Histograms: Bin Width Changes the Story

Full transcript

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

0:00Twenty-eight numbers ▶

Here are twenty-eight numbers: how many minutes each student in one class needs to get to school. Read them for a moment, and notice that they tell you almost nothing. A list of a thousand would tell you even less. By the end of this video you will read the shape of any pile of numbers in one look. And you will know what one hidden choice can do to that shape.

0:29Is a histogram just a bar chart? ▶

Most people think a histogram is just a bar chart with the gaps closed up, and the difference matters. A bar chart counts categories: apples, pears, bananas, each bar its own thing. A histogram takes one line of numbers, cuts that line into pieces, and counts how many values land in each piece. Where you make the cuts is a choice, and nobody mentions that when they hand you the picture. Stay with me and you get three things. How to read one, how to choose the cuts, and what a histogram shows that the average hides.

1:09Dots on a line ▶

Start with the simplest picture there is: a line from zero to sixty minutes, and every student on it as one dot. Now the twenty-eight numbers stop being a list and become a shape. Two clusters appear on their own: a big one around eleven minutes and a smaller one in the thirties. Between them lies an empty stretch where nobody lives. Dots have a limit, though: with three thousand students the line turns into a smear, so we need a way to count them.

1:42Bins: drop them in, stack them up ▶

Cut the line into bins of five minutes: zero to five, five to ten, and so on. Drop every student into the bin where their number lands, and stack them. That stack is a histogram, and the height of each bar means one thing only: how many students landed in that bin, nothing else. The tallest bar, ten to fifteen minutes, holds thirteen students. The second hill, over in the thirties, peaks at five. Two hills, and the valley between them is empty. It is the same shape the dots showed, but now it scales to twenty-eight thousand students just as well. There is only one thing I did without asking you.

2:27The hidden choice: bin width ▶

I chose five minutes, and nobody told me to; I could have chosen one, or ten, or twenty. Watch what a bin of one minute does to the same twenty-eight students. The two hills shatter into a comb of seventeen thin bars, and the shape is gone. Narrow bins show you every bump in the road, including the bumps that are only noise. So let us go the other way, and before you watch, make a guess. If I make the bins twenty minutes wide, what happens to the two hills?

3:03Same data, one slope ▶

They are gone: one slope, sliding down to the right, and what is left is seventeen, ten, one. The empty stretch where nobody lives has vanished, and not because anyone moved. The wide bins swallowed the gap whole, and nobody changed a single number. Same twenty-eight students, same line, and the picture now tells a different story: one group, spread out. Wide bins are calm, and calm can be false. So the obvious question is which width is right.

3:40Which width? Try three ▶

There is no single right answer, but there is a rule of thumb and a habit. The rule of thumb: take the square root of how many values you have. Twenty-eight students gives a square root of about five, so aim for five or six bins, which over sixty minutes means bins of roughly ten. The habit is better than the rule: try three widths and look for what survives. The two hills are there at five, still there at ten, and gone at twenty. A shape that survives two widths is worth believing; a shape that only lives at one width is the bins talking, not the data.

4:21What the mean hides ▶

Now ask the question every report asks: what is the average travel time? Add them up, divide by twenty- eight, and you get twenty minutes. Put that on the histogram, and it lands in an empty bin, right between the two hills. It belongs to neither group: the cyclists are around eleven, the bus students around thirty-four, and the mean is the class, not a student. The median, the middle student, is thirteen minutes, sitting in the tallest bar. That is what a histogram is for: before you trust any single number, look at the shape it is supposed to summarise. Two hills and one average is a warning sign, and you only see it here.

5:06One look back ▶

One quick look back first: two videos ago you saw three hundred incomes, and their histogram did something this class does not. Which of the two centres got dragged away from the crowd, into the tail? The mean did, and the shape leaned to the right to show it. Same tool, different shape, same lesson: the picture first, the number second.

5:31In one picture ▶

So here it is in one picture. A histogram cuts a line of numbers into bins and stacks what lands in each, and height is the count and nothing else. The bin width is your choice, so try three and trust the shape that survives. Look at the shape before you trust any single number, because a mean can sit where nobody is. Which raises the next question: two hills or one, how far do these students actually spread —

6:02Next: the spread ▶

— and can one number say it? That is next time: one ruler for the spread of any pile of numbers, and why it divides by one less than you would expect. And if you want to keep learning, subscribe — the next video will find you. Until then: happy learning.