Z Score and Probability

Introduction:

The purpose of this assignment is to investigate the foreclosures in Dane County using z-scores, probability, and add field tool. The officials in Dane County are concerned about the increasing amount of foreclosures from 2011 to 2012 and would like spatial analysis to be done to find patterns. The questions asked for this assignment are what is the pattern of change from 2011 to 2012 and what foreclosures will look like in 2013 if the trend continues.

Methodology:

To answer the first question the Dane County officials asked, the Dane County tracts feature class needs to be added to the work space. The tracts feature class contains information on the foreclosures for the years of 2011 and 2012 but does not have information for the differences between the two. However, the difference can be found with the information that is given. To do this, a new field needs to be created and the field calculator is used to make the field values equal to "2012 foreclosures - 2011 foreclosures". This will give each tract a new attribute that is the differences from 2011 to 2012. Once this is complete, a map can be made that shows the change for each tract from 2011 to 2012.

For the next question, z scores are needed. A z score is a specific value for a observation that represents the exact number of standard deviations an observation is from the mean. It is calculated by taking the observation value and subtracting the mean from it, then take that number and divide it by the standard deviation. This will give the z score. The larger the z score is, the farther from the mean the observation is. The smaller the z score is, the closer to the mean the observation is. For this assignment, the z score for three tracts was found for 2011 and 2012. Map 1 below shows all the tracts in Dane County. The three counties that z scores were found for were tract 122.01, 31, and

map 1. Dane County Wisconsin Tracts

114.01. Tract 122.01 in 2011 had a z score of -0.61 and a z score of -0.64 in 2012. This means that this tract is within -1 - 1 standard deviation from the mean and these counties have an about average amount of foreclosures. Tract 31 has a z score of 1.44 and .58 in 2011 and 2012 respectively. This shows that from 2011 to 2012 the amount of foreclosures  has gotten a lot closer to the mean. Tract 114.01  had a z score of 2.35 and 2.70 in 2011 and 2012 respectively. This tract is a lot more of an outlier because it is over 2 standard deviations from the mean. This means that there are a lot more foreclosures in tract 114.01 than there are in other tracts.  Z scores can be used with a z score table to find out the probability that the observation will happen. The z score and z score table are used to answer the next question about how the foreclosures will look in 2013 if current trends continue. The first thing that needs to be done is the Dane County tracts feature class needs to be added to the work space. The differences field that was created above is used to create another field to predict what the foreclosures in 2013 will look like. This new fields values are found by adding the "2012 foreclosures" to the "differences" in the field calculator. This will give a new attribute for each tract that is the estimated foreclosures for 2013. After the new attribute is added, the z score table can be used to find out the amount of foreclosures a tract needs before the foreclosures are in the top 70% and top 20% of all tracts.    

Results:

The results from the first question can be seen below in map 2. Map 2 shows the change in amounts of foreclosures form 2011 to 2012. Tracts that are blue had more foreclosures in 2012 than 2011 and red tracts had more foreclosures in 2012 than 2011. The darker the color, the more the foreclosures

map 2. Dane County foreclosure differences from 2011 to 2012

have changes from 2011 to 2012. If a tract is neither blue nor red, then the tract did not have substantial change from 2011 to 2012. For the next question, the results are in map 3 below. Map 3 shows the probability that a tract in 2013 will be in the top 70% of foreclosures and the top 20% of foreclosures. For the predictred foreclosures for 2013, the data has a mean 13.2 foreclosures and a standard deviation of 13.4.  The tracts that are in green are predicted to be in the top 70% of foreclosures with a z score of 6.2 foreclosures. This means if a tract has more than 6.2 foreclsoures it is in the top 70% of tract. The cross hatched tracts are going to be in the top 20% of foreclosures with a z score of 24.5. This means that if a tract has 24.5 forclosures it is in the top 20% of tracts in Dane County. Naturally, all the cross hatched tracts are also green. These green cross hatched areas are where the most expected foreclosures will be located. Most of these areas are in large tracts that

map 3. Dane county projected foreclosures in 2013 by tracts.

have more houses and more opportunity to get foreclosed. Most of the green crosshatched areas are also blue areas on map 2 where they gained a lot of foreclosures from 2011 to 2012. This is most likely due to areas that gained a lot of foreclosures between 2011 and 2012 are projected to gain the same amount of foreclosures in the next year causing the projected foreclosures to be exaggerated.

Conclusion: 

Most of the foreclosures are located in the larger tracts on the outside of the county and the counties that gained a lot of foreclosures from 2011 to 2012. This implies that more rural areas have a lot greater chance of having a foreclosure and larger tracts also have a greater chance of having a foreclosure. A recommendation to solve this problems is to investigate how easy it is to get a loan for these home and maybe put higher restriction on home loans and mortgage so only people who can afford homes and pay for them can get homes so they do not get foreclosed on.