Assignment 3- Z-Scores and Probability

Introduction:

A foreclosure is the action of taking possession of a mortgaged property when the mortgagor fails to keep up their mortgage payments (Weintraub, 2017). Foreclosures are viewed as an indicator of economic decline, and are thus a cause for worry among elected officials.  In Dane County, Wisconsin, county officials are concerned over their increase in foreclosures from 2011 to 2012.  Although the reason behind the foreclosures cannot be determined, the spatial pattern of foreclosures in the county and the likelihood foreclosures will increase in 2013 can be determined using basic statistics.  This report uses the number of foreclosures in 2011 and 2012 to explain the pattern of foreclosures in Dane County and based on previous years’ data determine the probability, or the likelihood of an increase in foreclosures in 2013. 

Methodology:

The pattern of foreclosures from 2011 to 2012 was determined by calculating the change from 2011 to 2012 and then  based on that chang ecreating a map of standard deviation in ArcMap.  Standard deviation is a statistic that describes how tightly the data clusters around the mean, with large negative or positive standard deviation values indicating the data point lies far from the mean.  
 First, the addresses of the foreclosures in 2011 and 2012 were geocoded, meaning x and y coordinates were attached to each address and mapped. Next they were added to the census tracts for Dane County.  Then a new field was created in the attribute table subtracting the number of foreclosures in 2011 from the number of foreclosures in 2012.  A standard deviation map was created from this field. 

To better understand the map and the data six z-scores were calculated from three census tracts: 120.01, 108, and 25 with one in both 2011 and 2012 for each tract. Z-scores are standardized numbers from data values converted by their relative position to the mean to allow comparison to a normal or bell-shaped distribution curve. The mean, median, and standard deviation for the whole county in 2011 and 2012 were obtained from the ArcGIS software to calculate z-scores.  The z-scores were calculated using the formula below (Taylor, 2014).

Figure 1. Z-score formula.

Finally, based on 2012 data, the number of foreclosures that will be exceeded in 2013 80% and 10% of the time were calculated using the mean and standard deviation obtained above and the appropriate z-scores from a z-score and probability table. The calculations for all these steps is pictured below. 

Figure 2. Mean, median, and standard deviation for foreclosures in 2011 and 2012, z-scores in 2011 and 2012 for census tracts 120.01, 108, and 25, and the number of foreclosures likely to be exceeded 80% and 10% of the time in 2013.

Results:

The map in figure 3 shows most census tracts seeing increases and decreases between 2.5 standard deviations from the mean.

Figure 3. Standard deviation map of the change in foreclosures from 2011 to 2012.

However, census tracts 120.01, 116, and 105.01 all had increases in foreclosures greater than 2.5 standard deviations from the mean. In figure 3, a decrease in foreclosures is indicated by the blue and green colors while an increase in foreclosures is indicated by the orange and red colors. The map in figure 3 shows that most tracts in the center had very little change in foreclosures while the outer ring of tracts showed more significant increases and decreases in foreclosures.  Census tract 120.01 had a z-score of 1.78 in 2011 but a z-score of 3.0 in 2012.  This increase is the most notable of the 3 census tracts with calculated z-scores with a change of greater than 2.5 standard deviations from the mean.    Census tract 108 had z-scores of 2.01 and 1.48 in 2011 and 2012 and tract 25 had z-scores of -0.61 and -0.94 in 2011 and 2012 respectively.  Both are shown in green in figure 3, which reflects the decrease in foreclosures in both.  However census tract 108 moved closer to the mean in 2012 while tract 25 moved farther from the mean in 2012. 

The final calculations determined that 80% of the time approximately 4 foreclosures would be exceeded in Dane County while 10% of the time approximately 25 foreclosures would be exceeded in Dane county.  These numbers are recorded in figure 2. Based on figure 3 and the number of foreclosures in 2012, the area that will exceed approximately 25 foreclosures will be the following census tracts: 120.01, 119, 116, 105.01, 118, and 30.01 all shown in figure 4. These tracts had an increase in foreclosures from 2011 to 2012 and had at least 23 foreclosures in 2012. 

Figure 4. Census tracts that will exceed 25 foreclosures.

The area that will most likely exceed approximately 4 foreclosures will be most of the map except for the following census tracts: 25, 130, 17.05, 9.02, 101, 12, and 4.01.  These tracts, pictured in figure 5, all have less than four foreclosures in 2012 and had a decrease in foreclosures from 2011 to 2012. 

Figure 5. Census tracts that will not exceed four foreclosures.

  Finally, the mean and median for 2011 were 11.39 and 11 respectively and 12.30 and 10 for 2012 respectively.  These values indicate that in 2012 there was a large outlier that influenced the mean in 2012 because the mean and median had a difference of 2.3 while 2011 only had a difference of 0.39.  This would indicate that some county had a significant increase in foreclosures in 2012 from 2011.  This finding supports the previous discussion on census tract 120.01 which showed dark red in figure 3 and had a large z-score in 2012. 

Conclusions:

There were both increases and decreases in foreclosures in Dane County from 2011 to 2012.   Z-scores calculated on three different census tracts mirrored the results in figure 3 and validated the data.  Although county officials were concerned about the increase in foreclosures from 2011 to 2012, only 60% of census tracts showed an increase in foreclosures, while the rest saw none or a decrease in the number of foreclosures.  In addition, the mean number of foreclosures from 2011 to 2012 went from 11.39 to 12.30, a 0.91 increase, not even by one foreclosure.  The median number of foreclosures actually decreased from 2011 to 2012, going from 11 to 10.  This difference can be explained by the previously mentioned 3 census tracts with an increase of foreclosures, which increased the mean.  These results indicate that although a slight majority of census tracts saw an increase in foreclosures, the median number went down (a number that isn’t influenced by large outliers) which indicates an overall decreasing trend. 

 The results suggest than county officials should focus their foreclosure reduction efforts in the counties with a chance of exceeding 25 foreclosures shown in figure 4 because these counties are at the most risk of continuing to see foreclosures as opposed to the previously mentioned tracts that will not exceed even four.  The median of 2011 and 2012 also suggest that foreclosures are not increasing for the whole county but rather for a select few census tracts. More data is needed to determine the cause of the spatial patterns.  The general pattern that can be derived from figure 3 and the calculations suggests that the likelihood of an increase in foreclosures in 2013 is not equal for all census tracts. County officials should not be as concerned over the increases for the county and instead focus on decreasing the rate of foreclosures in the select counties that will likely see increases.

Sources:

Weintraub, E. (2017, February 26). What is a foreclosure: how do foreclosures work. Retrieved from https://www.thebalance.com/what-is-a-foreclosure-1798185
Taylor, C. (2014, May 21). Z score formula. Retrieved from https://www.thoughtco.com/z-score-formula-3126281
Census tract and foreclosure data provided by Dr. Ryan Weichelt of the University of Wisconsin- Eau Claire.