# The Use of Abscissae in Graphs

By Mike Turnbull BAppSc(Distinctions)(QUT) MAppSc(CQU)
Lecturer
School of Information Technology
Faculty of Informatics and Communication,
Bundaberg Campus
Central Queensland Seismic Research Group Central Queensland University

# Introduction

An oft occuring academic debate, concerning the correct usage of the abscissa when graphing data from any of the disciplines of mathematics, but in particular the physical sciences, is discussed in this document. The opinions are that of the author.

The central issue of the debate is whether the abscissa of a graph should always be used to represent the independent data. This is a view held by some of the author's scientist colleagues. In order to approach the debate in a rational manner it is necessary to define the words "abscissa", and "independent", and then undertake a review of the traditional and current accepted usages.

# Definition of "abscissa"

The Oxford English Dictionary (2005) provides the following definition of "abscissa" in relation to its geometric usage.

"Pl. -es; more commonly in the L. form abscissa (æb'sIs), pl. abscissae; also Eng. abscissas.

1. Geom. Literally, a line or distance cut off; spec. the portion of a given line intercepted between a fixed point within it, and an ordinate drawn to it from a given point without it.
In Conic Sections: the segment (or segments) of a diameter (or in a hyperbola, a diameter produced), intercepted between the point where it is cut by an ordinate, and the bounding curve. In Rectilineal Coördinates, the segment of a given line, x, intercepted between the point where it is cut by another line, y, and that in which it is cut by a line parallel to the latter drawn from a given point without it, and called the ordinate." Emphasis is not the author's

The meaning of this definition is rather obscure, and requires some knowledge of geometry, and mathematics in general, in order to understand it. In common palance this definition indicates that an abscissa is a line drawn in any general orientation, to represent any quantifiable data, that is intercepted by another line (called the ordinate). A specific case of an abscissa being used to model the diameter of a conic section is referred to. The definition leaves open the possibility that more than one ordinate may be associated with any particular abscissa. Apart from the implication that the abscissa is some how different from an ordinant, the definition ascribes no other importance to the role the abscissa plays in geometric representation of quantifiable data. Indeed, the definition implies that the role of the abscissa and the ordinate can be reversed, because, just as the ordinate is "drawn to" the abscissa "from a givem point without it", so is the abscissa related to the ordinate.

The Oxford English Dictionary (OED) is renouned for, and claims to present the earliest English usage of the words it lists. It also effectively tracks the common evolutionary usage of words in the English language. This being the case, the author regards the OED as being the pre-eminant etomological authority for common English, and defers to its authority in regard to any debate on the common usage of the word "abscissa" in mathematics, in the context of its usage by English speaking communities.

# What is an Independant Variable

There is an opinion amongst some scientists that an independent quantity is one that can be considered in isolation, as being dependent on no other quantity for its intrinsic value. The author contends that there is no quantity that can be so defined. Nature is such that all physical quantities are necessarily interdependent on some other converse quantity. Indeed the Heisenberg Uncertainty Principle formally states this. There is no physical quantity that can be measured independent of some other converse quantity. It may be debated on a philosophical level that some quantity may exist that is intrinsically independent of the rest of the Universe, but this contention is not supported by scientific evidence.

As an example of what is currently understood by an independant mathematical variable, the Colorado State University Mathematics Department (2004), as part of its Mathematics Placement Examination Review, provides the following definition. (Emphasis added by the author.)

An independent variable is one that may be changed at will, that controls a process, or that determines the value of the other variables. A variable which responds to or whose value is determined by the value(s) of the independent variable(s) is called the dependent variable."

The author accepts this definition as being valid. In particular the author accepts, and agrees, that an independant variable is any variable that, in the context of its usage, may be varied at the will of the practitioner to determine the value of some other variable that is pertinent to the physical situation under consideration. StatSoft (2004) define an independent variable, in the statistical sense, as follows.

Dependent vs. independent variables. Independent variables are those that are manipulated whereas dependent variables are only measured or registered. This distinction appears terminologically confusing to many because, as some students say, "all variables depend on something." However, once you get used to this distinction, it becomes indispensable. The terms dependent and independent variable apply mostly to experimental research where some variables are manipulated, and in this sense they are "independent" from the initial reaction patterns, features, intentions, etc. of the subjects. Some other variables are expected to be "dependent" on the manipulation or experimental conditions. That is to say, they depend on "what the subject will do" in response. Somewhat contrary to the nature of this distinction, these terms are also used in studies where we do not literally manipulate independent variables, but only assign subjects to "experimental groups" based on some pre-existing properties of the subjects. For example, if in an experiment, males are compared with females regarding their white cell count (WCC), Gender could be called the independent variable and WCC the dependent variable.

This definition supports the accepted concepts that an independent variable implies all of the following usages.

• A number representing the value of the variable quantity that is being manipulated by the practitioner during an experiment.

• A variable that the practitioner believes might influence the measured outcome of an experiment. (Simon, 2005).

• A variable in an equation that may have its value freely chosen without considering values of any other variable. For equations such as y = 3x – 2, the independent variable is x. The variable y is not independent since it depends on the number chosen for x. Formally, an independent variable is a variable which can be assigned any permissible value without any restriction imposed by any other variable. (Mathwords, 2004)

# First Review

It is clear from available definitions of the words "abscissa" and "independent variable" that they each convey totally different meanings in mathematical terms, especially in the disciplines of geometry and statistics.

An abscissa is a line used to display geometric and statistical data in a graphical manner, in conjunction with other lines referred to as ordinates. This implies that the values on the ordinates may be derived from the values on the abscissae, either by means of well defined transfer functions, or due to some relative causitive effect - but this is not a necessary relation. Both the abscissa and the ordinate values may be obtained from experimental measurements, with no assumption of cause and effect relationship between the values.

The values modelled by an independant variable (which may or may not be depicted on an abscissa) do not necessarily exist independently and in isolation of other variable quantities (which may or may not be depicted on an ordinate). The role of the independent variable has to be interpreted in the context of the universal data set being modelled. This role may be ambiguous, and can be determined by the practitioner.

# Case Studies

## Thermometer Calibration

The National Institute of Standards and Technology (2004) provides the following information concerning the procedure used to calibrate some commonly used thermometers.

Industrial Thermometer Calibration Laboratory

The Industrial Thermometer Calibration Laboratory calibrates liquid-in-glass thermometers, industrial platinum resistance thermometers, thermistors, and thermocouples in the temperature range -196 ºC to 550 ºC. Calibrations in this laboratory are made by comparison with a calibrated standard platinum resistance thermometer in stirred liquid baths. Fixed-point measurements at the Ga melting point (29.7646 °C) [for a thermometer with an outer diameter of less than 3.6 mm], the water triple point (0.01 °C) and the ice point (0 °C) are available.

 Thermometer Temperature Range IPRT's -196 °C to 550 °C Thermistors -196 °C to 100 °C Thermocouples -196 °C to 550 °C Liquid in glass -196°C to 400°C Digital -196 °C to 550 °C

The process requires that the measured quality of the thermometer being calibrated (be it voltage, current, resistance, length, or whatever) be "compared" with the measured quality of a standard platinum resistance thermometer under identical, controlled, thermodynamic environmental conditions; the standard thermometer having been previously calibrated by subjecting it to reproducable thermodynamic environmental conditions of known effect.

During the process of calibrating the standard thermometer the value representing the reproducable thermodynamic environmental conditions of known effect may be considered to be the independent variable in that process - after all, it is the reproducable thermodynamic environmental conditions of known effect that that number represents, that is causing the measured output value of the standard thermometer to change, not the other way around.

During the process of using the standard thermometer to calibrate a working thermometer, the two instruments are subjected to the same thermodynamic environmental conditions, with the standard instrument being used to quantify the environmental effect as the effect is changed. In practice the output of the standard thermometer is observed while the practitioner alters the environmental conditions to achieve the required output value. In this case it is the output of the standard thermometer that is being used to control the environmental condition, not the other way around. Therefore the output of the standard thermometer can be considered, in the context of usage at that time, to be the independemt variable. It is also of interest to note that, as the environmental conditions are being changed in a controlled manner, the output value of the working thermometer is also being recorded, and compared to that of the standard thermometer. This comparison is generally depicted as a graph, with the measured values obtained from the standard thermometer displayed on the abscissa as temperature (directly tracable to the original calibration chart of the standard instrument), and the measured values obtained from the working thermometer displayed on the ordinate, in voltage, current, resistance, length, or whatever physical quality being exploited. So the values on the ordinate are not directly related to the values on the abscissa; and the values on the abscissa are not in fact the values being measured on the output of the standard instrument at all.

It is clear from this case study that in practical, everyday usage, the perceived independance of data obtained during the calibration and operation of a thermometer changes from process to process. As the perceived independence changes the practitioner's choice as to whether the data is modelled on the abscissa or the ordinate of a display graph also changes.

As an epilogue to this case sudy, it is noted that, in using a working liquid-in-glass thermometer, it is the length of the liquid column that is measured to determine the current ambient temperature. The length of the column is the known quantity, and the temperature is the unknown quantity. When graphing this relation it is customary to put the known quantity on the abscissa and the unknown quantity on the ordinate - even though an adhereance to the causative relation would require the converse on the grounds of the independancy of data being attributed to the temperature.

## Earthquake Epicentral Distance

When an earthquake occurs it generates two distinct energy waves that radiate out from the point where the earthquake ocurred in all directions. These two waves are known as the P wave and the S wave. The P wave travels faster than the S wave. At a monitoring station some distance from the point on the surface beneath which the earthquake ocurred (known as the epicentre) the P wave will arrive before the S wave. Therefore, as the distance from the epicentre to a monitoring station increases, the S-P delta time will also increase. In this sense it can be said that the S-P delta time depends on the epicentral distance, not the other way around.

By recording the measured S-P delta time and associated known epicentral distances of numerous earthquake events, a graph can be drawn to depict the relationship between the two sets of data. The purpose of drawing this graph is so that, when a given S-P delta time is observed for a particular future event, the epicentral distance to that event can be simply read off the graph. Consequently, even though during the calibration phase both the epicentral distance and the S-P times are known values, with the S-P delta time being dependent on the epicentral distance, during the usage phase, it is the epicentral distance that is the unknown quantity. Therefore, during the calibration phase it may be appropriate to model the epicentral distance on the abscissa. Later, during the usage phase, it may be more appropriate to model the epicentral distance on the ordinate.

## Abstract Mathematical Relation

Consider the hyperthetical functional relation defined by the equation:
`d(s) = ms - k      ...     (Eq1)`
In the above equation d is the dependent variable, and s is the independent variable in the context of the function so defined.

By algebraic manipulation Eq1 can be transformed to Eq2.

`s(d) = pd - q      ...     (Eq1)`
In Eq2 it is s that is considered to be the dependent variable, with d being the independent variable in the context of the function so defined.

It is clear that, in this example, any concept of independency of data is a philosopical idology imposed by the observer, that is not suported by the underlying abstract nature of the problem.

# Second Review

In practical usage it is not always the data with perceived independance that is modelled on the abscissa. In the context of purpose it is usually the known data (that is, the data that can be readily measured) that is modelled on the abscissa. In such cases the unknown data (that is, the values that can be derived from the known data) are modelled on the ordinate or ordinates.

In the case of a working thermometer the length of the liquid column is the known quantity not the ambient temperature; even though the column length is physically dependent on the ambient temperature.

In the case of earthquake epicentral distance determination, it is the S-P delta time that is the known quantity; even though the S-P delta time is physically dependent on the epicentral distance.

If a relation is treated as a pure mathematical abstraction the independence of the variables may often be interchanged by mathematical transformation. It is only when an externally perceived reality is imposed on the relation that any concept of independence emerges. This imposition may in some instances enable the observer to give physical meaning to the relation. On the other hand, such an imposition may cause the observer to become blind to the abstract potential of the relation.

# Conclusion

It has been demonstrated that the concept of independence and dependence of related variables is imposed by an external concept of reality. That concept of reality is unique to the observer and is ambiguous depending on the observer's bias. Although it is an established custom to model the perceived independent data on the abscissa of a graph, the ambiguity imposed by different practitioners will necessarily result in conflicts of opinion.

In common practice it is the known values that are usually modelled on the abscissa, with the unknown values being modelled on associated ordinates. Whether the value of a particulat quality is known or unknown changes from process to process. Consequently, a quality that is modelled on the abscissa in a process may be modelled on the ordinate in the converse process.

If modelling your data set on the abscissa makes sense to you, then don't feel bad about it.

# References

Oxford English Dictionary: The definitive record of the English language, OED Online, available at http://www.oed.com/, (Accessed Feb 24 2005).

Colorado State University Department of Mathematics, Mathematics Placement Exam, Review Materials, Section I: Algebra, Dec 15, 2004 (Accessed Feb 24, 2005).

Simon, S., Steve's Attempt to Teach Statistics: Definition: Independent variable, Children's Mercy Hospitals & Clinics, Available onlinr at http://myschoolonline.com/page/0,1871,2491-182599-2-56401,00.html, (Accessed Mar 01, 2005).

StatSoft, Inc. (2004). Electronic Statistics Textbook. Tulsa, OK: StatSoft. WEB: http://www.statsoft.com/textbook/stathome.html. Available online at http://www.statsoft.com/textbook/stathome.html. (Accessed Mar 01, 2005).

Mathwords, 2004. http://www.mathwords.com/i/independent_variable.htm. (Accessed Mar 01, 2005).

National Institute of Standards and Technology, (2004) Indusrial Thermometer Calibration Information, http://www.cstl.nist.gov/div836/836.05/thermometry/calibrations/industrialcal.htm (Accessed Mar 01, 2005).