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This contour plot shows the relationship between the finished square feet of a home and the tax-assessed value for homes in a County.

a contour plot of 2 variables

Things I couldn't understand:

  1. what does it mean when lots of bands are close to one another? And when they are very far to one another?
  2. what does the weird(or asymmetric) shape of a contour band represent?
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  • $\begingroup$ Could you explain what you mean by "grouped"? I don't see any apparent examples of combining or grouping contour levels in your example. $\endgroup$
    – whuber
    Commented Sep 9, 2020 at 18:42
  • $\begingroup$ Lots of bands close to each other, as can be seen from 1000 to 2000 of x-axis. $\endgroup$ Commented Sep 10, 2020 at 3:09

1 Answer 1

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tldr: Denser implies steeper gradient.

On a contour plot, the value on the continuous plot is the same. What might be helpful in your plot is to give numbers to the contour lines.

Imagine a Z-axis, rising out of the screen. If we plot,

$z^2 = x^2 + y^2$,

For a circle with radius 1, the circle will be created a unit distance out of the screen. For radius, 2 it will be 2 unit distance out of the scree and so on. If we project these circles in the air on to the X-Y plane and give them number we would get, concentric contours.

It would be safe to assume, that the incremental contours as we head out keep increasing (they could be decreasing but just to explain assume they increase). Eg. Inner-most is 1, then the next one 2 and so on.

The densely crowded regions indicate, that the gradient is steeper in that region. The gradient is usually referred to in the direction perpendicular to the contours. (In the circle case it will be the the radial direction).

For eg. For Finished Square feet = 1000, if tax assessed goes from 150000 to 170000 (roughly), we change around 4-5 contour lines. i.e.

$abs(\frac{\delta{z}}{\delta{y}}_{x=1000, y= approx 160k} )= \frac{5}{20k} $ (5 is because roughly 5 contours crossed in 20k change). z is contours, y is tax-assessed value and x is Finished square feet.

If you perform similar calculation for x=1000 and y between, 300k to 320k, $abs(\frac{\delta{z}}{\delta{y}}_{x=1000, y= approx 310k} )= \frac{1}{20k} $ (1 is because roughly only 1 contour crossed in 20k change).

Note: Values are rough eyeballing and just to give sense of magnitude.

So denser is steeper gradient.

Notice, I have used, absolutes. This is because, going towards the central portion is opposite in direction to going out and we just wanted to demonstrate magnitude of gradient. Based on value of the contours increasing or decreasing the sign can be determined. Hope this helps.

Further references:

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