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Odd Intersection Property

 
What it is:
  • The odd intersection property states that the two median levels, of any two given points sets will cross an odd number of times.  We can see this by drawing a diagram below (they cross 3 times).
  • Or, we can see it in the fact that if at one line is above another, then they will have to cross an odd number of times for it to be below.


As, we can see from the diagram the two median levels (denoted by the darker red and blue lines) cross an odd number of times. (denoted by the green dots)
 
  • This means that if we take any interval over the entire diagram, then the odd intersection property holds if the median levels flip positions in that interval!
  •  For an easy way to determine if the odd intersection property holds for a certain area, we can use the formula:

    (L1(x1) - L2(x1)) * (L1(x2) - L2(x2)) < 0

  • Where
    • L1(x1) is the red median level at x1
    • L2(x1) is the blue median level at x1
    • L1(x2) is the red median level at x2
    • L2(x2) is the blue median level at x2
  • If this formula is true then the odd intersection property holds!

 
  • Let's take a look at an example:
    • Let x1 = 5, and x2 = 12
    • Let the red median levels be equal to: L1(5) = 7, and L1(12) = 13
    • Let the blue medians be equal to: L2(5) = 9, and L2(12) = 5

    L1(5) - L2(5) = 7 - 9 = -2
    L1(12) - L2(12) = 13 - 5 = 7

     

  • Using the formula we get:

    -2 * 7 = -14, which is less than 0.

    So the odd intersection property holds for this interval!