Assignment 2 Geography 370

Goals and Background

The goal of the following assignment is to become familiar with a variety of statistical methods including Range, Mean, Median, Mode, Kurtosis, Skewness, and Standard Deviation. Furthermore, we will familiarize ourself with more programs such as MS Excel and ESRI Arc Map to help compute data we were given.

Definitions

Range

Range is the difference between the the largest value and the smallest value in a set of data. Example) say you had a data set of 1,3,5,7.  The range would be 7-1, which is 6.

Mean

Mean is average of all the numbers in a given data set. The mean is calculated by adding all of the numbers together and dividing the total by the total numbers of values in the data set. Example) 1,3,5,7.  1+3+5+7= 16.  16/4=4.  The mean is 4 in this data set.

Median

Median is the number which falls in the middle of a data set when put in order from smallest to largest. Example) 1,2,3,4,5.  3 is your median in this case.  If your data set has an even number of observations, you simply take the middle two values, add them together, and divide by 2.  Example) 1,2,3,4.  2+3=5.

5/2 =2.5.  2.5 would be the median in this case.  

Mode

Mode is the number which occurs most often in a data set. Example)  1,2,3,3,3,4,4,5,6.   In his case, 3 is the mode.  *Note* If there are two numbers that occur at the same frequency, then there can be two modes.

Skewness

Skewness describes the balance of the histogram compared to a normal distribution. There are 3 types of skewness: positive, no skew, and negative.  Positive skew is when the outliers in a data set are on the positive, or right side  of the mean (Fig.1). Negative skew is when the outliers in the data set are on the left side of the mean (Fig.1).  No skew means there is an even distribution of the data and the graph looks like a bell-shaped curve. 

Figure 1: Images containing postive and negative skewness http://study.com/academy/lesson/skewness-in-statistics-definition-formula-example.html

Kurtosis describes the shape of the histogram whether its "steep," or "flat" o compared to the "normal distribution.There are 3 different ways to describe Kutosis: Leptokurtic, Mesokurtic, and Platykurtic.   Leptokurtic is a description of a very peaked or "steep" distribution. Mesokurtic is deemed to have a  "normal" distribution. Platykurtic is a short or a "flat" distribution. Additionally, Platykurtic is described as negative Kurtosis and Leptokurtic is positive Kurtosis. Figure 2 below is an example of the three types of Kurtosis. When analyzing Kurtosis calculations anything greater than 1 is Leptokurtic and below -1 is Platykurtic.

Figure 2: The forms of Kurtosis.  http://mvpprograms.com/help/mvpstats/distributions/SkewnessKurtosis

Standard Deviation

Standard Deviation is a statistical measurement which describes how spread out the numbers in a data set are from the mean. "1 Standard Deviation" from the mean is equal to 68.2% of the values in a data set. "2 Standard Deviations" from the mean is equal to 95.4% of the values in a data set.  "3 Standard Deviations" from the mean is equal to 99.7% of the values in a data set. 

Assignment Description - Part 1

We were given the following scenario by our professor:

Cycling is often seen as an individual sport, but it is actually more of a team sport.  You are looking to invest a large sum of money into a cycle team.  While having a superstar is nice and brings attention, having a better team overall will mean more money in your pocket.  In the last race in the TOUR de GEOGRAPHIA, the overall individual winner won $300,000, with only 25% going to the team owner, but the team that won, gained $400,000 in a variety of ways, with 35% going to the team owner.  

Using the incredible set of knowledge learned in your Quant Methods class at UWEC, you decide to put it to good use.  You have data (total time for entire race) for teams and individual racers over the last race held in Spain. To begin your investigation you are to analyze the race times of members from the team. Traditionally Team ASTANA has typically produced the race winner (meaning the rider that finishes first), but an up and coming group named Team TOBLER has been making waves on the cycling circuit.  

The questions that will be answered are as follows:

Should you invest in Team ASTANA or gamble on Team TOBLER?   Why did you pick one team over another?  What descriptive statistics do you think best help explain your answer?  Please explain your results using the statistics to support your answer **Please explain results in hours and mins. 

Methods

For this assignment I had to calculate the Range, Mean, Median, Mode, Kurtosis, Skewness, and Standard Deviation for the race times provided to me.The standard deviation was to be calculated by hand and shown on paper, as figures 3 and 4 provide the proof below.  The rest of the results could be calculated in MS Excel itself.

I first calculated the standard deviation of each team (Fig. 3-4) using the population standard deviation formula.  Figure 3 and 4 provide the work I did for both Team Tobler and Team Astana.  

 I then copied the provided data and imported it into Excel so I could sort the numbers in descending order.  After  that, I was able to calculate the Range, Mean, Median, Mode, Kurtosis, Skewness for each team using Excel and the results are shown below in table 1.

Figure 3: Calculating the Standard Deviation of Team Tobler

Figure 4: Calculating the Standard Deviation of Team Astana

Table 1 Results

Table 1

Discussion and Answer

It's hard to answer whether I'd invest in team Astana or gamble on Tobler.  Based on the results of the data set given, I would likely invest with team Astana because the mean is a very safe number.  However, this is the results of one race, and one race results isn't enough for me to make a solid decision on which team to invest money into.  
Some more reasons I would choose Team Astana over Team Tobler is that the total time (sum)  for Astana was equal to 569 hours and 10 minuets versus Tobler which had a total time of 571 hours and 21 minuets.  Team  Tobler had a lower standard deviation, meaning the team had fewer outliers and most the team finished around the same time.  In spite of the lower standard deviation, the majority of team Tobler finished behind team Astana. Team Astana had a higher standard deviation because of Racer K, who possibly had a bad day running, which in turn made Team Astanas standard deviation higher.  




Assignment Description - Part 2

For part 2 we are to calculate the mean center and weighted mean center for the population of Wisconsin by county for 2000 and 2015.  Before that, I will provide the important definitions for part 2 below.  

Mean Center

The mean center is the average location of points which have an X and a Y value and are plotted on a graph or Cartesian Plane.  

Weighted Mean Center

The weighted mean center is a set of points that is adjusted in order to influence a second value associated with each point.  The difference between this and mean center is that the points have weights or "frequencies" attached to them (Fig. 5)


Figure 5: Geographic Mean Centers of Wisconsin Population from 2000 and 2015.

Discussion and Conclusion

In the above map, you can see the mean center is centrally located in the state.  The mean center is calculated off of the center point of each county, which is why it is located in the center of the state. The weighted mean centers are still in central Wisconsin, but further south from the mean center.  That is because the weighted mean center takes the population of counties into account.  The more populated counties are all in the southern and southeast part of the state, specifically Milwaukee County and its' surrounding area.  You can see that the weighted mean center from 2015 is slightly west of the 2000 weighted mean center.   It appears that there has been a population shift from 2000 to 2015, possibly to get closer to other working centers such as La Crosse, or the Twin Cities in Minnesota.