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INTRODUCTION TO MAPPING & GIS: EXERCISE 8 - CLASS INTERVALING TECHNIQUES NAME: ___________________________________________________ OBJECTIVES: This exercise gives you "hands-on" experience with the basic class intervaling techniques used most commonly in thematic mapping. When you have finished, you will know the fundamental principles of class intervaling and you will be able to calculate the mean and standard deviation. You will also be able to calculate break points using the following methods: [1] equal value range, [2] mean and standard deviation, [3] nested mean, [4] quantile, and [5] geometric progression. PRINCIPLES: Class intervaling is a classification process used to reduce a large number of quantitative values to a smaller number of ordered categories. In thematic mapping we class the data so we can see the forest rather than be overwhelmed by the trees. Classing is one step in creating maps that enable the clear depiction of data that in their raw form would be nearly incomprehensible. It is one means of carrying out the KISS principle. There are many techniques from which you can choose in setting up data classes, but in all cases you must observe two fundamental principles: 1. 2. Each of the original [unclassed] data values must fall into one of the classes None of the original [unclassed] data values may fall into more than one class. A short way of putting this is to say that the classes must be mutually exclusive and exhaustive. MEAN To complete this exercise you will need to know how to calculate the mean or average of a set of numbers. There should be no mystery involved here. Certainly you are familiar with concepts like grade point average, mean income level, the average height of members of the Philadelphia Seventy Sixers baseball team and the like. The mean is simply one of several measures that express, in a single number, the "central tendency" of a set of data. To calculate the mean first count how many numbers there are. Call this number "N.” Next, sum all of the values and divide this sum by N. The result of the division is the mean. In doing all of this you might find certain conventions useful. Often we will find that it is convenient to refer to any member of a set of numbers X, as Xi. For instance, the set of numbers X has the following members: { X1, X2, . . . , Xi, . . . , Xn} We refer to any one of those numbers as Xi. The other convention you need to know about is the large or uppercase Greek letter sigma, which looks like this: Σ and means "take the sum of whatever follows." The formula for the mean is: STANDARD DEVIATION One way in which to summarize a set of data is to report a measure of central tendency. The mean is an example of such a measure. Common sense will tell you that there is much more to a set of data than the mean value. Two sets of numbers can have the same mean and yet be very different from each other in the amount of dispersion or divergence from the mean. Measures of variability provide a statistical indication of dispersion in a set of data. The standard deviation is one such measure. Others include the range, the interquartile range, and the variance. For now, focus on how to calculate the standard deviation. 1. To begin, you calculate the mean [see above]. 1 2. Next, subtract the mean from each of the original data values, Xi. This will give you a number of differences equal to N. 3. Now, square each of these N differences. 4. After you square the differences, sum them and then divide the sum by N. 5. The result of this division is a common measure of dispersion called the variance. To get the standard deviation simply use a calculator, computer, or square root table to take the square root of the variance. The formula for the standard deviation is: PRACTICE Just to make sure that you understand and to help you get a feel for the standard deviation let's calculate the mean and standard deviation for two simple sets of data both of which have the same mean value, but which vary in their dispersion about the mean. The data are hypothetical incomes for residents of two towns: one is Sameville and the other is Variton [see the table]. Calculate the mean and standard deviation for the income of each place and briefly discuss your results in the space provided. TABLE 1.1. INCOME DATA FOR THE RESIDENTS OF SAMEVILLE AND VARITON [NUMBERS IN THOUSANDS OF DOLLARS] SAMEVILLE VARITON 7.0 7.5 8.0 8.5 9.0 11.0 11.5 12.0 12.5 2.0 3.0 5.0 7.0 9.0 11.0 13.0 17.0 20.0 Show the results of your calculations in the spaces provided in Table 1.2. Also, show your calculations in the space provided below Table 1.2. TABLE 1.2. INCOME DATA FOR THE RESIDENTS OF SAMEVILLE AND VARITON [NUMBERS IN THOUSANDS OF DOLLARS] SAMEVILLE VARITON MEAN SD 2 SHOW YOUR CALCULATIONS IN THE SPACE BELOW AND COMMENT 3 INTERVAL CALCULATION: This section requires that you use the classing techniques discussed in the readings and in class to create class limits for some real world data. There are two data sets: 1. Data on total population for the counties of New Jersey. The column, "sorted population" will be used in calculating quantile classes. [Table 1.3] 2. Data on percentage of population male for counties in Kansas [Table 1.4] Each classing problem will refer you to the appropriate table. TABLE 1.3 TOTAL POPULATION FOR NEW JERSEY COUNTIES COUNTY POPULATION SUSSEX COUNTY PASSAIC COUNTY BERGEN COUNTY WARREN COUNTY MORRIS COUNTY ESSEX COUNTY HUDSON COUNTY HUNTERDON COUNTY SOMERSET COUNTY UNION COUNTY MIDDLESEX COUNTY MONMOUTH COUNTY MERCER COUNTY BURLINGTON COUNTY OCEAN COUNTY CAMDEN COUNTY GLOUCESTER COUNTY SALEM COUNTY ATLANTIC COUNTY CUMBERLAND COUNTY CAPE_MAY COUNTY 144166 489049 884118 102437 470212 793633 608975 121989 297490 522541 750162 615301 350761 423394 510916 508932 254673 64285 252552 146438 102326 SORTED POPULATION 884118 793633 750162 615301 608975 522541 510916 508932 489049 470212 423394 350761 297490 254673 252552 146438 144166 121989 102437 102326 64285 Average =400683 Standard deviation =245685 At this point review your notes on each of the classing techniques that we discussed in class. For most of these techniques you will also find some discussion in the readings. 4 EQUAL VALUE RANGE CLASSES: Divide the data in Table 1.3 into five classes of equal value range. To provide better looking classes use an artificial minimum of zero and an artificial maximum of 400. Show your results in the space below. CLASS LOWER LIMIT UPPER LIMIT 1 __________ ___________ 2 __________ ___________ 3 __________ ___________ 4 __________ ___________ 5 __________ ___________ SHOW YOUR WORK IN THE SPACE BELOW: 5 MEAN AND STANDARD DEVIATION CLASSES: Using the real minimum and maximum as lower and upper limits, divide the data in Table 1.4 [Percent of the population male] into four classes using mean and standard deviation class breaks. CLASS LOWER LIMIT UPPER LIMIT 1 __________ ___________ 2 __________ ___________ 3 __________ ___________ 4 __________ ___________ SHOW YOUR WORK IN THE SPACE BELOW: 6 QUANTILE CLASSES: As you can see the data in Table 1.3 are quite skewed. Sometimes quantile classes are useful in the case of skewed data. Establish the break points for quintile class limits. To help you I have sorted the data for you [see sorted column in table]. CLASS LOWER LIMIT UPPER LIMIT 1 __________ ___________ 2 __________ ___________ 3 __________ ___________ 4 __________ ___________ 5 __________ ___________ SHOW YOUR WORK IN THE SPACE BELOW: 7 Table 1.4 Kansas Data on Percentage of Population Male FIPS 20001 20003 20005 20007 20009 20011 20013 20015 20017 20019 20021 20023 20025 20027 20029 20031 20033 20035 20037 20039 20041 20043 20045 20047 20049 20051 20053 20055 20057 20059 20061 20063 20065 20067 20069 20071 20073 20075 20077 20079 20081 20083 20085 20087 20089 %MALE 48.09 48.16 48.76 47.99 48.25 47.05 47.92 49.12 49.55 48.63 47.67 47.89 48.47 48.45 46.98 49.49 47.86 48.46 48.02 48.57 48.15 49.08 49.78 48.27 47.82 48.99 51.76 50.76 50.46 48.43 51.3 49.64 49.08 49.95 49.09 49.66 48.53 47.49 48.18 48.46 50.46 49.56 49.06 50.27 49.52 SORT 55.08 55.04 51.96 51.76 51.3 50.9 50.85 50.81 50.76 50.55 50.54 50.48 50.46 50.46 50.27 49.95 49.78 49.78 49.71 49.68 49.66 49.64 49.63 49.56 49.55 49.52 49.49 49.43 49.39 49.35 49.32 49.3 49.22 49.2 49.17 49.14 49.12 49.09 49.09 49.08 49.08 49.06 49.06 49.02 48.99 8 20091 20093 20095 20097 20099 20101 20103 20105 20107 20109 20111 20113 20115 20117 20119 20121 20123 20125 20127 20129 20131 20133 20135 20137 20139 20141 20143 20145 20147 20149 20151 20153 20155 20157 20159 20161 20163 20165 20167 20169 20171 20173 20175 20177 20179 20181 20183 20185 20187 48.33 50.81 48.6 47.9 47.97 50.48 55.08 47.66 49.06 49.3 48.82 48.64 47.89 48.85 49.09 49.17 48.3 47.2 48.77 49.2 49.39 47.96 49.22 51.96 48.68 47.85 48.42 50.55 48.56 49.78 48.41 49.35 49.14 47.81 47.77 55.04 48.25 48.26 48.25 48.31 49.63 48.96 50.54 48.22 50.9 48.89 48.03 48.15 49.68 48.99 48.96 48.89 48.85 48.82 48.77 48.76 48.74 48.68 48.66 48.64 48.63 48.6 48.57 48.56 48.53 48.52 48.47 48.46 48.46 48.45 48.43 48.42 48.41 48.33 48.31 48.3 48.27 48.26 48.25 48.25 48.25 48.22 48.18 48.16 48.15 48.15 48.09 48.03 48.02 47.99 47.97 47.96 47.92 47.91 47.9 47.89 47.89 47.86 9 20189 20191 20193 20195 20197 20199 20201 20203 20205 20207 20209 48.99 48.66 48.74 49.32 49.43 50.85 49.02 49.71 47.91 48.52 47.69 47.85 47.82 47.81 47.77 47.69 47.67 47.66 47.49 47.2 47.05 46.98 MEAN SD 49.00047619 1.297314722 Part II - Mapping Various Classing Techniques After working out these classing techniques by hand, you now have a really good idea what is taking place when different classing techniques are utilized in mapping. Now you can create maps of New Jersey’s population by county using the techniques demonstrated above. Copy the exercise8 folder to your local C: drive and open up the map document file exercise8.mxd. You will see four maps of New Jersey municipalities. Your objective is to color the maps by four classing methods – natural breaks, quantile, equal interval and standard deviation. Right-click on the layer name of one of the layers in the table of contents and open up the properties dialog box. Click on the Symbology tab. In the “show” window select Quantities and Graduated colors. In the Value field choose POP2000. Click on the Classify button to open up the classify window. This window is where you can change the classify method and any parameters. Spend some time playing around with the classification functions of ArcGIS. Each time you change a parameter click OK and then apply to see how it changes the way the map is drawn. See if you can figure out how the chart that opens in the classification window depicts the categories. Try moving the blue lines by dragging them. How does that change the map. Once you are comfortable with the classification functions, color each of your four maps with a different classification method. Add a legend. Write a source statement (NJ DEP & US CENSUS). Be sure to add titles for each classification type. Also add your name and any other necessary or desired graphic elements, print out the map and hand in with the rest of this lab. 10