ILS 4430 Histogram of Non Symmetric Data Set Presentation

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Symmetry Paper (Non-Symmetric Data Set)
Section I: Histogram of Non-Symmetric Data Set
μ = 26.66086957
σ = 17.37556611
Section II: Calculations for Non-Symmetric Data Set
Mean of Data Set: μ = 26.7
Standard Deviation of Data Set: σ = 17.4
Calculated Mean of Sample Means (samples of size 10, 30, 100): μx̅ = μ = 26.7
σ
Calculated Standard Deviation of Sample Means (samples of size 10): σx̅ =
√n
Calculated Standard Deviation of Sample Means (samples of size 30): σx̅ =
√n
σ
Calculated Standard Deviation of Sample Means (samples of size 100): σx̅ =
=
=
σ
√n
𝟏𝟕.𝟒
√10
𝟏𝟕.𝟒
√30
=
= 5.5
= 3.2
𝟏𝟕.𝟒
√100
= 1.7
Section III: Histograms of Sample Means and Actual and Calculated
Statistics of Sample Means for Non-Symmetric Data Set
Histogram of Sample Means for Samples of Size 10:
Actual Statistics
μx̅ = 24.9
σx̅ = 4.6
Calculated Statistics
μx̅ = 26.7
σx̅ = 5.5
Histogram of Sample Means for Samples of Size 30:
Actual Statistics
μx̅ = 25.7
σx̅ = 3.3
Calculated Statistics
μx̅ = 26.7
σx̅ = 3.2
Histogram of Sample Means for Samples of Size 100:
Actual Statistics
μx̅ = 26.1
σx̅ = 0.8
Calculated Statistics
μx̅ = 26.7
σx̅ = 1.7
Section IV: Discussion of the Central Limit Theorem for NonSymmetric Data Sets
A sampling distribution is constructed by taking several of the same sized samples
from the population data, calculating the sample mean of each of the randomly selected
samples, and creating a histogram of these sample means.
The Central Limit Theorem states that the sampling distribution of sample means will
approximate a normal distribution regardless of the population distribution shape, given
that a sufficiently large sample size is used. Sufficiently large is generally assumed to
describe a sample size greater than 30. In other words, the symmetry of sampling
distributions of sample means is predicted by the Central Limit Theorem to increase as the
sample size used to construct the distribution increases. The average of the sampling
distribution is always equal to the mean of the population, and the standard deviation of
the sampling distribution decreases as the sample size increases.
Additionally, the CLT provides the following equations for the mean and the standard
deviation of the sampling distribution:
Mean: μx̅ = μ
Standard deviation: σx̅ =
σ
√n
Section V: Discussion of Results for Non-Symmetric Data Set
The original data contained 115 data points, which had a mean of 26.7 and a
standard deviation of 17.4. A histogram of this population data shows the distribution
following a shape that is moderately uniform and skewed to the right. The majority of the
data lies within the first 3 bin’s of the histogram, with only a few observations in the
remaining 2 bin’s. However, when sampling distributions (samples of size 10, 30, 100) for
this data are constructed, histograms show the shapes of the three distributions converge
to a relatively normal, symmetrical curve. Furthermore, the shape of the distributions
becomes more normal as a larger sample size is used. This phenomenon can be explained
by the Central Limit Theorem.
The first sampling distribution (n=10) was generated for the sample means of
samples of size 10. The shape of this distribution had a clear center but is still skewed
slightly to the right. Both the average and standard deviation of this sampling distribution
are slightly lower than the calculated theoretical values for the distributions, as well as
lower than the mean and standard deviation of the population distribution.
The second sampling distribution (n=30) follows more of a symmetrical shape than
the first sampling distribution generated. The distribution has a unimodal, slightly normal
shape that includes one gap. The mean of this distribution is closer to the population mean
than the first sampling distribution. As expected, the standard deviation for this sampling
distribution is much less than the population standard deviation. It is also less than the
standard deviation for this sampling distribution with a sample size of 10. The actual
standard deviation for the sampling distribution is very close to the estimated calculated
sampling distribution standard deviation value.
The third sampling distribution (n=100) displays the most symmetrical, relatively
normal shape of the three sampling distributions. Similarly, the average of this distribution
is the closest to the theoretical sampling distribution average of the three. This distribution
has a very small standard deviation when compared to the other sampling distributions and
the population distribution.
Ultimately, as the sample size of the sampling distribution increased the shape of the
distribution became more normal and symmetrically shaped, the estimates of the sampling
distribution means increased and became more accurate, and the estimates of the sampling
distribution standard deviations decreased and became more precise.
Section VI: Discussion of Unexpected Results for NonSymmetric Data Set
Although the shapes of the histogram begin to converge to an approximately normal shape
as the sample size increases, there is a larger amount of observations within the first bin of
the histogram than one would expect for the final histogram (n=100).
Some possible explanations for this unusual result are:
1. The population distribution was extremely non-normal.
Since the shape of the population distribution was extremely non-symmetrical and
skewed to the right, the sampling distributions based off this original dataset would
follow a shape that are similarly skewed, just to a lesser degree than that of the
population distribution.
2. The sample size used was not large enough.
Although the rule of thumb is that the CLT applies for sample sizes greater than 30,
stronger skewed distributions can require larger sample sizes to approximate a
completely normal shape.
3. The random samples had a larger variability or amount of outliers than expected.
The randomly selected samples used to construct the sampling distribution had – by
chance – a larger number of outliers or a greater amount of variability, which caused
the shape of the sampling distribution to remain more skewed.
Symmetry Presentation – Instructions
Throughout this term, we have studied symmetry in mathematics. We have
considered translational and rotational symmetry, wallpaper and frieze patterns,
tiling’s, the symmetry of regular polygons, the role of groups in the mathematics of
symmetry, symmetry in numbers, symmetry in statistics and the self similarly of
chaos.
For this presentation, which is worth 10% of your final grade, you will discuss
symmetry in an academic field or area of interest outside of mathematics, and the
connection that this symmetry has to the mathematical symmetries that we have
studied throughout the term. Examples of academic fields or areas of interests that
you may consider for your presentation include, but are not limited to: logic,
physics, biology, ecology, chemistry, economics, sociology, psychology, paintings,
sculptures, architecture, music, literature, and poetry.
Presentations may be done individually, or in groups of two, should be between
five and ten minutes long, and should consist of at least ten power point slides. The
following presentation format is recommended, but not required:
Part I: Make some opening remarks. Introduce the academic discipline or area of
interest that you have chosen to discuss symmetry in. Explain why you have
chosen this academic discipline or area of interest.
Part II: Give a general definition of symmetry.
Part III: Define symmetry in the context of the academic discipline or area of
interest that you have chosen.
Part IV: State at least three different types of symmetry found in the academic
discipline or area of interest.
Part V: Give some examples from the academic discipline or area of interest that
illustrate the type(s) of symmetry you discussed in Part IV, and if possible, make
connections between these symmetries and the symmetries of mathematics. When
applicable, use figures, charts, tables, etc. to illustrate your examples.
Part VI: Make some concluding remarks.

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Histogram of NonSymmetric Data Set

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