# 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

1. Four particles end at −3, −1, +1 and +3 micrometres from where they started.
2. The second drawing puts those positions along the bottom, with a bar one particle tall above each.
3. The bars stand two on each side of zero, at the same distances out. The picture is symmetric, so the average position is 0.
4. Yet no bar stands at zero: every particle moved. The graph shows at a glance what the average hides.

## 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.

## This lesson builds on

- [A sign records direction](/foundations/bridge-negative-numbers-direction/)