Part 1: Calculations of Data by Hand
In this part of the lab a sample of test scores from the juniors of two high schools, Eau Claire North and Eau Claire Memorial, in the Eau Claire School District are analyzed. The fact that Eau Claire North high school does not have the highest test score and historically the highest test score comes from Eau Claire Memorial high school has led to concerns from the public leading to criticism of the teaching methods at Eau Claire North high school. The understanding of the public is that the teachers at Eau Claire North should be fired because the test scores at that high school are so much lower.
Figure 1. The given sample of test scores from the two high schools.
The 7 different analyses conducted:
Range: the difference between the highest and lowest value in a data set.
Mean: the average value of the data set.
Median: the value that appears in the middle of the spread of data when it is organized in numeric order.
Mode: the number that appears most commonly in a data set.
Kurtosis: in a distribution curve of the data set, the sharpness of the peak of the data.
Skewness: the measure of the symmetry of the distribution curve of the data set.
Standard Deviation: the amount the data set differs from the calculated mean value.
There are two different calculations for standard deviation.
1. Population Standard Deviation: used when the whole data set is being used. An example would
be using 12 months of weather data when looking at weather data of a whole year.
2. Sample Standard Deviation: used when only a sample of a data set is being used. An example would be if you are looking at weather data for a whole year but only use 6 months of the data.
Figure 2. The calculated results for the 7 analyses conducted for each high school.
Figure 3. The hand written data calculations Figure 4. The hand written data of calculations
for the standard deviation of the test scores for the standard deviation of the test scores of
of Eau Claire North high school. Eau Claire Memorial high school.
Figure 3 and Figure 4 show that in the standard deviation calculation for both high schools, the sample population standard deviation equation was used because only a sample or small portion of the test scores from the two high schools. Every test score of every junior who took the test is not used.
The teachers at Eau Claire North should not worry about not having the highest test grade. The statistics calculated suggests that the test scores at Eau Claire North are overall better than at Eau Claire Memorial. The teachers at North should not be having loosing their jobs threatened.
The results in Figure 2 highlight the statistics that prove that Eau Claire North has better test scores. The most common test score (mode) is higher at Eau Claire North. The average (mean) score is better at Eau Claire North. Meaning that overall there were higher than low scores at Eau Claire North. The standard deviation at Eau Claire North was higher also, which means that the data deviated less from the mean than Eau Claire Memorial. The test scores at North were more concentrated around a higher mean value than at Memorial where test scores were less concentrated around a lower mean. The range is smaller at Eau Claire North than at Eau Claire Memorial, while Memorial may have the highest score, it also has the lowest score. Eau Claire North has the larger middle number, median, suggesting that there are less low test scores than at Memorial. Both high schools have negative kurtosis meaning that the distribution of the test scores at both schools is flat and not peaked. This means both schools have broad distributions of test scores. Both high schools have negative skew. This suggests that both schools have a higher concentration on test scores on the higher end of the mean. North has a skew value that deviates from 1 more than memorial meaning that there is a larger concentration of high test scores than at Memorial. Overall the statistics show that Eau Claire North has better overall test scores as Eau Claire Memorial, even though Eau Claire Memorial has the highest test score. The teachers are North should not be worried about losing their jobs, and the public should stop criticizing their hard work.
Part 2: Calculating Mean Centers and Weighted Mean Centers
For part two of the lab population data from 2000 and 2015 for Wisconsin Counties was used to calculate and compare mean centers of Wisconsin. Three different mean centers were calculated by using ArcMap: geographic mean center of Wisconsin, weighted mean center of population from 2000, and weighted mean center of population from 2015. In ArcMap in ArcToolbox under the Statial Statistics Tools in the Measuring Geographic Distributions tool set the Mean Center operation was used. Mean Centers are calculated by using the average x and y coordinates in a selected geographic area. By weighting the mean center by population changes in population between 2000 and 2015 can be tracked and compared.
Figure 5. Distribution of calculated mean centers in Wisconsin.
The geographic mean center for Wisconsin was very different from the two weighted calculated mean centers. The geographic mean center resides in the exact middle of the state. In Figure 5 it can be seen that the geographic mean center is farther north and west than the two mean centers weighted by county populations. This can be explained by larger, more heavily populated cities in Wisconsin like Racine, Milwaukee, and Madison being located in the more southern and eastern part of the state. That moves the population based mean centers closer to these heavily populated cities and farther away from the geographic mean center. The large city of Green Bay is also located on the eastern side of the state, moving the population mean centers eastward. The mean centers that were weighted based on county population data are very similar. The mean center based on the Wisconsin county populations of 2015 is more western than the mean center based on the Wisconsin county populations of 2000. I think this shift between the two mean centers occurred because cities in the western part of the state grew in population's size, like the City of Eau Claire. Overall all of the mean centers are located in the central region of Wisconsin, but the weighted mean centers are located more in the southeastern part of the state than the geographical mean center.
Citations:
Kurtosis Formula. (n.d.). Retrieved October 03, 2017, from http://www.macroption.com/kurtosis-formula/.
Skewness Formula. (n.d.). Retrieved October 03, 2017, from http://www.macroption.com/skewness-formula/.
