This page contains six posts in the thread 'Defining "variable"' that were posted to the sci.stat.edu UseNet newsgroup (= e-mail list EdStat-L) between June 28 and July 5, 1996. The authors are Robert Frick (2 posts), Herman Rubin (3 posts) and William Oliver (1 post).

Date: Fri, 28 Jun 1996 11:45:02 -0400 Message-Id: <7395BF061E@psych1.psy.sunysb.edu> Reply-To: RFRICK@psych1.psy.sunysb.edu Originator: edstat-l@jse.stat.ncsu.edu Sender: edstat-l@jse.stat.ncsu.edu From: ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu> To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu> Subject: Defining "variable" I thought Don Macnaughton's definition of variable was brilliant. Last semester I gave my class the problem to name a varible that usually takes the value 1. While this question perhaps can be faulted, and two Ph.D. psychologists solved it slowly, nonetheless they both came up with good answers. Anyway, for most of the class, their answer was not even a variable. Same thing for another class with the easier question to name a variable that usually takes the value yes. So, I suspect that variable is a tough concept for students to grasp. Rereading the standard definitions, they don't seem to make any sense unless you first know what a variable is. The following is my attempt to paraphrase Macnaughton for the research methodology textbook I am writing. People conceptualize the world as containing "things". These things can be physical objects, such as people, buildings, hammers, or cities. They can also be processes, such as driving to school. Each thing has associated with it a collection of properties, and the goal in learning about a thing is to learn its properties. For example, my daughter is 23 pounds, has blond hair, and is cautious; my drive to school takes 15 minutes and has 3 stop lights. For a *collection* of things, the property in question is called a variable. For example, for a collection of children, the properties of weight, hair color, and amount of cautiousness are variables; for a collelction of drives to work, the properties of duration and number of stoplights are variables. For the collection of people in an experiment, whether or not they run and their GPA are variables. For each thing in the collection, the variable takes a particular value. Bob Frick rfrick@psych1.psy.sunysb.edu

Date: Sun, 30 Jun 1996 09:10:25 -0400 Message-Id: <4r4194$2v1k@b.stat.purdue.edu> Reply-To: hrubin@b.stat.purdue.edu Originator: edstat-l@jse.stat.ncsu.edu Sender: edstat-l@jse.stat.ncsu.edu From: hrubin@b.stat.purdue.edu (Herman Rubin) To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu> Subject: Re: Defining "variable" In article <7395BF061E@psych1.psy.sunysb.edu>, ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu> wrote: >I thought Don Macnaughton's definition of variable was brilliant. >Last semester I gave my class the problem to name a varible that >usually takes the value 1. While this question perhaps can be >faulted, and two Ph.D. psychologists solved it slowly, nonetheless >they both came >up with good answers. Anyway, for most of the class, their answer >was not even a variable. Same thing for another class with the easier >question to name a variable that usually takes the value yes. >So, I suspect that variable is a tough concept for students to grasp. > Rereading the standard definitions, they don't seem to make any >sense unless you first know what a variable is. >The following is my attempt to paraphrase Macnaughton for the >research methodology textbook I am writing. > People conceptualize the world as containing "things". These > things can be physical objects, such as people, buildings, > hammers, or cities. They can also be processes, such as driving > to school. Each thing has associated with it a collection of > properties, and the goal in learning about a thing is to learn > its properties. For example, my daughter is 23 pounds, has > blond hair, and is cautious; my drive to school takes 15 minutes > and has 3 stop lights. > For a *collection* of things, the property in question is > called a variable. For example, for a collection of children, > the properties of weight, hair color, and amount of cautiousness > are variables; for a collelction of drives to work, the > properties of duration and number of stoplights are variables. > For the collection of people in an experiment, whether or not > they run and their GPA are variables. For each thing in the > collection, the variable takes a particular value. This is still not too good for random variables, where it is more appropriate. I would define a random variable as something which can be computed knowing the real world situation; it need not be a number. The tie-in with the usual "definition" found in statistics books is that, for every sample space representation, if the representation is adequate for that random variable, the variable is represented by a function on the sample space. You are using the function here as the variable, and this does lead to problems. But for variable in mathematics, a far better "definition" is a pronoun. One might say that a restricted variable is a common noun, which in many respects is closer to pronoun than to proper noun. It is this which should be taught in first grade. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558

Date: Sun, 30 Jun 1996 19:55:41 -0400 Message-Id: <william.oliver-2906962152260001@tele-anx0148.colorado.edu> Reply-To: william.oliver@colorado.edu Originator: edstat-l@jse.stat.ncsu.edu Sender: edstat-l@jse.stat.ncsu.edu From: william.oliver@colorado.edu (Bill Oliver) To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu> Subject: Re: Defining "variable" > In article <7395BF061E@psych1.psy.sunysb.edu>, > ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu> wrote: > >I thought Don Macnaughton's definition of variable was brilliant. > >Last semester I gave my class the problem to name a varible that > >usually takes the value 1. While this question perhaps can be > >faulted, and two Ph.D. psychologists solved it slowly, nonetheless > >they both came > >up with good answers. Anyway, for most of the class, their answer > >was not even a variable. Same thing for another class with the easier > >question to name a variable that usually takes the value yes. > <snip> I've missed this thread, so perhaps I've missed out on why the following informal definition of Kolmogorov's is lacking. "A *random variable* is the name given to a quantity which under conditions S may take various values with specific probabilities. For us it is sufficient to consider random variables that may take on only a finite number of different values. To give the *probability distribution*, as it is called, of such a random variable, it is sufficient to state its possible values..." I find this definition quite clear, but perhaps it's wrong in some way. -Bill -- william.oliver@colorado.edu

Date: Sun, 30 Jun 1996 23:23:19 -0400 Message-Id: <4r56l2$2777@b.stat.purdue.edu> Reply-To: hrubin@b.stat.purdue.edu Originator: edstat-l@jse.stat.ncsu.edu Sender: edstat-l@jse.stat.ncsu.edu From: hrubin@b.stat.purdue.edu (Herman Rubin) To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu> Subject: Re: Defining "variable" In article <william.oliver-2906962152260001@tele-anx0148.colorado.edu>, Bill Oliver <william.oliver@colorado.edu> wrote: >> In article <7395BF061E@psych1.psy.sunysb.edu>, >> ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu> wrote: >> >I thought Don Macnaughton's definition of variable was brilliant. >> >Last semester I gave my class the problem to name a varible that >> >usually takes the value 1. While this question perhaps can be >> >faulted, and two Ph.D. psychologists solved it slowly, nonetheless >> >they both came >> >up with good answers. Anyway, for most of the class, their answer >> >was not even a variable. Same thing for another class with the easier >> >question to name a variable that usually takes the value yes. ><snip> >I've missed this thread, so perhaps I've missed out on why the following >informal definition of Kolmogorov's is lacking. >"A *random variable* is the name given to a quantity which under >conditions S may take various values with specific probabilities. For us >it is sufficient to consider random variables that may take on only a >finite number of different values. To give the *probability distribution*, >as it is called, of such a random variable, it is sufficient to state its >possible values..." >I find this definition quite clear, but perhaps it's wrong in some way. First of all, to understand random variables, we must eliminate the connection with probability distribution as essential. Kolmogorov was coming at the whole problem as a means of "defining" probability, and it is this which causes problems by itself. Probabilty, TO BE USED, must be something in the real world, which is only represented in the mathematical world. However, it is only in the mathematical world which we can understand the ideas of calculation. So it is necessary to carefully model the real world in the mathematical world. The representation is not what is being represented. An oversimplified representation may, in some cases, be a reasonable approximation for the solution of a particular type of problem, but it blocks understanding of the general problem. So: There is no "unique" sample space for a situation. Different representations are appropriate, and one can even have representations for aspects of a problem; one can move between representations as is convenient. If the various representations are compatible, any answers will not change. The representation of a random variable, which need not be numeric, corresponds to a function on the sample space used, if that sample space is adequate. This is not a matter of measurability; in a problem of tossing a coin more than once, the result of the first toss is not representable on the sample space consisting of the non-negative integers, where the representation is the number of heads. When one looks at a fixed function on a fixed space, there is nothing random or variable about it, but it can represent a random variable. The randomness comes from the fact that there is a mapping of the real world phenomenon into the sample space. This mapping, which is essentially a dictionary, is what is not stressed in teaching, but which is important in understanding. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558

Date: Tue, 2 Jul 1996 13:11:23 -0400 Message-Id: <D4F8874150@psych1.psy.sunysb.edu> Reply-To: RFRICK@psych1.psy.sunysb.edu Originator: edstat-l@jse.stat.ncsu.edu Sender: edstat-l@jse.stat.ncsu.edu From: ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu> To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu> Subject: Re: Defining "variable" Bill Oliver asked what was lacking with the standard definition: > "A *random variable* is the name given to a quantity which under > conditions S may take various values with specific probabilities. For us > it is sufficient to consider random variables that may take on only a > finite number of different values. To give the *probability distribution*, > as it is called, of such a random variable, it is sufficient to state its > possible values..." > > I find this definition quite clear, but perhaps it's wrong in some way. I once thought that my definition of "experiment" was perfectly clear to students, until I discovered that I was using "manipulate" in a technical sense and the students weren't. (Suppose in an experiment testing if music can reduce sadness, I first make all of my subjects sad. Students tend to think, quite appropriately, that I have manipulated sadness.) So "Something that can take two or more values" makes perfect sense to me, but I know what a variable is. What would a student make of this definition? Speculations: 1. One good example would be the price of a car. One price is listed, but the salesman is willing to sell it for a second price. Well, perhaps "price of car" is a variable, but not for the student's reason. 2. Another good example would be a compass. Normally a compass has no value, but if you were lost, it would be very valuable. I think these examples fit the definition fairly well. I don't know what subjects actually make of the standard definition, I just felt that I was having trouble teaching the concept of variable using the standard definition and that after reading Macnaughton's definition, the standard definition felt shallow and incomplete. Perhaps importantly, I was trying to teach the definition of a variable in the context of an experiment, which might be different from the context of algebra. Bob Frick rfrick@psych1.psy.sunysb.edu

Date: Fri, 5 Jul 1996 22:27:21 -0400 Message-Id: <4rj36c$v3c@b.stat.purdue.edu> Reply-To: hrubin@b.stat.purdue.edu Originator: edstat-l@jse.stat.ncsu.edu Sender: edstat-l@jse.stat.ncsu.edu From: hrubin@b.stat.purdue.edu (Herman Rubin) To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu> Subject: Re: Defining "variable" In article <D4F8874150@psych1.psy.sunysb.edu>, ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu> wrote: >Bill Oliver asked what was lacking with the standard definition: >> "A *random variable* is the name given to a quantity which under >> conditions S may take various values with specific probabilities. For us >> it is sufficient to consider random variables that may take on only a >> finite number of different values. To give the *probability distribution*, >> as it is called, of such a random variable, it is sufficient to state its >> possible values..." >> I find this definition quite clear, but perhaps it's wrong in some way. >I once thought that my definition of "experiment" was perfectly >clear to students, until I discovered that I was using "manipulate" >in a technical sense and the students weren't. (Suppose in an >experiment testing if music can reduce sadness, I first make all of >my subjects sad. Students tend to think, quite appropriately, that I >have manipulated sadness.) >So "Something that can take two or more values" makes perfect sense to >me, but I know what a variable is. What would a student make of >this definition? This is one of the places where attempts to make things too simplistic result in confusion. There is too much of a tendency to try to DEFINE, rather than to CHARACTERIZE. Now some things must be defined, to start out and have an adequate language. But the adequate language here is mathematics, and the idea of representation of a "real world" situation by a mathematical model. Speculations: >1. One good example would be the price of a car. One price is >listed, but the salesman is willing to sell it for a second price. >Well, perhaps "price of car" is a variable, but not for the >student's reason. >2. Another good example would be a compass. Normally a compass has no >value, but if you were lost, it would be very valuable. >I think these examples fit the definition fairly well. I don't know >what subjects actually make of the standard definition, I just felt >that I was having trouble teaching the concept of variable using the >standard definition and that after reading Macnaughton's definition, >the standard definition felt shallow and incomplete. Perhaps >importantly, I was trying to teach the definition of a variable in >the context of an experiment, which might be different from the >context of algebra. There are too quite different uses of "variable" here. One of them is the use as a key part of the language of mathematics; it should be added to the more customary languages as a means of precise communication. The other is as "random variable". This is very definitely NOT the same. And in neither case is the above intuition even self-consistent. After the restrictions got worked out, the linguistic use of variable is essentially that of pronoun. Mathematics is a grammatically simple language, with a few linguistic constants, and the rest consists of names for "objects" (this is not a good term) upon which restrictions MAY be imposed, but are not inherent. These names can be changed; this is the use of substitution. In some cases, it can be shown that a particular name refers to exactly one mathematical object; this is the general introduction of constants, they are variables which can only take on one value. I repeat that this is linguistic, but it is the most important part of mathematics for the non-mathematician. The applied problem, translated into this terminology, becomes a problem on which the full power of mathematics can be unleashed, just as the typesetter and press operator do not need to understand the book. Now what about random variables and related items? There is a real world situation, which calls for probabilistic or statistical interpretation and advice for action. Again, this needs to be modeled, and there is no one way of modeling. In fact, one may want to use different models, with different amounts of coverage, in the same problem. If something can be computed in different correct models, the results are equal. One should start out with a sample space, which is a purely abstract set, together with a dictionary, such that each real world outcome corresponds to exactly one point of the set. Insisting that each point of the set is attained is neither necessary nor desirable. It may not even be known if a point is attained, but including it does absolutely no harm. Now a random variable is something which can be computed from the state of the real world. It can be REPRESENTED by a function on a sample space, if all real world occurrences giving rise to any point in that sample space give the same value to the random variable. The function doing the representation is a fixed function on a fixed space; the randomness in the real world is contained entirely in the representation. And it is even possible that the random variable is a constant. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558

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