The following figure shows counts of behavior and portrays those counts graphically in sets. The counts can represent any learner behavior the teacher may wish to measure (e.g., interrupting a conversation, volunteering to do a classroom chore, writing antonyms or factoring trinomials). The dots represent the physical amount for each count (e.g., for 2, the two dots show 2 written antonyms, or two competed chores).
When displayed on charts, the counts measured in the above Figure transform into an ordered arrangement of data. The resulting visual pattern provides information to the chart reader beyond just examining numbers or counts alone. When displayed on an arithmetically scaled line chart, Figure below, the differences between 1 and 2, 2 and 4, and 4 and 8 all have disparities in the amount of space given to represent the change. The change from one to two has the smallest space because it has the least amount of total change. Conversely, going from 4 to 8 we see the largest amount of space showing the greatest amount of total change. With arithmetically scaled charts the relationships between quantities will always appear as shown in this example because the architecture of the graphic (i.e., arithmetically scaled line chart) has the specific design for showing absolute amount changes. Arithmetically scaled line charts exist to show how much more (amount) or less (amount) a quantity changes. Teachers who use these charts place all of their analytical ability on knowing only whether an intervention produced a greater or lesser amount of the targeted behavior.
The displays on ratio charts, or charts with semilogarithmic or multiply/divide scales, illustrate the proportionality between the differences between 1 and 2, 2 and 4, and 4 and 8. The amount of space appears equal on the cross section for the charted data with all three data sets because all have a proportional change of x2.0, or a doubling. Semilogarithmically scaled or ratio line charts will always show absolute changes but focus the eye of the chart reader on relative and proportional effects. The objective of the resulting visual pattern between the two types of time series graphics, arithmetically scaled and semilogarithmically scaled line charts, seems at odds. How can a substantially larger amount of change not always convey greater significance when compared to a smaller absolute amount of change? The example in the differences between a cross-section of an arithmetically (above) and a semilogarithmically scaled or ratio line chart (below) help to answer the question.
It follows that analysis of visual data occurs as a principle result of the dimensions inherent in the graphical display. All the arithmetically and semilogarithmically scaled line charts will have critical features that impose order to the data set such as scales, data labels and an aspect ratio. The resulting data picture significantly affects the chart readers analysis, interpretation, judgement, and subsequent decision making. Decisions based on only what provides more or less (the middle figure) provides very different information from a visual display (above) showing how much things change but also within the context of what they change from (relative change) and how they change proportionally. Contrasting the two visual displays goes to the heart of how we determine significance of changing quantities.
4 to 8 is +4 while 1 to 2 is +1. If we only care about which provides more then we value 4 to 8 change more than 1 to 2 change. But 1 to 2 change = 100% change and 4 to 8 = 100% change. On arithmetically scaled line charts, visually people see the 1 to 2 change the same as as 7 to 8 change (both show +1) but 1 to 2 = 100% percentage change while 7 to 8 = 14% percentage change. In the add world, and with arithmetically scaled line charts, we value the change of 1 to 2 as much as we value the change of 7 to 8. We should get very excited, however, at the 1 to 2 change, and not so much with 7 to 8 change when we look at the data change in the multiply world and the ratio line charts. And if you don’t think relative change is more valuable or significant than absolute change, answer this question: if you invested your money which return rate would you rather have – a 100% return rate or a 14% return rate?
People always, always answer that question the same way, they say they want a 100% return rate over the 14% return rate. How do we calculate return rates? We must use the first number, where we start from, to the second number. Yes, if you said to me “Rick do you want $1,000 or $10 I will say “I want $1,000.” But if you say to me, “Well you have to give me $10,000 if you want $1,000” I might not have $10,000. But I do have $100 so I can get the $10 deal (both of these figures, the $1000 and $10 come from a 10% calculation). When the fields of education and psychology place so much stock in only absolute amount of changes, epitomized by arithmetically scaled line charts, we fail to appreciate true significance of changes and only look for what produces the biggest changes ignoring the facts of percentage changes.
The technical features (i.e., showing relative and proportional change) of a ratio or semilogarithmically scaled chart line chart make one wonder why it has never taken hold in scientific disciplines like education and psychology. According to Schmid and Schmid (1979, p.99), “The semilogarithmic chart is unequaled for many purposes, especially in portraying proportional and percentage relationships. In comparison with the arithmetic line chart, it possesses most of the advantages without the disadvantages. This type of chart not only correctly represents relative changes, but also indicates absolute amounts.” In their book Engineering graphics, Giesecke et al. (2001, p.846) reinforce Schmid and Schmid and other authors when they write, “Semilog charts have the same advantages as rectangular coordinate line charts (arithmetic charts)…. When rectangular coordinate line charts give a false impression of the trend of a curve, the semilog charts will be more effective in revealing with the rate of change is increasing, decreasing, or constant.”
Consider the ratio chart. And if you consider the ratio chart, consider one of its most sophisticated forms – the Standard Celeration Chart. You will see significance of data change in a whole new light.
Giesecke, F. E., Mitchell, A., Spencer, H. C., Hill, I. L., Loving, R. O., Dygdon, J. T., & Novak, J. E. (2001). Engineering graphics (8th ed.). Upper Saddle River, NJ: Prentice Hall.
Schmid, C. F., & Schmid, S. E. (1979). Handbook of graphic presentation (2nd ed.). New York: John Wiley & Sons.