Descriptive Statistics and Mean Centers
Statistical
Analysis of Test Scores
I have been asked to analyze the test scores from two high
schools in the Eau Claire area. These schools are Eau Claire Memorial and Eau
Claire North. As test scores are usually used as an indicator on the success of
a school, by comparing the two, people can get a sense of which school is doing
better. In the past, Eau Claire North has always scored less on standardized
test compared to Memorial which leaves some people in the community wondering
if Eau Claire North has an issue. To fix this, members of the community have
suggested that the teachers are to blame and should be replaced.
It is my job to look further into the issue and either
confirm or deny the public's allegations. To do so, I will examine a sample of
each schools test scores and use various statistical methods to make my
analysis. These tools are the following:
Range: The
amount between the lowest data point and the highest data point. To find this,
you subtract the lowest from the highest. The resulting value is your range. For
the test scores, that would be the lowest score and the highest score.
Mean: The
statistical average of your dataset. It is found by taking the sum of your data
points and dividing it by the number of values. This is the average score on
the test for each dataset.
Median:
The exact center of your data. Once ordered from smallest to largest the middle
value or the middle of the two middle values is your median. In regards to the
test scores, this may not be one of the scores, but rather the middle of the
two center ones.
Mode: The
value that repeats the most in a data set. The test score that is most commonly
gotten.
Kurtosis: The
shape your date makes when it is distributed and the change in shape compared
to that of a normal distribution. This is usually a measure of if the shape is
wider or thinner than normal.
Skewness:
The amount the data points differ from the mean, higher or lower. When more
data points fall on one side or the other of the mean, the data is skewed
either positively or negatively, often showing outliers in your data.
Standard
Deviation: Value that displays how the data is located around the mean
and the distance that a data point is from the mean.
Figure 1: Data spreadsheet
After doing the calculations for both samples, figure 1
displays my findings. There are some key comparisons to consider when trying to
figure out which school does better. First, when looking at the range, you can
see that EC North’s range is lower than that of EC Memorial’s. This means that
EC North’s scores are closer together in the span of 83 points. EC Memorial’s
scores are distributed more sparsely covering 91 points. North’s scores are
thus more consistent than Memorial’s, but not by much. Secondly, when you examine each schools score
means, you can see that North averages a better score on standardize tests by
roughly 2.5 points. That value debunks the idea that Memorial does better on
these tests. Additionally, the median for North’s data set is also higher than
Memorials, another indicator that North’s students’ scores are over all higher.
Even the mode, a near pointless measure is higher for EC North. From those four
figures, it is already apparent that EC North does better on standardize
testing than EC Memorial. Next, looking at the kurtosis values, North is within
the normal distribution while Memorial has a platykurtic, or wider
distribution. This is important and relates back to range showing that the
Memorial scores are less consistent. The 6th measurement is
skewness, and while they are both negatively skewed, EC North has a greater
skew meaning that more of their scores compared to Memorial are above the mean.
Lastly, when we look at the standard deviation, EC North has a smaller value of
23.635. This means that 68% of the scores fall within 23.635 points from the
mean as opposed to 27.158 points for Memorial.
Now
that is a lot of figures and values and to many it does not mean much. The
overall point of the analysis is that the public’s assumption that Eau Claire
North scores worse than Memorial is false. Yes, Memorial has high scores and
seemingly more of them, but when it is broken down and analyzed, the statistics
point to North as the stronger school. Regarding the proposal to fire teachers
from Eau Claire North, they should feel safe in knowing that they are having
success in their methods and it is showing on standardized tests. The biggest
indicators of this in my opinion are the range, mean and skewness. North’s
range is smaller than Memorial’s showing that as a whole, their scores are
closer together. That is even more important when you look at the mean. First,
the mean is higher than Memorials’. Simple as that, their scores are better. Lastly,
skewness shows us that North had fewer scores below the mean than above it.
Most people in the sample score above it.
Part II
Mean Centers and
Weighted Mean Centers
Tasked with finding a couple of mean centers for the state
of Wisconsin, I joined my dataset with a shapefile of Wisconsin using ArcMap.
The data used is from the United States Census Bureau. A Mean Center is the middle of a geographical area based upon the x and y values of that area. A Weighted Mean Center is a mean center that uses specific data to find the middle of. For this assignment, that is population, meaning the weighted mean center will find the center of the state based on the distribution of population in the state. I was asked to
find 3 mean centers, the geographical mean center, the mean center weighted by
the population in the year 2000 and the mean center weighted by the population
of 2015. Each point is located on the map below.
As you can see in figure 4, the geographical mean center, as
indicated by the green star is located in the center of the state in eastern
Wood County. Used as a reference point, the differences from the geographical
center compared to when you weight it by population is drastic. The mean center
weighted by the 2000 population is symbolized by the red circle. When you take
the center based off population it becomes clear that the majority of the state’s
population is located in the southern counties. Additionally, more people live
in eastern Wisconsin than western. This is shown by the location of the mean
center for the 2000 population, located in Green Lake County, noticeably shifted
south west from the geographical center. This is not surprising as Wisconsin’s
two largest cities, Milwaukee and Madison are both in this region of the state.
Next, when you add the mean center based on the population in 2015, we see
another shift, this time south of the 2000 mean center. While the shift is
minor, it does indicate some changes throughout the state. In that 15 year
period, the southern part of the state became even more concentrated with
population.
With these changes in mean center, we can infer they have
changed for a number of reasons. First, in the middle of the time period
examined, 2000 to 2015, the United States had a major recession. This recession
hit rural areas especially hard. People struggled with money and many lost
their homes or relocated to find new jobs. As the northern part of the state is
largely rural, the shift in where population is located even farther south
could have to do with the recession. As more and more people relocate to the
south to find jobs, the mean center shifts south, following them. Another
possibility is the trend that it is becoming increasingly harder to make a
decent living off agriculture. This trend follows the same outcomes as the one
stated above, forcing framers to move to more urban areas for jobs. A major
reason for the population centers to be shifted towards the east is the urban development
of the eastern part of the state. Areas like Milwaukee County and Brown County have
been steadily growing, meeting the demands of the national economy.
While these reasons are all just speculation, it goes to
show that changes in mean centers weighted by population are subject to a wide
range of variables. I’m sure that these moves could be seen in many states and
that the trend of population shifting from rural to urban is not limited to
Wisconsin, but seen across the nation. Using a map to visualize where
population is focused gives the reader a lot of insight to the makeup of the
state. You wouldn’t even need to know much about Wisconsin to make assumptions
based off the mean centers.
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