I'm a fan of the New Jersey Devils. Laugh, point, sneer, do as you will. But besides being a fan of a team with a less than "family-friendly" fan-base and an inability to score goals, the Devils are also known for one thing, goaltending. Having grown up watching the greatest goalie ever, Martin Brodeur, I know have the distinct pleasure of watching Corey Schneider play in goal. My function blog will track Schneider's save percentage over the last six seasons.
Year SV%
'08-'09 .877
'09-'10 .915
'10-'11 .929
'11-'12 .937
'12-'13 .927
'13-'14 .924
2. In order for a relationship like the one above to be a function, it must meet a simple requirement. Per every input, there must be only ONE output. Now, while a goalie has numerous stats, focusing in on save percentage over six years does leave us with a function relationship. In other words for every year (the input of the relationship) we have a distinct save percentage (the output of the relationship).
4. Like I just said above the one-to-one relationship above makes this a function. X=f(t) =>
SV%= f(Y). For every year, we have one and ONLY ONE output for save percentage, giving us a function!
5. It is pretty obvious just by looking that the numbers that we don't have a linear function. But I'll use the Y2-Y1/X2-X1 formula just to prove it.
.915-.877/2010-2009= .038/1 or .038.
Now
.929-.915/2011-2010= .014/1 0r .014
For clarification, a lower number like .014 while proving this function IS NOT LINEAR, is a good thing for a fan like me, who wants small changes in a save percentage from their goalie.
7. The math I did above proves this isn't a linear function. A linear function is identified as a function who's average rate of change (the change from output to output) is the same between each input level. But as I explained above, the average rate of change is mostly decreasing from level to level, and not staying the same. So this is not a linear function.
8. This isn't a mathematical model, even if it meets some of the requirements. For a function to be a mathematical model, it must 1. Explain a real world event and 2. Have an exact one input to one output relationship. Moreover, it must prove accurate when used in equation form. While it is true for this function, there is one issue. A mathematical model should be able to predicate future results, by which I mean, add in a new input and the model should give you the correct output. But, the data here can't be predicted ahead of time. Schneider's current save percentage may not even be now what it will be what it is at season's end. And since we can't plug in a Y value in the future and get the right "x" value, it isn't a mathematical model.
Part B:
This will be quick, but hopefully informative. Outside of the one output per input law, you can also visually test a graph to see if it is a function using the vertical line test. This test states that a graph can be recognized as a function if a vertical line drawn through it only bisects the graph one time. Also, both input and output need to be dependent on one another, you should not be able to derive one without knowing the other.
This article,
http://www.ala.org/bbooks/frequentlychallengedbooks/top10, lists the most challenged books per year. These books have been banned, or brought up for discussion as unsuitable for at least some group of readers for multiple reasons. Every year acts as the input, plugging that in to an equation would yield all of the books most commonly challenged that year. And, most of the books when plugged into the equation will yield only one year in which they were challenged heavily. It should be noted that there is some dependency within the variables, at least for this function.
But here's the thing, every year has more than one book connected to it. In fact, each year (acting as the input) yields multiple challenged books (meaning multiple outputs) so it fails to follow the one to one rule. And furthermore, if we graphed this, even one year's worth of data wouldn't pass the vertical line test.
With that in mind, the data in the article does not yield a function!
I like the way you explained every single thing.
ReplyDeleteOne, I am so sorry for your lack of a season
ReplyDeleteIts disappointing that the rest of the team can't get their act because the numbers have shown that Schneider isn't playing terrible. I am always a fan of hockey references so I like the use of his goal stats to show what is a function. You also did a good job showing the ROC. I also like the example of a non-function with the books but it seemed a little rushed, so it was hard to follow at some points, but it puts multiple inputs and non-functions into an awesome and fun light
grant,
ReplyDeletereally good job on both of these examples. i love that you used hockey as an example for the first. i, too, am a hockey fan of a team that no one really thinks about, the avs. your explanations for the first example are very well explained, especially the part about why it is not a mathematical model.
i thought and thought about your second example. i love the topic...i'm all about banned books. there are several ways to look at the relationships happening in your second example that would actually make the relationships separate functions. but if we look at it the way that you have explained it, i believe that you have an argument for saying that it is not a function. i couldn't find a counter-argument, so good job!
professor little