While the Standard Deviation indicates the spread or cluster of data about the Mean value, Quartiles and Percentiles are used to perform a similar function to estimate the dispersion around the Median value. You can estimate the dispersion around the Median in two ways:

• Mathematically: calculate the Inter-Quartile Range (see below)
  
• Graphically (e.g: spatial distribution on a map):

1. Locate the Median Center (not very precise)
2. Visualize the Inter-Quartile Area (more precise) (see below)
   
3. Calculate the Index of Dispersion to measure the dispersion you see on the map (see below)

Inter-Quartile (or Percentile) Range (IQR)

This is a simple and quickly obtained measure of dispersion, but is a much coarser (less precise) one than the standard deviation.
Use Excel, or the method below:
The data must be ranked and divided into four groups called quartiles, each containing 25% of the values. The quartile containing the top 25% values is the upper quartile, while the one with the bottom 25% values is the lower quartile:

• Q0 is the minimum value
  
• Q1 is the value between the lower and 2nd quartile (25% of the values in the sample are below)
• Q2 is the value between the 2nd and 3rd quartile, which is also the Median value (50% of the values in the sample are above/below this value)
• Q3 is the value between the 3rd and 4th quartile (25% of the values in the sample are above)
• Q4 is the maximum value
  

You can use the simple Excel formula to find the quartiles of your sample: e.g. for Q3 use the formula =QUARTILE(sample,3)
The Inter-Quartile Range IQR = Q3 - Q1. A high IQR means the data is very dispersed, while a low IQR means the data is less dispersed.

Note that the listings may be divided into any number of equal parts each containing the same number of percentage of the number of values. In this case the categories are called percentiles. The IQR is then called the Inter-Percentile Range.

Example: the SAT scores of a group of students are given below, ranked from best to worse:

1800 1790 1770 1750 1720 1700 1700 1650 1600 1540 1320 1200 1200 1180 1100

Q0 (minimum value) = 1100
Q1 (25% of values are below this number) = 1260
Q2 (median value) = 1650
Q3 (25% of values are above this number) = 1735
Q4 (maximum value) = 1800

IQR = Q3 - Q1 = 1735 - 1260 = 475
The IQR can then be compared to another set of students to assess the dispersion of the sample.

Quartiles are frequently represented using boxplot (or "box-and-whiskers") graphs. Follow the steps indicated below to create box-and-whiskers graph on Excel 2007.

• The IQR is the area in yellow
  
• The upper "whisker" represents the upper quartile (top 25%)
  
• The upper box represents the 3rd quartile (25% above the median)
  
• The horizontal line between the two boxes represents the median
  
• The lower box represents the 2nd quartile (25% below the median)
• The bottom "whisker" represents the lower quartile (bottom 25%)

Steps to create a box-and-whiskers graph on Excel 2007:

1. List the values of the sample in column A (they do not have to be ranked in order) (here the same SAT scores as above are used)
   
2. Insert the formula listed in cells B1 thru B5 into cells C1 thru C5 to calculate the values needed for the graph
3. Select cells B2 thru B4 (yellow), click "insert" and select "column charts": select the 2nd 2-D column chart from the left (=stacked bar graph)
4. In the tab "Design", click "switch row/column" to obtain a vertical stacked bar graph
   
5. Boxes: select and delete the key and the horizontal axis of the graph, and select. Point your mouse to the bottom box of the bar graph, right click and select "Format Data Series" (or use the pull-down selection menu on the top left under the "Format" tab): under the "Fill" category, select "No Fill" to make the bottom box "invisible". Select the upper two boxes and do the same thing, but instead of "no fill", choose a common color for both boxes (in this example, yellow). You can also use the "Format Data Series" function to format the border style/color of the boxes
   
6. Bottom whisker: select the "invisible" bottom box (the one with "no fill"), click on the "Layout" tab, then in "error bars" (under "Analysis). At the bottom of the pull-down menu, select "more error bars options": under "display", select "minus" and under "error amount", click on "custom" and "specify value". Do not change the "positive error value" and select cell C1 (blue) for the "negative error value". 
7. Top whisker: select the top (yellow) box, click on the "Layout" tab, then in "error bars" (under "Analysis). At the bottom of the pull-down menu, select "more error bars options": under "display", select "minus" and under "error amount", click on "custom" and "specify value". Do not change the "negative error value" and select cell C5 (red) for the "positive error value". Done!
   

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Inter-Quartile Area (Q) and Index of Dispersion (Id)

This graphic method is more precise than the Median Center. It is a variation of the principle above which can be used to illustrate the distribution of phenomena spatially (e.g. clustering/dispersion of rural settlements or of high-rises in a CBD). It serves a similar function to the Nearest Neighbor Analysis but is quicker and easier to operate, although the results are coarser (less precise). IMPORTANT: there must be a MINIMUM of 8 observed points in the sample for this method to work.

Method:

1. Mark the location of the observed points (e.g. high-rises in a CBD) onto a base map (see sample below) and define the boundaries as you would for the Nearest Neighbor Analysis.
2. Count up and number the total number of points (n) from the left to the right of the diagram.
3. The horizontal and vertical mid-points are defined as follows:

                * If there is an even number of points: the mid-point is halfway between point n/2 and the following one by counting left to right (horizontal) or                 top to bottom (vertical)
                * If there is an odd number of points: the mid-point is at point (n+1)/2 by counting left to right (horizontal) or bottom to top (vertical)

1. Draw a red line vertically through the horizontal mid-point (ie: counting from the left), and draw a red horizontal line through the vertical mid-point (ie: counting down from the top). Note: these two mid-points are not necessarily the same point on the diagram. You should now have an equal number of points on both side of the horizontal line, and on both side of the vertical line.
   
2. Using the same principle for determining the mid-points, divide the map up into 16th by drawing two more (green) lines vertically and two more (green) lines horizontally. Again, you should have an equal number of points in each vertical section (or quartile), and in each horizontal section (or quartile).
3. The Inter-Quartile Area (Q) is the shaded area of the rectangle formed in the center of the base map. Use a ruler to calculate the size and area of the rectangle Q. A small Q suggests clustering while a large Q suggest dispersal.
4. It is possible from this method to derive a simple Index of Dispersion (Id) which relates to the dispersal points in relation to a central point, and is useful to compare the dispersal/clustering of two sets of points.
   

                    0 = maximum concentration (clustering) around the central point
                    1 = maximum dispersal from the central point
                    Formula: Id = Q / A (Q = Inter-Quartile Area, A = total area mapped)
                    In the example below, with 9 points, if the map is 10 cm x 6 cm, Q = approx. 15 sq cm., and Id = 15/60 = 0.25 (=moderate concentration).