Assignment 3: Z-Scores and Probability

Introduction

This assignment looks into foreclosures in Dane County Wisconsin between the years 2011 and 2012.  The purpose of this assignment is to spatially analyze and look for patterns in the foreclosure data through the changes between the two years.  The probability of foreclosures in all of Dane County pf the patterns holding were calculated for exceeding 80% and 10% of the time.  Also predictions for the chance of foreclosures increasing in 2013 were made.  Data was organized into census tracts for Dane County giving the number of foreclosures per tract.     

Methodology

To accomplish the goals of the assignment several methods were used.  Z-scores were used to measure the number of standard deviations specific foreclosure counts for specific census tracts in Dane County for 2011 and 2012.  Data from the 2011 and 2012 census bureau was used to define the number of foreclosures per census tract in Dane county was used in this analysis.

Z-Scores:

Z-scores were calculated to help analyze the spatial changes in foreclosures in Dane County between the years 2011 and 2012.  Z-scores are the measure of how many standard deviations an observation is above or below the mean of the population of observations.  To calculate z-scores the standard deviation and mean of the data set need to be known.  The equation to calculate z-scores is in Figure 1.   The mean and standard deviation were calculated in ArcMap for the number of foreclosures in Dane county in 2011 and 2012.  The recorded mean and standard deviations for the 2011 and 2012 data can be seen in Figure 2.  Figure 3 is of the specific equations used to calculate the z-scores for 2011 and 2012 census foreclosure data.  Z-scores were calculated for three census tracts in Dane County: Census Tract 108, 25, and 120.01.  Figure 4 shows the number of foreclosures for the three selected census tracts and the calculated z-scores.   
       

Figure 1. Equation to calculate z-scores. 

Figure 2. The mean and standard deviation measures for the counts of foreclosures in Dane County for 2011 and 2012.  These measurements were used to calculate z-score for the foreclosure data.

Figure 3. The specific equations used to calculate the z-scores for the 2011 and 2012 foreclosure data.  The observation values for the equations can be found in Figure 4.

Figure 4.  The calculated z-scores for the three selected census tracts for foreclosures in 2011 and 2012.  The number of foreclosures for each selected tract in 2011 and 2012 is also displayed.

Probability: 

Z-scores can be used to find the probability of occurrence of a value by using a normal distribution as a guide.  Two different probabilities for Dane County were calculated.  The first was the probability that the number of foreclosures for all of Dane County will be exceeded 80% of the time if the patterns of 2012 continue.  The second probability was that the number of foreclosures for all of Dane County will be exceeded 10% of the time if the patterns of 2012 continue.  To determine these probabilities the equation used to find z-scores in Figure 1 was used again only in a slightly different way.  To solve for probability the z-score is calculated and then the z-score chart in Figure 5.  To use the chart a z-score is found using the outer x and y axis of the table and the probability is found by cross referencing the two axis values.     

Figure 5. Z-score chart used to calculate probability.

In this example of using probability the two probabilities are given and the z-score needs to be found from the table is Figure 5.  By using the z-score equation in Figure 1 the number of foreclosures at that probability can be found.  

To solve when the probability is 80%, first 80% was converted to a decimal, 0.80.  Next, the z-score on the chart in Figure 5 was found by using 0.80 as the probability.  The z-score found was 0.84.  Because 80% is larger than 50% meaning it is on the area to the right of the normal distribution curve the z-score was made negative, -0.84.  Now the equation in Figure 6 was used to find the number of foreclosures that can be expected 80% of the time using 2012 expectations.  The number of foreclosures that will be exceeded 80% of the time is about 4 foreclosures.      

Figure 6. Equation used to find the number of foreclosures that will probably occur 80% of the time in Dane County using the expectations from 2012

The same methodology used to solve for the probability of foreclosures being exceeded 80% of the time was used for the probability of foreclosures being exceeded 10% of the time in Dane County.  First 10% was converted to a decimal and the table in Figure 5 was used to find a z-score of 1.29. Because 10% is less than 50% meaning it is on the area to the left of the normal distribution curve 1.29 will stay a positive value.  Using the equation in Figure 7, the number of foreclosures that can be expected 10% of the time using 2012 expectations was found.  The number of foreclosures that will be exceeded 10% of the time is about 25 foreclosures.     

Figure 7.  Equation used to find the number of foreclosures that will probably occur 10% of the time in Dane County using the expectations from 2012

Results


The analysis of foreclosure data for census tracts in Dane County, WI for 2011 and 2012 presented some interesting results. The change between the number of foreclosures in the census tracts in Dane County between 2011 and 2012 can be seen in Figure 8.  The census tracts with the largest changes in increases of foreclosures occurred in the northwest, northeast, and south east corners of the county.  There were also some large increases in the number of foreclosures around the outskirts of the center of the county where the capital of Wisconsin, Madison is located.  This could be due to people not being able to afford rent living so close to a large city as the location become more valuable with the increasing economic importance of Madison between the years 2011 and 2012.  Most of the census tracts with zero change in the number of foreclosures are located in the middle of Dane County where Madison is located.  With such a large city being present there, there would be a small chance of economic deficit and foreclosures occurring.  Properties would most likely be sold or tenants be removed not properties foreclosed on. Madison is a large college town where students can come into loans or rent apartments to friends in the case of not being able to afford rent.   

Looking specifically at the census tracts the z-scores were calculated for in this assignment: census tract 25, 120.01, and 108.  Referring back to Figure 4 with the z-score calculations and looking at the changes in foreclosures in these census tracts in Figure 8 intriguing results.  In 2011 and 2012 tract 25 appeared below the mean number of foreclosures for Dane County and in 2011 and 2012 tracts 108 and 120.01 appeared above the mean number of foreclosures in Dane County.  Tract 25 while appearing below the mean decreased in the number of foreclosures between the two years resulting in 2012 it occurring even farther away from the mean.  Tract 25 and tract 108 decreased their number of foreclosures between 2011 and 2012, while tract 120.01 underwent a large increase in the number of foreclosures between 2011 and 2012.         

Figure 8. The change in the number of foreclosures in the census tracts in Dane County between 2011 and 2012.

The results from calculating the probability of the number of foreclosures that will occur 80% of the time and 10% of the time resulted allowed the census tracts with the most foreclosures to be easily identified.  In Figure 9 it can be seen which census tracts have an 80% and 10% probability for the number of foreclosures.  In Figure 9 it can be seen that the census tracts in Dane County in the middle of the county and on the west side of the county have the lowest number of foreclosures.  The census tracts with the greatest number of foreclosures are located on the east side of the county in Figure 9.  This can be explained by the capital of Wisconsin, Madison, being located in the center of Dane County.  There would not be a large number of foreclosures in the largest and most heavily populated city in Wisconsin.  Looking towards the east side of the county in Figure 9 there are large numbers of foreclosures.  It is uncertain why there large quantities of foreclosures are occurring, if a future study it would be interesting to see if large numbers of foreclosures are occurring in the east side neighboring county Jefferson County.  This could help determine if the large number of foreclosures in this area is a county problem or a regional problem.  Dane County should apply aid and additional resources to census tracts on the east side of the county.  The county should supply extra funding to the homeless shelters or other care facilities so the people that are struggling there can receive the help they need.     

Figure 9. The census tracts of Dane County that have a probability of achieving a number of foreclosures 80% and 10% of the time.

Lastly, for this assignment predictions for 2013 were made.  I predict that even more foreclosures will occur in census tract 120.01.  This county had a large change in the number of foreclosures between 2011 and 2012 and was in the top 10% for having the largest number of foreclosures in the county.  Dane County should prepare to supply more aid and assistance to this census tract.  I would expect the census tracts in the southwest part of Dane County to continue to decrease in the number of foreclosures they had a decrease in the number of foreclosures between 2011 and 2012 and they are in the 80% of all tracts in the county for having the lowest numbers of foreclosures.  I would also expect the census tracts in the center of Dane County to continue to have zero change in the number of  foreclosures like there were between 2011 and 2012 due to the large city of Madison being present there.   

Conclusions

The results of the analysis of foreclosure data for census tracts in Dane County, WI for 2011 and 2012 suggested that the east side of Dane County needs the most relief aid and help with increasing numbers of foreclosures and the greatest numbers of foreclosures in the county.  A specific census tract that should be paid special attention to is tract 120.1.  There was a large increase in foreclosures in this tract between 2011 and 2012 as well as a high probability the values in this county will greater than 25 foreclosures in 2013.  Census tracts in the center of Dane County are of minimal concern for foreclosures due to the fact of the city of Madison being present there.  I recommend the east side of the county be under surveillance for housing market changes in the value of homes decreasing do to the large number of foreclosures in the area.