Line graphs, and what change over time looks like

0:00Here are twelve numbers. The highest temperature recorded each month, for a year, in one place near the coast. Drawn as columns, one bar to a month, they look like this. Drawn as marks joined by straight lines, they look like this. Same twelve numbers. Same twelve heights. Nothing has been added and nothing has been taken away. So which is right? Neither, because both hold exactly the same information.

0:31But the second is not the first one tidied up. It quietly says something the first never did, and this whole video is about what that something is. Look at one segment. March to April, thirty-two point five up to thirty-three. The two marks are measurements. Somebody recorded them. The line between them is not. Nobody measured halfway through March, and yet the line says thirty-two point seven five as if they had.

1:05That is the claim. Every segment asserts that the quantity travelled smoothly from one value to the other, passing through everything in between. The columns never said that. They gave you two heights and stopped. Ask the line about mid-March and it answers instantly. Ask the data, and the honest answer is: nobody knows. So a line graph is only honest when there IS a between. Months have one. Time runs left to right whether you draw it or not, and mid-March exists.

1:42Now put six cities along the bottom instead, and their rainfall above them. Join the tops. The segment between the second city and the third now claims there is a city between them, with a rainfall of its own. There is not. Shuffle those cities into another order and you get a different picture of the same data. If reordering the axis changes the shape, the shape was never a fact.

2:11That is the test. Before drawing a line, ask whether moving right along the bottom means anything. Reading one you did not draw takes two passes, and most people skip the first. Pass one asks only what is given. What the upright axis measures and in what units. What runs along the bottom. What one interval is worth. What the colours and marker shapes mean. Notice the marker shapes. Circles for one place, squares for the other. Shown without colour, or read by somebody who cannot separate red from blue, the shapes are the only thing keeping them apart.

2:56Pass two, and only then, asks what it means. Do those in the wrong order and you will read a shape off a scale you never checked. Because the scale decides what flat looks like. Nought to forty in tens is five labelled lines, and both places fit inside it comfortably. Zoom the axis to twenty-eight up to thirty-four, stepping by two - four labelled lines now - and the coastal year that looked almost level fills the whole height and looks dramatic.

3:30Nothing moved. The same twelve numbers. And start the axis at twenty instead of nought and January vanishes off the bottom, because January was nineteen. A line that looks flat is a claim about the numbers AND about the axis, and you have to read both. Here is the second place, inland, on the plain. Its steepest climb of the whole year is the very first segment, January to February. Six and a half degrees in one month.

4:04And here is the trap. January is nineteen and February is twenty-five and a half. Those are the two SMALLEST numbers on the entire picture. The steepest segment sits at the coldest end of the year. Steepness is the size of a CHANGE, never the size of the quantity. A tall mark and a steep segment are two different facts. And two segments here fall by exactly five - October to November, and November to December. There is no steepest fall. Asking for one asks a question the data does not answer.

4:44Now a question almost nobody asks. Where did one of these twelve numbers come from? Each of these places has several weather stations. This month they recorded thirty-one, thirty, thirty-seven, twenty-eight point five and thirty-three. The number on the graph is the LARGEST of them. Thirty-seven. Not the average, and not the middle one, which is thirty-one. The figure sits six degrees above the middle of what was actually recorded.

5:17So one hot afternoon at one station sets the value for a whole place and a whole month, and nothing here reports a typical day. The line is honest. It is just not answering the question you thought it was. Put both places on one axis and the comparison does itself. The coast climbs to thirty-three in April, dips to twenty-nine in July, and ends about where it started. Highest minus lowest: four degrees, all year.

5:52The plain starts at nineteen, climbs to thirty-eight in June, and falls back to twenty-three. Highest minus lowest: nineteen degrees. So the plain is the more variable of the two - colder in winter AND hotter in summer. A single average for each would have hidden all of it. One foot in ice and one in the fire, and on average all is well. Why choose the line at all, if the columns carry the same numbers?

6:24Two places over twelve months is twenty-four columns. Against two lines. Four series over thirteen years would be fifty-two columns. Against four lines. Fifty-two bars is not wrong. It is unreadable. To follow one series your eye must hop over three neighbours to reach its next bar. The line does that hopping for you. So the choice between the forms is a fact about the reader, never a fact about the data.

6:57Now build one. Two rows: how many customers walked into a shop each day of the week, and how many of them bought something. Sixteen, nineteen, ten, fourteen, twenty, twenty-two, thirty-five walked in. Ten, eight, seven, eleven, twelve, sixteen, twenty-six bought. Pick the interval first. The largest number is thirty-five, so an axis to thirty-eight stepping by two gives twenty labelled lines and holds both rows. Plot, join, and look at what neither row shows on its own: the gap between the lines.

7:37Six, eleven, three, three, eight, six, nine. Tuesday is the widest gap of the week, and Tuesday is not the busiest day. Only eight of nineteen bought anything. Thursday turned eleven of fourteen into buyers. A hundred and thirty-six came in that week and ninety bought, and the forty-six who did not are the gap. Here is the price of a bag of salt, taken every January for ten years, in one region.

8:10Over the decade it rose by six dollars thirty-five. That is the whole story, if a number is all you keep. The line tells a different one. It DOUBLED in the very first year, six to twelve, and then it wobbled. It stood still for three years running, peaked in the middle of the decade at fifteen, and came down again. Three years it rose, three it fell, three it did not move.

8:39Lowest to highest is nine dollars - larger than the decade's rise, because the ends are not the extremes. The rise is not the shape. It is the two ends of the shape with everything between them deleted. Two last readings, both done without measuring anything. Monthly births in one country, every mark between one point four and one point nine five million. How many births in a year? You need not read twelve marks off a crowded axis. Twelve numbers in that band must total between sixteen point eight and twenty-three point four million. Twenty million is inside. Thirty is not.

9:25Now sunrise and sunset for one place, both curves on a clock. The band between them is the day, and the band is the point. Six in the morning to half past four is a ten and a half hour day at the eastern edge. Further north, ten hours in January and fourteen and a half in June, a swing of four and a half hours. Further south the day barely moves all year.

9:53Last one. Moonrise slips fifty minutes later every day. By the fourteenth it rises at six in the evening as the sun sets - a full moon. Keep going and it runs off the top of the clock and restarts at the bottom, which is the axis and not the moon. Fifteen days on it rises at half past six in the morning, with the sun. Marks are measurements. Segments are claims. Read the axis before the shape, and read steepness as change and never as size.