Statistics Assignment: Descriptive Statistics and Probability Analysis

This statistics assignment is divided into two parts. Part 1 involves probability calculations using z-scores to determine probabilities for different scenarios, such as the probability of a value exceeding a certain threshold and the probability of a value falling within a range. Part 2 focuses on descriptive statistics, analyzing property taxes per capita across 50 states. It includes calculations of mean, median, mode, standard deviation, skewness, and kurtosis, along with visual representations like histograms and scattered graphs. The assignment highlights the application of these statistical measures to understand the distribution and variability of property tax data, providing insights for business decision-making. The analysis includes a conclusion summarizing the findings and the utility of descriptive statistics in drawing valid conclusions.

Statistics Assignment

Table of Contents
PART 1............................................................................................................................................1
PART B............................................................................................................................................1
Introduction:.....................................................................................................................................1
Conclusion.......................................................................................................................................5
REFERENCES................................................................................................................................7

PART 1
a). Under the given question, probabilities for randomly selected individual is calculated by
using: (X-Mean)/SD.
Z=(25-21)5=+0.8
Area in standard normal curve right to the z value represents the required probability.
P(X>25)=P(z>0.8)=0.2119.
b). z1= (10-21)/5= -2.2
z2=(20-21)/5= -0.2.
Area which requires the standard normal curve between these two z values mention the required
probability.
P(10<X<20)=P(-2.2<z<0<0.2)=0.4861 (area corresponding to z1)- 0.0793 (area corresponding to
z2)= 0.4068.
c). Middle 95% is depicted by 0.9500 area about mean.
Which implies which 0.9500/2= 0. 4750. area lies on accurate side of the mean. The z value
corresponding to 0.4750 area is 1.96 henceforth, required X values are mean +/-(z*SD). 21+/-
9.8
Middle 95% of the amount of cash spent would be reduce between 11.2 and 30.8.
PART B.
Introduction:
Property taxes are the key major revenue source which is used by the state and the local
government. Nowadays, state property tax format covers only a subset of total income: mainly
homes and business real estate and, under few states, cars and the business property instead of
the real estate. Regressivity of property taxes is totally based on the issues under the control of
the policy makers, like- implementation of the exemptions, tax credits, and preferential tax rates
for homeowners, and on external components. Mean, median, mode are used almost entire levels
of myth and statistics1. Although, there are abnormal types of descriptive statistics which are
most crucial. Under part B, various business tools are used such as entire descriptive statistics
that are able to help various business decisions regarding property taxes among 50 states.
1 Ghadle, K. P., and Y. M. Muley. "Revised Ones Assignment Method for solving assignment
problem." Journal of Statistics and Mathematics 4, no. 1 (2013): 147.
1

Under the given research, descriptive coefficients summarize which summarize
highlighted data set, that could be either a representation of the whole population or a sample of
it. Descriptive statistics are segregates into measure of central tendency and measure of
variability, or spread. Measure of central tendency breaking down into the measure of central
tendency and measures of variability, or spread. Measures of central tendency covers the the
mean, median and mode 2. On the other hand measure of central tendency covers the mean,
median and mode, while measures of variability covers standard deviation or variance.
Under the given case, this has been seen that the property taxes per capital is determined,
and these are apportioned accordingly. The below mentioned table is defined as follows:
property taxes per capital
($) Number of State
200 0
400 2
600 7
800 6
1000 11
1200 11
1400 4
1600 4
1800 2
2000 3
2200 1
2 Peters, Jochen, Evgeny Matusov, Carsten Meyer, and Dietrich Klakow. "Text
segmentation and label assignment with user interaction by means of topic specific language
models and topic-specific label statistics." U.S. Patent 8,200,487, issued June 12, 2012.
2

Mean 1040.862745098
Standard Error 60.0073918297
Median 981
Mode 963
Standard Deviation 428.5384939353
Sample Variance 183645.240784314
Kurtosis -0.1113554964
Skewness 0.6042971823
Range 1732
Minimum 367
Maximum 2099
Sum 53084
Count 51
3

4

From the above mentioned histogram and scattered graph, this has been seen that
property taxes per capital for 50 states were different. Almost 2 states who have property taxes
per capital is $400. while 7 states are paying property taxes per capital that is $600. 6 states have
$800 per capita income. 11 states and 12 states have consecutively have $1000 property taxes. 4
states have 1400 and 1600 property taxes. While 2 states have 1800 property taxes per capital. 3
states have 2000 property taxes and only 1 state have $2200.
This above mentioned calculation reflects the descriptive data. Which is used by the researcher
in order to draw a valid conclusion. From the above mentioned calculation, this has been seen
that 50 states, property taxes per capita is different. The mean is the average of the set of
variances of 50 states' capita income and calculated 1040.87. Standard error demonstrates the
standard distribution of statistics3. This is a statistical term which analyse the accuracy of the
overall states' actual mean of per capita income; which might lead to deviates standard error. The
standard error of this entire calculation is 60.
Median is the middle number of the set of variances which has been arranged for
magnitude. The value calculated does not completely affected by the outliers and skewed data.
Middle value of the property taxes as per capita of entire states' is 981.
Mode represents the most frequent number in firm data set. On the above mentioned
histogram graph, this reflects the biggest bar under a bar chart or histogram. However,
sometimes opting the mode as being the most frequent option.
Standard deviation: It is the measure which summarises the amount under which each
value within a dataset changes from the mean. In an efficient manner, this demonstrates how
roughly the values in the dataset are bunched around the mean value. This is the most robust and
highly implemented measure of dispersion. Which is used to take into account on each variable
in the dataset. The standard deviation is mainly reflected in conjunction with the mean and is
3 Schepaschenko, Dmitry, Ian McCallum, Anatoly Shvidenko, Steffen Fritz, Florian
Kraxner, and Michael Obersteiner. "A new hybrid land cover dataset for Russia: a methodology
for integrating statistics, remote sensing and in situ information." Journal of Land Use Science 6,
no. 4 (2011): 245-259.
5

measured within the same units4. This mainly reflects the risk of the total set of variances.
Standard deviation of 50 states was 428.53.
Sample variance: It refers as those variance which is expressed into numerical values
that is derived from average of the square out of total mean. It is used to measure that how far
every number in the set is come out from the mean values. It is calculated by taking difference
from each set of observations. It is calculated by squaring up the values from standard deviation.
A sample is a chosen set of value of items those are selected from a given population. There are
various steps can be used to determine variances such as, Firstly calculate the mean, after that
deduct it from the mean and square the total outcome. The last step is to work out those
differences which are collected from average of given observations. The sample variances of the
total population is 183645.
Skewness: In the statistical distribution it is known as asymmetry. Under which the
presented curve come out as skewed from left or from the right side. It is used to determine
various aspects in the presented distribution system. It consists of different impact such as:
 Positive skewness: It refers as those observation which are derived from number of
population collected as a results of the calculation come as more than Zero, it is said to be
positive skewed.
 Negative Skewness: If the possibility of the outcomes are less than zero is considered as
negative skewed.
 Normal skewed: If the results are equal to zero then mean of that population is known as
symmetric or normal skewed.
From the above calculated skewness it has been found that the value come from the
population is -1.1 which is below to zero so it is considered as negative skewness.
4 Kavuluru, Ramakanth, and Zhenghao He. "Unsupervised medical subject heading
assignment using output label co-occurrence statistics and semantic predications."
In International Conference on Application of Natural Language to Information Systems, pp.
176-188. Springer, Berlin, Heidelberg, 2013.
6

Kurtosis: It is Greek word, in which is said to be bulginess. It is used to measure the
shape of frequency curve which is derived from the total population5. It also determine the
degree of prime of a frequency arrangement. It is divided into three types:
I) Platykurtic: It is said to be flat curve which is presented from the total observation.
Under this distribution are having narrow tails. The value is negative when it comes to be
more than kurtosis.
II) Mesokurtic: It refers as normal curve derived from the number of population mean.
III) Leptokurtic: These are having data on the X-axis that is fixed on the clustered and hence,
high kurtosis.
Range: This is the tool in which a set of observations are analysed and then in that range
is calculated which represents the limit within which activities of company shall lie. In this two
calculations are made which are maximum and minimum points. Minimum represents the lower
values of the set of observations while maximum represents the higher value of the set of
observation. These are used for knowing the minimum and maximum value of the set of
observations. In the given situation, analysation of property tax that is paid by 50 states and on
the basis of this, range is determined. It has been found that minimum point is represented by
Alabama which pays 362 as tax per capita and maximum range is identified in New Jercy at the
amount of 2099 per capita. So it shall be ensured that tax is to be maintained within these values.
Conclusion
From the above mentioned report, this has concluded that descriptive measure are used
by the researcher in order to get the fruitful results so that they could able to make their business
operations effective. However, this also been seen that the these defines and assess the features
of a particular data set, by rendering short summary about the sample and assessment of the data.
Under this report, mean, median, mode, standard deviation, kurtosis or skewness are used.
Which terms as measures of variability. Under this report, histogram, tables and general
discussion are used in order to assist people for knowing the meaning of the data which is being
assessed. Under this report, various statistical tools are used in order to have the fruitful
conclusions accordingly.
5 Ghadle, K. P., and Y. M. Muley. "Revised Ones Assignment Method for solving
assignment problem." Journal of Statistics and Mathematics 4, no. 1 (2013): 147.
7

REFERENCES
Books and journal:
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