Assignment 2-Descriptive Statistics and Mean Centers

Purpose:

The purpose of this assignment is to practice using a variety of statistical methods and computer programs in real world scenarios. The first scenario involves analyzing test scores for two high schools in Eau Claire, Wisconsin to determine whether or not the school's teaching methods are effective. The second scenario involves using population data to find the Geographic and Population Mean centers for the state of Wisconsin.

Scenario 1.

I have been given a series of test scores to analyze by the Eau Claire Area School District. Traditionally one high school, Eau Claire Memorial, has had higher scores that the other school, Eau Claire North. This has led some to believe that the teaching methods at Eau Claire North are inferior and that Teachers who's students have lower scores should be fired. It is my job to use a variety of statistical methods to determine if this is the case

The statistical methods used in this analysis are as follows

Mean: The mean is a measure of the average of a set of numbers. It can be find by adding the numbers in a set and then dividing by the total amount of observations

Median: The median is the middle number in a number set. It can be used as a reference point to analyze higher or lower observations

Mode: The mode is the number that appears most frequently in a number set

Range: The range is the range of numbers between the lowest and highest observation. It can be found by subtracting the lowest number in the set from the highest.

Skewness: Skewness is a measure of how data is influenced by outliers, or observations that do not fit in with the average. A positive skew would be the result of an observation far greater than the average. A negative skew would occur if an observation is far below the average.

Kurtosis: Kurtosis is a measure of how peaked a distribution is. A peaked distribution would be known as leptokurtic and indicates that you have a higher concentration of observations in one part of the distribution (Several similar test scores and a few outliers). A flat, or platykurtic kurtosis indicates a more even distribution (wider range of scores between lowest and highest)

Standard Deviation: The standard deviation is a measure of how observations deviate from the average of a number set. It is found using the standard distance equation. For each observation, the average for all observations is subtracted and the difference is squared. These numbers are then averaged and the square root of the average is the standard deviation. Almost all observations will fall within 3 standard deviations of the average. For example, Memorial high the average score of 161 with a standard  deviation of 26. This means that one standard deviation of 161 is plus or minus 26 points.

Below are the calculations and results for this analysis. These were performed in Microsoft Excel with the standard deviation being calculated by hand.

Eau Claire North High School Test Scores

Mean: 4184/26=160.92

Range: 194-111= 83

Mode: 170

Median: 164.5

Skew: -.57

Kurtosis: -.55

Standard Deviation:

(111-160.92)^4=2492

(120-160.92)^2= 1674.44

(130-160.92)^2= 956.04

(135-160.92)^2=671.84

(142-160.92)^2=357.96

(145-160.92)^2= 253.44

(149-160.92)^2=142.08

(153-160.92)^2=62.72

(154-160.92)^2=47.88

(162-160.92)^2=1.16

(164-160.92)^2=9.48

(165-160.92)^2=16.64

(170-160.92)^2=82.44

(175-160.92)^2=198.24

(176-160.92)^2=249.64

(180-160.94)^2=363.28

(182-160.92)^2=364.04

(184-160.92)^=444.36

(188-160.92)^2=532.68

(189-160.92)^2=133.32

(192-160.92)^2=788.48

(194-160.92^2=965.96

SqRt(Sum of Values/25)=23.45

Eau Claire Memorial Test Scores

Standard Deviation:

(107-161.42)^2=2961.53

(120-161.42)^2=1715.616

(135--161.42)^2=698.01

(137-161.42)^2=596.3364

(140-161.42)^2=458.8164

(145-161.42)^2=269.61

(148-161.42)^2=180.09

(149-161.42)^2=154.25

(154-161.42)^2=55.05

(165-161.42)^2=12.81

(167-161.42)^2=31.13

(175-161.42)^2= 184.41

(182-161.42)^2=423.53

(184-161.42)^2=509.85

(189-161.42)^2=160.654

(190-161.42)^2=816.81

(193-161.42)^2=997.29

(194-161.42)^2= 1061.45

(198-161.42)^2=1138.09

Sum= 16966.35

Sum/25 (N-1) = 678.65

SqRt678.65=26.05

Mean: 161.4

Mode: 145

Median: 166

Range: (198-107)= 91

Skew: -.35

Kurtosis:-.91

Results/Interpretation

Based on these results I feel that the two schools are not as far apart as first predicted. Looking at the average test score, Eau Claire Memorial's average is only higher by 1 point. The median scores are also similar. The statistics that provide the best insight are the skewness values for each school. Eau Claire north has a negative skew value. This indicates that the average is being affected by lower scores which are outliers. This does not mean that all students scores are lower on average. In fact, Eau Claire Memorial, which traditionally has higher test scores, also had the lowest score of 107 out of 200 points. If the average score were lower with a skewness value closer to zero, this would indicate lower overall scores. However since the averages are similar I believe the teachers at North should not be fired. 

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Part 2

The second part of this assignment focuses on finding mean centers. A mean center shows the geographic average location of data. For example, a geographic mean center for a states area would be the geographic center of the state. The mean center for a population would be average spot where people are located in the state. By comparing population mean centers over time one can get an idea of how the population overall has migrated.

This example looks at the geographic and population mean centers for the state of Wisconsin

As can be seen on the Map, the geographic mean center for Wisconsin is located in Wood County. This represents the geographic center of the state. The population mean centers are located in Green Lake county. The population mean centers represent the average location where population is centered in the state.

It makes sense that the population mean center would be in Green Lake county as it is surrounded by three of the largest demographic areas in the state. To the northeast is the Fox Cities area and Green Bay in Outagamie, Calumet, Winnebago and Brown Counties. To the southeast is the Milwaukee metro area, the most populated area in the state. Finally, to the south is Madison and areas surrounding the state capital. One trend that may be noticed is that between 2000 and 2015 the population mean center has shifted slightly to the southwest. This would indicate a larger population in that general direction that is influencing the average. I would guess that this caused by an increase in  population in the Madison/Sun Prairie area of Dane county. Other than being the state capital, Madison is home to the flagship school of the University of Wisconsin system as well as being home to several industries. The combination of the two would draw students and young professionals to the city and surrounding areas which would shift the overall average.