Scale and Origin in a Graph

In this lesson you will: Write an equation in two variables from a situation and decide which quantity goes on each axis; Label both axes with a quantity and a unit, and choose a scale the data actually fits; Compute a comparison from the numbers on the axis, never from how tall the picture looks.

Two variables, two axes, two decisions

A babysitter is paid $14 an hour plus a flat $10 travel payment. With h hours worked and E dollars earned, the model is E=10+14h.

Graphing it takes two decisions before a single point is plotted. Which quantity goes on which axis — hours is the one you choose, so it goes across — and what one step along each axis is worth.

Hours run from 0 to about 8, so one tick per hour is comfortable. Earnings run from $10 to $122, so one tick per dollar would need a hundred and twenty rows. Twenty dollars a tick fits the same information in six.

Picking a scale is arithmetic, not taste

Divide the range of the data by the number of ticks you have room for, then round up to a friendly number — 1, 2, 5, 10, 20, 25, 50, 100 and their relatives.

Rounding down is what forces you to redraw the graph when the last point will not fit. Choosing something unfriendly, like 7 per tick, makes every later reading an arithmetic problem.

A moved origin changes the picture, not the numbers

A club's membership went from 48 to 60. That is a rise of 12, and 60 / 48=1.25, so membership is up a quarter.

Draw it with the vertical axis starting at zero and the second mark sits a little higher than the first. Draw the same two numbers with the axis starting at 45 and the marks stand 3 and 15 units above the bottom of the picture — the second looks five times the first.

Nothing was falsified. Both graphs plot 48 and 60. What moved is the origin, and with it the only thing a reader's eye actually measures: height above the bottom line.

So read the axis before you read the picture

Three questions, in this order. Where does the vertical axis start? What is one tick worth? What is being counted?

Only then look at the shape. If the axis does not start at zero, any statement about “twice as tall” has to be recomputed from the labels, because the heights on the page are measuring distance above a line somebody chose.

Common mistake — Reading a truncated axis as if it started at zero

A chart of quarterly sales has bars that look roughly four times one another, and a caption is written claiming sales quadrupled.

The eye is doing exactly what it was trained to do. In every graph most people meet, the bottom of the bar is nothing, so bar height reads as amount. That habit is what a truncated axis exploits.

The bottom of this axis is 4.4, not 0. The bars stand for 4.8 and 6.0, so the visible parts are 0.4 and 1.6 — a ratio of 4 — while the actual ratio is 6.0 / 4.8=1.25.

A quarter is a good quarter. It is not a quadrupling, and the gap between those two claims is the entire reason this trick keeps getting used.

How to catch it: Find the number printed where the vertical axis begins. If it is not 0, do not compare heights at all — take the two values printed on the axis and divide. The picture is allowed to be a summary; the ratio has to come from the numbers.