Reading graphs: axes, scales, and the comparison that matters

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Translate a graph into quantities and comparisons that stay valid when its display changes.

About 14 minutes, including the worked examples and checks.

Read a study and its figures · Lesson 4 of 6 · Continue in this path

Concept: read the coordinates before the shape

A curve is a way of displaying measurements. Its apparent steepness depends on the units, axis ranges, and scale used to place those measurements on the page. Before deciding what a graph means, translate a point into a sentence: “At this value of x, the measured value of y was this.” That sentence anchors the interpretation to quantities rather than to the picture’s proportions.

In an experiment, an x-axis often shows the variable researchers manipulated and a y-axis the response they measured. But axis placement alone does not establish manipulation or causation. A scatterplot can place two observed variables on the axes. Read the methods and caption to learn what was done. AAMC data-reasoning skills include interpreting relationships and evaluating whether a representation fits the observations.¹

Check four things before comparing curves: the quantity and unit on each axis, the tick values, what each mark represents, and the stated experimental comparison. A point might be an individual measurement, a mean, a ratio, or a model prediction. An error bar might describe variability or uncertainty. The symbol’s meaning must come from the figure, not from its familiar appearance.

Pattern: equal distances can mean equal differences or equal ratios

On a linear axis, equal physical distances represent equal numerical differences. The distances from 0 to 10 and from 10 to 20 are equal. On a logarithmic axis, equal distances represent equal ratios. On a base-10 log axis, the positions 1, 10, 100, and 1,000 are equally spaced because their logarithms are 0, 1, 2, and 3.

The tick labels usually retain the original units even though their positions use a transformation. “100 nM” is still a concentration of 100 nM; it is not a response value of 2. For positive x, log₁₀(x) is the exponent needed to obtain x from base 10. Zero and negative values cannot appear on an ordinary logarithmic axis because their real logarithms are not defined.²

The midpoint between 10 and 100 on a log₁₀ axis is therefore 10^1.5, approximately 31.6. It is not the arithmetic midpoint, 55. More generally, the midpoint between positive values a and b on a log axis is √(ab), their geometric mean. This follows by averaging log(a) and log(b), then converting back.

Application: one dataset, two x-axis scales

The table gives an illustrative assay response at four concentrations. “AU” means arbitrary signal units on the same calibrated assay scale. The points are supplied measurements; no variability or fitted biological model is provided.

In the linear display, the first three concentrations occupy only the first tenth of the full 0–1,000 nM range. Their x-coordinates are close together. In the logarithmic display, each tenfold concentration increase occupies the same horizontal distance, so the low-concentration observations become easier to separate. The transformation changes the display, not the observations: 100 nM still corresponds to 60 AU in both panels.

The response increases across all four measured concentrations. From 10 to 100 nM, it rises by 35 AU. From 100 to 1,000 nM, it rises by 30 AU. These changes occur across very different absolute concentration intervals, 90 versus 900 nM. Their average slopes on the original variables are 35/90 ≈ 0.389 AU/nM and 30/900 ≈ 0.033 AU/nM.

On the log-x display, those same intervals each span one decade, so their changes can instead be expressed as 35 and 30 AU per tenfold increase. That is a different comparison. A nearly straight segment on a log-x graph does not establish a constant response increase per nanomolar. Always name the denominator of the slope you are interpreting.

Rule: distinguish display, interpolation, and extrapolation

Lines joining observations can guide the eye without representing a fitted mechanism. A straight connecting segment is not evidence that the system truly behaves linearly between its endpoints. Likewise, a visually flattening curve does not by itself identify a particular enzyme mechanism, receptor number, or limiting process.

Interpolation estimates a value inside the measured range. For example, estimating the response at 300 nM would interpolate between the observations at 100 and 1,000 nM, but the numerical answer depends on a stated model. Linear interpolation in concentration and linear interpolation in log concentration give different estimates. If no model is supplied, acknowledge that dependence instead of treating a line on the page as an extra measurement.

Extrapolation extends beyond the observed range. Predicting a response at 10,000 nM is more demanding here because there are no observations above 1,000 nM. The data do not establish a maximum response of 100 AU or exclude a later decline. A numerical prediction may be reasonable if a model is independently justified, but it is a prediction under that model rather than another observed result.

Application: a changed baseline changes the visual impression

Suppose two bars represent 80 and 88 AU. A y-axis beginning at zero makes their difference look modest. An axis beginning at 78 makes their visible heights 2 and 10 display units above the baseline, producing a much more dramatic impression. The actual difference remains 8 AU, and the response ratio remains 88/80 = 1.10.

For a bar chart, readers commonly interpret bar length as representing magnitude, so a truncated baseline can make length ratios misleading. For a line or scatterplot, a restricted y-range may legitimately reveal small changes, provided the axis is clear and the interpretation uses the values. A nonzero baseline is a feature to inspect, not automatic proof that a graph is invalid.

Finish with an invariant conclusion: one that survives a faithful change in display. “The measured response at 100 nM is 60 AU” survives. “The high-dose point is three times as tall on the screen” depends on the display. Translating visual impressions into values, units, and stated comparisons is the step that makes graph reading reliable.

Check your reasoning

Choose an answer before opening its feedback. Each option explains the reasoning behind that choice.

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Read a study and its figures

Lesson 4 of 6.

• Previous in Read a study and its figures: Observational and experimental studies: what can we conclude?
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Related lessons

• Normalized data, percent change, and fold change
• Mean, median, and spread: describing a dataset
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Sources

1. AAMC: Scientific Inquiry and Reasoning Skill 4

2. OpenStax Precalculus 2e: Graphs of Logarithmic Functions