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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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Symmetry Paper

Histogram of NonSymmetric Data Set

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