LAB EXERCISE A: Mean center, standard distance, and Manhattan median

All of the questions in Lab 3A have to do with the point data distribution in Figure 1. The relevant reading in McGrew and Monroe is Ch. 4.

Figure 1: Distribution of C. foetidus, Sulphur Mt., Ventura Co., CA

                        quadrat grid in meters

         0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 
         ___________________________________________________ 
         |                            *                     | 0 
         |     *                                   *        | 1 
         |                             *                    | 2 
         |   *                                              | 3 
         |   *   *                                       *  | 4 
         |*          *              *                 *     | 5 
         |                                                  | 6 
         |       *                                          | 7 
         |    **                                            | 8 
  ^      |                                                  | 9 
  |      |                                                  |10 
  N      |                                                  |11 
  |      |  *    *             *                  **        |12 
         |*                                              *  |13 
         |                                                  |14 
         |                                                  |15 
         |                                                  |16 
         |                                                  |17 
         |                                      *           |18 
         |                                *           *     |19 
         |                          *            *          |20 
         |                                    *             |21 
         |                                                  |22 
         |                                                  |23 
         |          *                               *       |24 
          -------------------------------------------------- 25 

1. Using your spreadsheet, calculate the x-y coördinates of the mean center of the plants mapped in the quadrat. Mark the mean center on the map in a contrasting color or point design.
   • Mean x coördinate: _______________ in m from upper left (NW) corner

   • Mean y coördinate: _______________ in m from upper left (NW) corner

2. Calculate the standard distance for this point pattern. On the map, draw a circle with that radius on the map around the mean center.
   • Sd = _______________ m

   • What can normally be found within that circle? ___________________________________
     _________________________________________________________________

     What actually turned up in this distribution? _____________________________________

     _________________________________________________________________

3. Calculate the relative distance for this distribution. Helpful hints: Area of a circle = pi x squared radius. Area divided by pi gets you the squared radius. Take its square root and, voilà, the necessary radius for doing relative distance.
   • Rd = _______________

   • The relative distance is nearly the analogue of the co-efficient of variation, but in two-dimensional space. That is, the co-efficient of variation is to the standard deviation as, roughly, relative distance is to standard distance. The analogy is not exact, however. The co-efficient of variation is calculated by dividing the standard deviation by the mean. Why isn't relative distance calculated simply by dividing the standard distance by the mean center?
     _________________________________________________________________

     _________________________________________________________________

     _________________________________________________________________

     _________________________________________________________________

     What are some shortcomings of dividing the standard distance by the radius of a circle containing the same area as the study area? In other words, why do you have to be a little bit cautious about interpreting differences in relative distances among distributions measured in different areas?

     _________________________________________________________________

     _________________________________________________________________

     _________________________________________________________________

     _________________________________________________________________

4. Calculate the Manhattan median for the same points, using the orientation of the implicit grid provided (N/S - E/W). Mark the Manhattan median directly on Figure 1.

5. In your own words, discuss the difference between the general concepts of Euclidean distance and Manhattan distance. Which one pretty consistently produces a larger value than the other? Under which circumstances would the Euclidean distance and the Manhattan distance between two points be equal?
   _________________________________________________________________

   _________________________________________________________________

   _________________________________________________________________

   _________________________________________________________________

LAB EXERCISE B: Manhattan median and Euclidean median

The following questions have to do with Figure 2, which depicts the Census tract boundaries within the small railroad town of Escalpón, Nevada. The town is a nearly homogeneous middle income community of nearly 40,000 souls. Each of the six Census tracts has about 6,500 to 6,750 people (so there's no real point to weighting the tracts for this exercise) and encloses one square mile.

Figure 2:  Escalpón, NV


                            ‡  ^
                            ‡  | to Tonopah
                            ‡
                            ‡
                            ‡---------
                            ‡         |
                            ‡         |
                   ---------‡         |          
                  |         ‡A        |
     ^            |         ‡---------|--------- 
     |            |         ‡         | B       |
     N            |         ‡         |         |
     |            |---------‡         |         |
                  |        C‡         |         |
                  |         ‡---------|--------- 
                  |         ‡   D     |
                  |         ‡         |
                   ---------‡         |
                            ‡         |
     |---------|            ‡---------
       one mi.              ‡
                            ‡
                            ‡
                            ‡
                            ‡  | to Las Vegas
                            ‡  v

1. Calculate the Euclidean distance between the centers of each of the six Census tracts and each of the four locations labeled A through D. This will, then, entail 24 separate measurements. Please express your measurements in miles, using the scale provided, and round your calculations to two decimal places.

   
             A to 1:  __________mi.        B to 1:  __________mi.
                                                                 
             A to 2:  __________mi.        B to 2:  __________mi.
                                                                 
             A to 3:  __________mi.        B to 3:  __________mi.
                                                                 
             A to 4:  __________mi.        B to 4:  __________mi.
                                                                 
             A to 5:  __________mi.        B to 5:  __________mi.
                                                                 
             A to 6:  __________mi.        B to 6:  __________mi.
   
                           .............................
   
             C to 1:  __________mi.        D to 1:  __________mi.
                                                                    
             C to 2:  __________mi.        D to 2:  __________mi.
                                                                    
             C to 3:  __________mi.        D to 3:  __________mi.
                                                                    
             C to 4:  __________mi.        D to 4:  __________mi.
                                                                    
             C to 5:  __________mi.        D to 5:  __________mi.
                                                                    
             C to 6:  __________mi.        D to 6:  __________mi.
   
   

2. Which of the four locations minimizes the aggregate Euclidean distance between itself and the cores of the six Census tracts? That is, which of the four choices is closest to the unknown Euclidean median of this distribution of Census tract centers?
   _______________

   Now, assume that all roads in Escalpón run parallel or perpendicular to the railroad shown, that is, they form a perfect, Manhattan-style grid. Show the Manhattan median for the four labeled locations directly on Figure 2.

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   last revised: 09/11/98
   © Dr. Christine M. Rodrigue

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