Foundation lesson
Reading a graph
A graph puts one quantity along the bottom and another up the side, one point for each pair. How steeply the curve climbs is a rate; the area under it is a total.
Written for this edition, not translated from Einstein.
How does a curve show the relationship between two physical quantities?
Walk for a minute and note how far you have gone every ten seconds: 0 m at the start, 14 m after ten seconds, 28 m after twenty, and so on. The first drawing below puts time along the bottom and distance up the side, with a dot for each pair. That is a graph: each dot is one moment, read across for the time and up for the distance.
The dots of a steady walk fall on a straight line, and the steeper the line, the faster the walk. This one climbs 14 metres for every 10 seconds: 1.4 metres per second. A flat line would mean standing still, and a line that bends upward would mean speeding up.
The papers draw other pairs the same way. Along the bottom goes what you choose or wait for, such as a position or a time; up the side goes what you then find, such as how many particles sit there. A bell-shaped curve, high in the middle and falling away on both sides, says that most particles are near the centre and fewer are farther out.
Worked example: Four particles that average to zero
- Four particles end at −3, −1, +1 and +3 micrometres from where they started.
- The second drawing puts those positions along the bottom, with a bar one particle tall above each.
- The bars stand two on each side of zero, at the same distances out. The picture is symmetric, so the average position is 0.
- Yet no bar stands at zero: every particle moved. The graph shows at a glance what the average hides.
The two graphs this lesson describes
Both are drawn here rather than left to the reader. The first is the walk: each dot is one moment, read across for the time and up for the distance.
The dots fall on a straight line, so the walk is steady. The step drawn on it reads the rate off the line: 14 metres for every 10 seconds, which is 1.4 metres per second. A steeper line would be a faster walk, a flat one would be standing still.
The second is the worked example: four particles, and a bar one particle tall above each place one of them stopped.
What it shows, in words
In the first graph the seven dots of the walk lie on one straight line rising from the corner: 0 metres at 0 seconds, 14 at 10 seconds, and on to 84 metres at 60 seconds. Stepping 10 seconds across the line and 14 metres up it gives the same 1.4 metres per second wherever the step is taken, which is what makes the walk steady.
In the second graph four bars of equal height stand above -3, -1, 1, 3 micrometres. Two are left of zero and two are right of it, at matching distances, so the picture is symmetric and the average position is 0. No bar stands at zero: every one of the four particles moved. That is the thing the average leaves out and the graph keeps.
Where this lesson stops
Before reading a curve, read its two labels and their units, and ask whether its height is a count, a density or a running total.
If you came here from a passage, Back returns you to the exact place you left.