An Explanation of Ratio Scales in Graphs

by Harry Browne

A graph uses one of two basic types of scales a linear scale or a ratio scale.

With a linear scale, the physical distance on the scale from, say, $1 to $2 is the same as the distance from $2 to $3.

On a ratio scale, the intervals between amounts are determined by their ratio to each other. Thus the distance from $1 to $2 is the same as the distance from $2 to $4 or from $3 to $6 because each represents an increase of 100%.

A ratio scale makes it possible to compare at a glance the magnitudes of changes that occur at different ranges. For example, if an investment rises from $1 to $2 during one time period, and later in the graph it rises from $3 to $6, a ratio scale will make it obvious that each rise was of the same magnitude. A linear scale would make it appear that the second rise was proportionally three times as great as the first rise.

On a linear scale, a vertical inch (or any other distance) represents the same number of units (such as dollars) wherever it appears on the graph. On a ratio scale, a vertical inch represents the same degree of growth wherever it appears on the graph.

A ratio scale demonstrates comparisons, growth, relationships between items and between time periods. It is especially useful for plotting two or more items on the same graph, because it's the only way to compare the growth of two items starting at different levels. If you compare them on a linear scale, the apparent differences between them will be meaningless.

If a graph bears no indication of which kind of scale it's using, it probably is a linear scale. If it's a ratio scale, there should be a label on it somewhere saying ratio scale, log scale, or semi-log scale.

Why Ratio Scales

To see the reason a ratio scale is important, look at this graph. It shows a trend that appears out of control. After making a slow start, the subject of the graph appears to be growing faster and faster and near the end of the graph it is exploding.

What is this thing that's racing to the sky? Is it the federal debt careening on the road to national bankruptcy? Is it some investment in the last stages of a runaway bull market climbing toward its final blowoff?

Well, as a matter of fact, it's neither of those things. It's the yearly value of a 5% passbook savings account with interest compounded year after year. Hold on to the account long enough and the curve will rise with a steeper and steeper arc, even though each year's value is only 5% greater than the prior year's just as at the beginning of the trend.

Sooner or later, a linear graph will turn any constant growth rate into a picture of a skyrocket. It distorts comparisons because it makes the change from 100 to 120 appear to be 20 times as large as the change from 5 to 6 even though each is, proportionately, a 20% increase.

Now look at this graph. The same 5% growth rate is plotted on a ratio scale. Each horizontal gridline represents the same percentage growth from the previous gridline. Since the growth is a constant 5% per year, the plot line is perfectly straight because 5% represents the same degree of gain towards the end of the graph as it did at the beginning.

Economists and mathematicians almost always use ratio scales whenever rates of growth are important.

Ratio scales are appropriate for investment prices, sales figures, income, or any other absolute amounts being plotted over a period of time. They should not be used to plot anything in which a relationship is already inherent in the amounts such as percentages (like the inflation rate), ratios between two items (such as a gold-silver ratio or price-earnings ratio) because the benefit provided by a ratio scale is already built into the figures being plotted.

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Harry Browne was the Libertarian Party presidential candidate in 1996 and 2000, and is now the Director of Public Policy for the American Liberty Foundation. You can read more of his articles at HarryBrowne.org.