on the logarithm.
For three weeks the linear offer wins by a mile: on day 20 it has 20,000, the cent 10,485.75. On day 22 the cent is already ahead. On day 30 it has over 10.7 million, while a thousand a day has only 30,000.
Exponential growth lasts as long as each new piece grows as freely as the first ones. The bacteria have enough food, the post still finds people who haven't seen it. But every resource is finite. The more of it is taken, the slower the growth, and the curve turns from an explosion into the letter S: almost exponential at first, then a bend and a level ceiling. This curve is called logistic. The danger is that at the start the S-curve and the exponential look the same, and the early data doesn't show where the ceiling will be.
“Deeper” is hidden on all puzzles.
N(t) = N₀ · 2t/T = N₀ · ert, r = ln 2 / T, T = ln 2 / r
These are two ways of writing the same curve. The first counts doublings: in time t there are t / T of them. The second uses base e and a growth rate r. If the quantity grows by p per step (for example, 10% = 0.1), then r = ln(1 + p), and the doubling time is T = ln 2 / ln(1 + p). This is where the "rule of 70" from the article on the logarithm comes from: for small p, ln(1 + p) ≈ p, so T ≈ 0.693 / p ≈ 70 / p%.
–
S = 1 + 2 + 4 + … + 2n−1, 2S − S = 2n − 1
Multiply the sum by 2: every term moves one square forward. Subtract the ordinary sum from the doubled one, and everything cancels except 2n and 1. So the first n squares hold 2n − 1 grains together, and the next, (n + 1)-th square holds 2n, exactly one more.
For the square on the slider above: –
exponential: dN/dt = rN · logistic: dN/dt = rN (1 − N/K)
N(t) = K / (1 + (K/N₀ − 1) · e−rt)
An exponential is growth whose speed is proportional to size. The logistic model adds a brake: the factor 1 − N/K, the share of the resource still free. While N is small compared with the ceiling K, the brake hardly acts. Compare the two curves for a post that doubles its viewers every hour, starting from one. The ceiling is the number of people who could ever come across it.
The ratio of the two curves is 1 + N₀(ert − 1)/K. So the exponential runs 10% ahead of the logistic curve as soon as it passes a tenth of the ceiling, whatever the ceiling is. Before that moment the data can't tell them apart.
The middle value: line everyone up in order, and half are below it, half above.
A millionaire walks into a bar: the average income shoots up, the median barely moves.
The sum of all values divided by how many there are: what everyone would get if it were shared equally.
Almost everyone has more legs than average: the average is a little under two.
The average result you can expect if you repeat the same thing very many times.
A die's expected value is 3.5, yet no face shows 3.5.
A number from −1 to +1 showing how closely two quantities move together.
In summer both ice cream sales and drownings go up. Ice cream isn't to blame: the heat drives both.
A measure of spread: how far values typically stray from the mean. The standard deviation is its square root, in the same units.
Two towns with the same average temperature: one stays near +15 all year, the other swings from −20 to +50.
A bell-shaped distribution: most values sit near the mean, and the further away, the rarer.
Every ball on a Galton board falls at random, yet thousands of balls always build the same bell.
The more repetitions, the closer the share of an outcome gets to its true probability.
The share of heads creeps towards 50%, while the gap between heads and tails in coins usually grows.
The chance of an event given that something else is already known.
Almost all NBA players are tall, but almost all tall people are not NBA players.
Events are independent if knowing about one doesn't change the chances of the other.
After five heads in a row tails is no more likely: the coin remembers nothing.
How many times you multiply the base by itself to get a number: log₂ 8 = 3, because 2 · 2 · 2 = 8.
A sheet of paper folded in half 42 times would be thicker than the distance to the Moon.
A sample some people get into more easily than others, so it describes everyone badly.
In 1936 a magazine polled millions of Americans and still picked the wrong election winner.
How often chance alone would produce a gap at least as big as yours if there were really no difference.
When there is no effect at all, p below 0.05 turns up as often as p between 0.50 and 0.55.
A range around a measured number, built by a method that covers the true value in a chosen share of samples, such as 95%.
Of twenty honest 95% intervals, on average one misses the truth, and none of them knows whether it is the one.
The probability that a test notices a real effect of a given size. A power of 80% means the test misses one working effect in five.
A weak A/B test that does find an effect almost always exaggerates it: a real +10% shows up on the dashboard as +24%.
How far an average or a rate computed from a sample usually misses the truth: σ divided by the square root of the sample size.
Halving the error takes four times as much data.
Counting how many ways things can be ordered or chosen: permutations n!, combinations C(n, k).
A well-shuffled deck is almost certainly in an order nobody has ever seen.
The sum 1 + ½ + ⅓ + ¼ + …: it grows without limit, but incredibly slowly.
Identical books can be stacked so the top one overhangs the table edge as far as you like.
The rule for updating confidence when new evidence arrives: prior odds × strength of the evidence.
Two weak pieces of evidence together can be strong, while one striking piece means little if the claim was unlikely.
A path where each next step is taken at random: forward or back, left or right.
A drunk man will always find his way home, but a drunk bird may never return.
A stream of independent random events at a constant average rate: calls, Geiger clicks, raindrops.
In 1944 Londoners thought German flying bombs were aimed at certain districts; it was ordinary random clumping.