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Especially in mathematical usage, do variable and parameter mean the same thing? If not, can you explain the difference in meaning and usage?

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Variable and parameter have clearly defined meanings in specific contexts. What these words mean in general English usage may not necessarily apply in mathematical, programming and other scientific/ technical contexts. As for 'mathematical usage', No, they do not mean the same thing. –  Kris Dec 26 '11 at 5:47
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So far, the answers are in the context of programming. For meanings in mathematics (unless someone provides one here) ask in a mathematics site... –  GEdgar Dec 26 '11 at 15:53
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The difference in mathematical statistics is very clear-cut. Suppose you have a set of data which you believe behaves according to a quadratic law

y = a x2 + b + c.

So you have observed lots of pairs of values (x,y)—with errors—and you are trying to estimate a, b and c. Then x and y are called variables and a, b, and c are called parameters.

In general mathematical terminology, the definitions are essentially the same. For functions, the inputs and outputs are variables, and other values in the definition of the function that might possibly change are called parameters. Here, however, you are not trying to estimate the parameters from observed values of the variables, so the definitions are somewhat fuzzier; you could treat the same object as a variable and as a parameter in two different sections of the same mathematical paper.

Note that the above definition does not apply to parametric equations, where the variables in a function are defined in terms of a parameter which does not appear in the function.

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Would you say than if I see y = y(x) then a,b,c are parameters, whereas if I see y = y(a,b,c) then x is a parameter? –  Kavka Dec 26 '11 at 20:30
    
@Kavka: I would. –  Peter Shor Dec 26 '11 at 20:33
    
Your answer seems only to add to the confusion. While you seem very clear about which is a variable and which a parameter in your example, you are thoroughly confused as to why they happen to be so, because you do not understand what actually defines a variable and a parameter in mathematical use. –  Kris Dec 27 '11 at 4:03
    
@Kris: then you should write your own answer. See the Wikipedia article on parameters, in particular, the section on Mathematical Functions. Except for the first paragraph of this section (in which I can't understand what they're saying) and the mention of parametric equations (where the word "parameter" is used for variables that play a somewhat different role), my definition is the same as theirs. –  Peter Shor Dec 27 '11 at 11:38
    
I have not attempted to answer because the answer is not related to the English language and not simple. However, misconceptions need to be avoided. Which is why I tried to clarify that the definitions are more complex than you imply. see: 'Note that the above definition does not apply to parametric equations, ...' from your own reference. –  Kris Dec 27 '11 at 12:19
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Note: This answer is with regards to programming, not mathematics.

Variable is a term used generically refer to a storage location, usually in memory, holding data for a program. (Sources: C# language definition, Wikipedia). I like this explanation from about.com:

Imagine a very large warehouse with lots of storage bays, tables, shelves, special rooms etc. These are all places where you can store something. Lets imagine we have a crate of beer in the warehouse. Where exactly is it located?

The variable is the location of the crate, given a symbolic name like "beerLocation" so you can easily reference it.

Of course, variable in the literal sense means "able to vary", so in some contexts this can be used as a means of differentiating between data that can be altered (variable) and data that is fixed (constant).

A parameter is a special type of variable, used in reference to data passed to functions/subroutines/macros. Sources: MSDN doc, Wikipedia. There is some subtle difference between parameters and arguments, but they are often used interchangeably.

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Variable refers to it has a changing values or context . A parameter is a special kind of variable, used in a subroutine to refer to one of the pieces of data provided as input to the subroutine.These pieces of data are called arguments. An ordered list of parameters is usually included in the definition of a subroutine, so that, each time the subroutine is called, its arguments for that call can be assigned to the corresponding parameters.

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A parameter is a variable? Always? Also, see @Apoorva. –  Kris Dec 26 '11 at 12:17
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Variable is a parameter whose value might change during the execution of code. Parameters are values that are passed to a function. It could be either variable or constant.

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Are all variables parameters? –  Kris Dec 26 '11 at 12:17
    
In programming sense i would say yes.. All variables are parameters but the reverse is not true i.e., all parameters are not variables.they might also be constants. –  Apoorva Dec 27 '11 at 12:49
    
A program may use its own internal variables in addition to parameters passed to it -- right? –  Kris Dec 27 '11 at 12:51
    
It depends on what parameters you would wish to pass to a particular method or a function. –  Apoorva Dec 27 '11 at 12:56
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I don't believe this answer is correct - parameters are normally used to refer to data passed into a given scope, whereas variables are data declared within a given scope. They are not parameters because they are not passed in. –  Lynn Dec 29 '11 at 15:36
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