Is it ok to scale a graph with "shifting" values?

[dethwing]

Kind of an obscure question, but it came up tonight when a student turned in his project. To quickly sum up, his horizontal axis included the points: 2, 4, 8, 16, 32, all with the same amount of space.

That is, it "looked" like each space was the same length, but the label indicated it wasn't.

Now, his vertical axis was done in the same way, and the points did indeed match up, but this strikes me as a cop out. Instead of making a reasonable scale for both directions, he just forced each point to be what he wanted. End result, the line "looked" like y = x, even though he was graphing y = 9x.

Now I told him I didn't think it was realistic and he went nuts on me, swearing and storming out, so I wanted to get some other opinions on it.

Thoughts?

[Lucresia]

I think it should be consistent; I disagree with this student. In my opinion, the inconsistency does not provide a true graphical representation of the data. I mean, yes it includes the correct data, but I think the realism and representation of the graph itself is distorted. It kinda ruins the point of having a graph imo..
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[Quailman]

Would an exponential scale have worked in this situation?

[Zag]

A logarithmic scale is reasonable for lots of applications. It was consistently logarithmic, so I think it should be ok.

[dethwing]

If it was an exponential, or logarithmic, function, sure. But this was linear.

[bgg1996]

[Dread Pirate Westley]

What Zag said. When the situation calls for it, I have been known to plot things on a log scale. We have at least one instrument at work for which it is the default. Granted, I usually use base 10.

[dethwing]

The equation in question was V = 9I [Volts and Current]. I'm not sure how a logarithmic scale is useful in that context.

[Thok*]

If y=9x, then ln y = ln x + ln 9, or in the new variables y'=x'+ln9.

So it's consistent, and it might save some space.

(32*9=288 units is a lot, and the other alternative is to compress the y axis by using a different vertical scale to the horizontal scale.)

[Zag]

You're going to be unhappy with me, here, but I'm going to say that it was exactly the right scale to use for that purpose (graphing voltage vs. current).

Obviously, when graphing algebraic equations, it doesn't actually matter what scale you use, because you aren't really intending to 'learn' something from the graph. (You're just supposed to learn algebra from the process, not anything from the graph itself.) So I got to thinking about what sorts of cases does it make sense to use a logarithmic graph.

The first case I thought of was for a linear-log scale, which is used when graphing something that you expect to have roughly constant growth, and what you are looking for is the deviations from that. For instance, here is the entire history of the Dow Jones Industrial Average, on a linear-log scale. (I guess our url command doesn't like that url -- you'll have to copy/paste.)

http://finance.yahoo.com/echarts?s=^DJI+Interactive#chart2:symbol=^dji;range=my;indicator=volume;charttype=line;crosshair=on;ohlcvalues=0;logscale=on;source=undefined

This is something that would have, in a perfect world, a straight line on a linear-log scale like this. It is the differences that are illuminating. There are two important effects of the scale: first, it makes exponential growth a straight line, so it is easy to see deviations from it. Second, it collapses the high end, so you can still see deviations at the low end. Consider if it were on a linear-linear scale and the straight line on the graph were there as a parabola. The deviations at the top end would still be visible, but the ones at the low end would be so tiny they'd be less than the width of your line.

So that got me thinking about what would be the purpose of a log-log scale like your student used. Something where the deviations, as a fraction of the value, were the important thing. So, if you were testing a voltmeter which is supposed to have a very large range and have, say, 5% accuracy in the entire range, this is exactly the sort of scale you would want. If it ranged from one-tenth volt up to 10,000 volts, then you would be making measurements in exponentially jumping increments. On a pure linear scale which could fit 10,000 volts, then the dozen or so measurements that you made under 120V would all be one tiny cluster of dots, and their error from the true value would be too small to see. So the scale you want would be exactly a log-log scale. You could see a 7% variation at the tenth-volt range and still be able to fit the 10,000 volt range on a reasonable piece of paper.

[Lepton*]

What Zag said.

I used to worry about scales on graphs a lot, because it is easy to see trends in lin-log or log-log graphs that aren't necessarily meaningful. Astrophysics is a good example of this, as many measurements are made in a way that is naturally polynomial to the variables in the underlying equations.

In a lab setting, I think I agree with the student's logic: it seems that s/he was asked to perform exponentially-increasing measurements, so a logarithmic scale would be appropriate. However, on finding that the slope of that log-log graph is approximately 1 (which means the relationship is y = ax^1) the next step is to see how well a straight line fits on a lin-lin graph.

And in real life, it's moot, because you'd use a least-squares algorithm (or something better) to do it all numerically.

[dethwing]

Looks like I'm wrong and owe my student an apology. Thanks everyone.

[Zag]

[Zag]

[Quailman]

[LordKinbote]

I think, more important of a question than "What scale does the graph use?" is "What does the graph tell me?"

If he can legitimately explain to you what kind of relationship the graph shows, then it doesn't matter. If the graph he drew confuses him into thinking that an addition on the X axis causes the exact same addition on the Y axis, then it definitely DOES matter.

I'm guessing you provided the values 2,4,8,etc and had the students calculate the voltage. And I'm guessing that no matter what numbers you provided for the student, he still would have evenly spaced the numbers provided on the graph. *Every* graph the student will ever draw will look like y=x if he uses his method, and therefore almost every graph will be worthless.

[Dragon Phoenix]

I would not accept a graph like that in a report by one of my staff. There is a difference between clearly marking an axis as ln(x) and just stating x on the axis with non-equidistant points. You make graphs to quickly convey a message - and by tempering with the graphs in this way you are conveying the wrong message.
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[L'lanmal]

I'm fine with non-linear scales and see them all the time. You certainly can buy log-log graph paper, and even more exotic ones like log-normal paper and Weibull paper.

However, if this were a math class, every graph submitted in a project must have clear axes and unit labels. And use of a non-linear scale is notable enough to need to be included in the axes labels. (Although if no standard method was used and he included the standardized test staple disclaimer of "This graph not drawn to scale", then we'd have another whole discussion.)

Also Thok's calculation raises a good point. If it was graphed on a consistant log ₂ scale on both axes, then the graph would not appear to go through the origin like y=x does.

[Courk]

I didn't understand all the technicalities about log scales and whatnot in this debate, but it seems to me that if the student is able to explain to you why he did it that way, with sound reasoning, then he should get credit. However, since his response to being told it was unrealistic was to swear and storm out, my gut reaction is that he cannot explain it, otherwise he would have already done so as his initial reaction.

[raekuul]

Or he's an irrational hormonal teenager that thinks he knows what he's doing. Same result either way, but we do need to get the motive and reasoning out of him to be sure.

[Jack_Ian]

Choosing the axis scale should be a deliberate action.
If the default linear scale is not chosen then there should be a reason and that reason should be defensible.
At minimum, the non-linear scale should have been highlighted on the graph by a label.

[Quailman]

[Zag]

It turns out that Excel can do that sort of graph. I just needed to poke around a little more.

Courk, consider the data set. The top graph is with linear scales. Note how only the highest couple of data points are useful. The bottom graph is with log-log scales.

[dethwing]

Follow up: Student came in the next day and was very apologetic, said he was out of line, etc.

Also, he's not a teenager. I teach mostly older people. People who have been out of school for some time.

[LordKinbote]