Semilogarithmic chart

Equal ratios receive equal space

On a base-10 logarithmic vertical axis, 1, 10, and 100 occupy evenly spaced major cycles. The movement from 1 to 10 is ×10, and the movement from 10 to 100 is also ×10. Equal ratios appear as equal vertical distances even though the additive increases are 9 and 90.

Values within a cycle are unevenly spaced because the axis represents logarithms. The distance from 1 to 2 is larger than the distance from 8 to 9. Read values from the labeled scale rather than estimating them as though the axis were linear.

Proportional and additive displays answer different questions

An equal-interval graph gives the same space to the change from 5 to 10 and 50 to 55 because each adds five. A semilog chart gives the same space to 5 to 10 and 50 to 100 because each doubles.

Use the semilog format when multiplication, division, percentage rate of change, or stability across a wide range matters. Use an equal-interval graph when absolute differences are central. Keeping the raw values permits both views without changing the evidence.

Zero needs a declared convention

The logarithm of zero is undefined, so zero cannot appear at an ordinary position on a logarithmic axis. A chart may use a separate zero line, a floor value, a special symbol, or another declared convention.

Never silently replace zero with a small positive number. That changes the data. Missing, ineligible, below-detection, and measured zero are different states and need different records.

When zero occurs often and is clinically important, a linear graph or paired display may be easier to understand.

A fictional frequency example

Lina practices a defined typing response in one-minute timings. Correct frequencies are 1, 2, 4, 8, and 16 per minute across five weekly probes. Each week roughly doubles the prior value.

On a semilog chart, the points form an approximately straight rising path because the proportional change is stable. On an equal-interval chart, the later increases look much steeper because the absolute gains grow from 1 to 8 responses per minute.

The straight path supports a description of this observed series. Five points do not guarantee continued doubling or identify a cause. Practice, instruction, measurement, and other concurrent changes remain part of interpretation.

Slope represents proportional change

With fixed chart conventions, a straight data path indicates a relatively constant multiplication or division rate across equal time periods. A steeper upward slope reflects faster multiplication; a downward slope reflects repeated division.

Celeration expresses change in frequency per unit of time, often with a multiplier such as ×2 per week or a divider such as ÷2 per week. State the chart and calculation convention because time bases can differ.

Reciprocal changes deserve careful wording. Moving from 4 to 8 is ×2, while moving from 8 to 4 is commonly described with a divider such as ÷2. Saying “minus 50%” introduces an additive percentage frame and can obscure the chart’s multiply-divide convention. State the starting value, ending value, and time period so another reviewer can reproduce the label.

Changing the time base also changes the celeration expression. A daily multiplier and a weekly multiplier can describe the same data with different numbers. Compare slopes only after confirming the same chart geometry and time unit.

Write the chosen time base beside the estimate.

Check the chart before interpreting it

Verify:

• logarithmic axis base and labeled cycles
• frequency, duration, and observation units
• horizontal time unit and spacing
• zero, missing, and below-floor conventions
• raw-value agreement
• phase labels and implementation dates
• whether automated trend or celeration lines use the intended points
• accessible data table and text explanation

Chart accuracy can be measured as required specifications matching the approved chart rule divided by specifications due for review. Interpretation quality needs a separate clinical review.

The chart does not create comparability

Two frequencies remain incomparable when their response definitions, timing durations, opportunities, or observation contexts differ. A common scale may make them appear orderly without fixing the measurement problem.

Likewise, a straight line does not establish a functional relation. Experimental design, fidelity, replications, and contextual evidence determine what conclusions the data can support.

The BACB BCBA Test Content Outline, sixth edition includes graphing and visual analysis. A precision-teaching synthesis describes equal-ratio charting and celeration, while a linear-versus-ratio graph comparison illustrates how scale format affects visual judgments.