What Is a Linear Scale?

A linear scale is a scale on which equal distances always represent equal differences in value. The tick marks go up by the same amount each step (0, 10, 20, 30…), so the gap from 10 to 20 is exactly as long as the gap from 90 to 100, and a point twice as far from zero is twice the value.

That is the whole meaning of "linear": position is a straight-line function of value. It is the default scale on almost every chart, and the one people assume when nobody says otherwise. Its main alternative is the logarithmic (log) scale, where equal distances represent equal ratios instead of equal differences.

How a linear scale works

Two numbers define a linear scale: where it starts (often 0, the baseline) and how many units each step represents (the interval). Every position along the axis then follows from one formula:

position = (value − minimum) ÷ (maximum − minimum) × axis length

Because that formula is a straight line, a linear scale has three useful properties:

• Constant intervals. The tick marks are evenly spaced and each step adds the same amount.
• The midpoint is the average. Halfway along an axis from 0 to 100 is 50.
• Lengths are proportional. If the axis starts at zero, a bar twice as long is twice the value — which is why bar charts must use a linear scale starting at zero.

Worked example: reading a linear scale chart

Suppose a vertical axis runs from 0 to 500 with ticks every 100, and the axis is 250 pixels tall. Each 100 units takes 50 pixels, so 1 unit = 0.5 pixels.

1. Find the interval: the labelled ticks are 0, 100, 200… so each step is 100.
2. Locate the point: a marker sits a little under halfway between the 300 and 400 ticks.
3. Interpolate: about 40% of the way through a 100-unit step is roughly 340.
4. Compare: a second marker at about 170 is half the height of the first, so it is about half the value. That proportional reasoning only works on a linear scale.

Faint gridlines extending from each tick make step 3 much easier.

Linear scale vs. log scale

A quick test: if the values you plot range from, say, 2 to 2,000,000, the small values will be squashed flat against the axis on a linear scale. That is the classic case for switching to a log scale. If they range from 40 to 90, stay linear.

When to use a linear scale

• Bar, column and area charts, where the length or area of a mark stands for quantity.
• Data within one order of magnitude (for example test scores, temperatures, monthly sales).
• When the absolute difference is the story: "sales rose by 1,200 units".
• For general audiences who should not have to decode a non-standard axis.

Common mistakes with linear scales

• Uneven tick spacing. Labelling 0, 10, 50, 100 at equal spacing breaks the scale — it is neither linear nor log, and every comparison becomes wrong.
• Truncated bar axes. A linear axis that starts at 80 instead of 0 makes a bar of 90 look twice the height of a bar of 85. See misleading charts.
• Different scales side by side. Two panels with different maximums look comparable but are not. Use the same scale or state clearly that they differ.
• Forcing huge ranges onto it. When one value dwarfs the rest, a linear scale hides all the variation among the small values.

Linear scales outside of charts

Many everyday devices use linear scales: a ruler, a tape measure, a kitchen measuring jug, a liquid thermometer, a bathroom scale dial. Each mark is the same distance apart and represents the same amount. By contrast, the decibel scale for sound and the Richter magnitude scale for earthquakes are logarithmic. On a map, a "linear scale" (or scale bar) is the bar that shows how far a given ground distance is on the map.