The Log Transformation: One Axis Trick

Full transcript

Word for word what is said in the video — the same text as the subtitles. Times are minutes:seconds into the video.

0:00The country that collapses ▶

Here is that leaning country of incomes again: 300 people, one dot each. Now invite one billionaire back in, and watch the page give up. Every single person collapses into one smear against the left wall. By the end of this video you will open that smear back into a bell — without changing a single number.

0:26Is a log axis a trick? ▶

Say "log scale" out loud, and someone will tell you it is a trick. An axis that squeezes the rich end until the gap looks smaller than it is. Take that seriously, because half of it is true: a log axis does change what your eye measures. Here is the other half: nothing in the data moves, and every dot keeps its exact value. By the end you will have the rule: when this ruler helps, and the three limits you have to respect.

1:00Two kinds of ruler ▶

Every ruler you have ever used makes one promise: each step adds one. One, two, three, four — equal steps, equal additions. There is a second kind of ruler, with a different promise: each step multiplies by ten. One, ten, one hundred, one thousand — equal steps, equal ratios. That second ruler is what people mean by a logarithmic scale — or just a log scale. Look at what it spends its space on: every tenfold jump gets exactly the same width. Hold on to those equal ratios, because they are the reason our smear is about to open.

1:43The log transformation ▶

Statisticians have a name for the move itself: the log transformation. Replace every value by its logarithm — by its number of tens. A thousand becomes 3, because a thousand is three tens multiplied together. 40,000 becomes 4.6. Draw those small numbers on an ordinary ruler, and the log ruler is built. There is a reason this suits incomes: the world changes in percentages, and a ruler of multiplications speaks that language. So before we swap them — what do you think happens to the smear? Does it just spread out, or does it change shape?

2:30Same data, new ruler ▶

Now watch one change of ruler do its work. Same dots, same values — only their places move. The squeeze on the left relaxes, the tail walks in, and the hill opens into a bell. Same data, new ruler — and the median lands dead centre of the bell, at 40,000. Look at the far right: even the billionaire fits on the page now. Three steps past the middle, a thousand times the typical income, and still an outlier. But swapping the ruler quietly swaps something else as well. On this axis, the average you have used all your life is not the one the picture balances on.

3:19Which average? ▶

So which average is the picture balancing on? Find the balance point on the log axis, then translate it back into money. You land on 40,000: the median itself, here, and an average with its own name — the geometric mean. The ordinary mean was 49,000, dragged into the tail by that single dot. Both are correct — they simply answer different questions. One totals a budget, the other captures a typical ratio.

3:54Spotting it in the wild ▶

So where will you actually meet this ruler? Here are the daily views of a video channel I made up, over its first 100 days, with two bad days marked. Both of them lost exactly 40 percent from the day before. On the ordinary axis the first one is a wobble you cannot see, and the second looks like a collapse. On the log ruler they are the same drop. And between those two days, the whole climb straightens into a line: steady percentage growth. Your ears read a ruler like this too: a rock concert carries about 100 million times the energy of a whisper.

4:37When not to use it ▶

So why not put everything on a log scale? The ruler has three honest limits. First, zero: every step down divides by ten, so the ruler approaches zero forever and never arrives. Second, negative values — a ratio ruler has no place for them either. And third, the quiet one: a chart only works if your audience can read it. If a log axis costs more explanation than it saves, reach for the tools from last time — the median, percentiles, and the boxplot.

5:14The bands that broke ▶

One promise from video one broke on skewed data — do you remember which? The three bands: 68, 95, 99.7. On the leaning data, two sigma reached below zero — a negative income. On the log ruler that same rule becomes a ratio: one sigma is a factor of 1.9, not a fixed amount. Slide them onto the log bell and count: 209 of our 300 people sit inside the first band. That is 70 percent, where the rule promises 68.

5:52In one picture ▶

So, in one picture: the same dots, on two rulers. On the ordinary ruler the lean crushes everything against the wall. On the log ruler — equal steps, equal ratios — it opens into a bell. A straight line means steady percentage growth, and the label tells you which ruler you are reading. Which brings us to two labels that get swapped all the time.

6:20Next: SD vs SE ▶

Standard deviation and standard error — they look alike and get mixed up in far too many papers. One describes the people in your data, the other describes how sure you are of your estimate. That difference decides what an error bar means. Tell me below: where did you last spot a log axis? And if you want to keep learning, subscribe — the next video will find you. Until then: happy learning.