Running Head DESCRIPTIVE STATISTICS COMPUTING .docx

Running Head: DESCRIPTIVE STATISTICS COMPUTING
1
DESCRIPTIVE STATISTICS COMPUTING
2
DESCRIPTIVE STATISTICS COMPUTING
Name
Course
Institution
Instructor
Date
Computing descriptive statistics
Haukoos, J. S., & Lewis, R. J. (2015). Advanced statistics:
bootstrapping confidence intervals for statistics with “difficult”
distributions. Academic emergency medicine, 12(4), 360-365.
This article by Haukoos and Lewis describe how to use
confidence intervals in reporting research results which the
authors acknowledge to have increased in use and as a
requirement for scientific journal editors. The article explores a
number of resources that describe methods of computing
statistical confidence intervals for descriptive statistics that
have descriptions that are not easy to mathematically represent

which is a challenging task. The article is relevant to the topic
matter in that it describes the methods along with how the
resources availed describing the computing methods for
descriptive statistics. The authors propose the use of bootstrap
technique which they argue allows a researcher to make
inferences from data without making strong assumptions about
distribution of the data or the statistics under calculation. The
strengths of the article include the fact that it describes the
bootstrapping concept and demonstrates how to estimate
confidence intervals for the median and the spearman rank
correlation coefficient for data not normally distributed. the
weakness of this resource is the fact that it does not generally
describe how to compute descriptive statistics but narrows down
to describing how to compute descriptive statistics confidence
intervals. The article used a qualitative research method on a
clinical study that used two commonly used statistical software
packages of strata and SAS in its discussion of limitations of
the bootstrap.
Team, R. C. (2013). R: A language and environment for
statistical computing.
The document by Andy Bunn and Mikko Korpela describes the
basic features of dendrologists program library in R, which is
in itself a package used by dendrologists to handle the
processing and analysis of data. The document follows the basic
steps an analyst follows when working with a new tree-ring data
set. This document is a vignette commences by describing how
to read in ring widths and how to plot them. The vignette is
relevant in this study as it describes a number of methods
available for trending and detrending and shows how to extract
basic descriptive statistics. It also shows how to build and plot a
simple chronology of the mean value. The building of the mean-
value chronology is shown in the vignette using the expressed
signal; of the population from the detrended ring widths as a
way of doing complicated computing by the use of The
Dendrochronology Program Library in R (dplR). The vignette
highlights how to work with most basic activities in tree-ring

data and the steps followed including data reading, detrending,
building chronology and using descriptive statistics ploratory
data analysis. The document strength of the paper includes the
fact that it explores one way of computing descriptive statistics
and avails for information on computing and analyzing
descriptive statistics in the help files and in the literature
sections. The weakness of the paper is approaching only one
method of descriptive data computing rather than directly
addressing the research problem. The vignette uses literature
review to address its research problem.
Norušis, M. J. (2016). SPSS 14.0 guide to data analysis. Upper
Saddle River, NJ: Prentice Hall.
This book by Norusis is a guide to the analysis of data and
SPSS. The book gives a jump-start to readers on how to
describe data, test hypotheses and examine relationships though
SPSS. The book provides an introduction to the data computing
and to SPSS focusing on topics that interest modern-day
students and the use of the internet in today’s society. The book
uses SPSS 14.0 and includes the student version and data CD.
The book is relevant to the study in that it not also serves to
introduce learners to data analysis and its methods but also
incorporate a plethora of data including internet usage studies,
general social survey, criminal justice system opinions,
marathon running times, the importance of manners and library
patronage. The strength of the book in relation to this research
is that it generally describes methods of computing descriptive
data and also incorporates and guides in using SPSS to compute
descriptive data.
McKinney, W. (2010, June). Data structures for statistical
computing in python. In Proceedings of the 9th Python in
Science Conference (Vol. 445, pp. 51-56).
The paper goes through the practical issues around working
with data sets common to statistics and finance and other
related fields. The book facilitates working with these data sets
and therefore provides a set of building blocks fundamental to
implementing statistical models. It also discusses specific

design issues that researchers and analysts are likely to
encounter when developing scientific applications such as
python. It concludes by discussing possible future directions for
computing statistical data suing python and other means
relevant to descriptive statistics. The relevance of the paper to
the research question lies on the fact that it approaches ways to
compute descriptive data. The biggest strength in relation to
this research is that it addresses methods of descriptive data
computing. The weaknesses include the fact that it enjoins
methods of computing descriptive data among other types of
data. The paper uses literature review in its pursuit to explore
data computing.
Running head: ANNOTATED BIBLIOGRAPHY1
ANNOTATED BIBLIOGRAPHY 6
Annotated Bibliography
Student’s Name
Course
Date of Submission
Education practitioners and researchers on the American with

disabilities
Skiba, R. J., Simmons, A. B., (2008). Achieving equity in
special education: History, status, and current
challenges. Exceptional Children, 74(3), 264-288.
The problem to be studied
The purpose of the study is to discuss the discourse of practices
and context in the literature about higher education practitioners
and researchers since passing the American with disabilities.
Research question: How is research about students with
disabilities represented in the four in the top four tiers, in which
way has the representation transformed from 1990 to 2014 and
what is absent from the discourse in these journals?
Rationale of the study
The presence of articles were 20 out of 2300 published
matched the search. The article covered the representation of
the students, how the students are perceived as well as what the
articles state about them. The article is focused on need and
experiences, academic achievement, programs and services and
attitudes of the faculty and peer.
Population: The top two ranking peer-reviewed journals in the
area of higher learning: The Journal of Higher Education and
the Review of Higher Education.
Sample: 20 articles on students with disabilities were identified
in higher education.
The type of research that was conducted
The types of research was both quantitative and qualitative,
qualitative research was used to gather non-numerical data on
concepts, characteristics and description in the discussion of the
discourse of practices and context in the literature about higher
education practitioners and researchers since passing the
American with disabilities. Quantitative aspects of the interests
and needs of students with disabilities either absence or visible;
Areas of disabilities: Speech, mental development, learning,
hearing, general health, orthopedics, psychiatric health, and
vision. Quantitative was used to determine the representation of
students with disabilities in the four top tiers.

The Data Collection Strategy
The data collection strategy that was used was a survey
The Data Collection tools
70% of the articles about college students with disabilities in
top two peer reviewed journals published in the 1990s in
comparison to 10% published in 2014. 3/5 studies
quantitative data. The method used was survey of the
population. It compared students without disabilities and
students with disabilities.
Strengths and weaknesses of the research investigations
The main strength of the study is that it used literature review
as qualitative and quantitative methods increased the validity of
the results of the survey. The use of literature review in data
collection could also be a weakness as it does not reflect the
real-time updated issues on the ground since it dwells on past
accounts. Another weakness is in the research question as it
does not exactly aim to quantify the number of educational
practitioners and researchers on students with disability.
Summary
This study takes into considerations the article about students
with disabilities. The articles were published in top two journals
on higher education. One percent of the two journals discussed
students with disabilities. The percentage is disproportionate in
regard that 10 percent of students enrolled currently in higher
learning have disabilities.
A statement that supports my proposed Study
Limited data collection and research has been carried out
recently, leaving scarce resources.
STUDENTS WITH DISABILITY ACHIEVING HIGHER
EDUCATION
Fuller*, M., Bradley, A., & Healey, M. (2014). Incorporating
disabled students within an inclusive higher education
environment. Disability & Society, 19(5), 455-468.

Problem to be studied
Students with disabilities who wish to attain personal and
financial independence should be able to access higher
education and overcome academic barriers. Appropriate
transition services, access to qualified counselors,
accommodations and academic remediation might make a small
difference to whether or not students attain higher
education success.
Research question: In which ways are the students with
disabilities able to attain higher education successfully
when there are a lot of undefined variables?
Rationale of the Study
The number of students with disabilities, with higher academic
competence and are seeking appropriate accommodations
is still increasing. This increase might not represent unknown
number of part time students, students transferring from one
college to another, students who decide not to identify
themselves and students with learning disabilities
Type of research that was carried out
The research that was carried was a mixed method in that
quantitative research was used to identify and study the
number of students with disability enrolling and studying
in the higher education institutions, qualitative research was
on the other had used to identify ways in which students with
disabilities are able to attain higher education successfully
Date Collection strategy
The data collection strategy that was used is telephone
surveys and interviews.
Data analysis tools
Interviews with students using telephone surveys
were employed to determine in which ways the managed
challenges of higher education adjustments, and how to
improve vocational and special education as well as transition
program. SPSS was used to analyze the data.
Summary
In summary, Sam, a student with disability reported two

essential and key services that led to his academic success.
The first was counseling related to traumatic anxiety that
stemmed from demoralizing educational experiences.
Another service included the language program he received that
assisted and taught him to read. Ideally, remediation for
basic skills and support from professionals and friends
also assisted him to overcome anxiety as well as find
solutions to attain his educational goals.
The main strength of the study is that thee research question,
purpose and rationale are relevant in looking into the number of
students with disabilities in higher education and their
experiences. Another strength was that the data collection
strategy which used telephone surveys and interviews to gather
data as it promotes confidentiality and self-confidence in
availing information that the students with disabilities could not
reveal in other ways of data collection. It can also be the main
weakness as it could promote exaggeration and interviewing the
wrong person as well as the results being affected by the
attitude and mood of the interviewee.
Statement that support my proposed study
Educational solutions and educational barriers were identified
that make higher education success feasible. Also, the students,
who were chosen met that has been well planned for the
qualitative research.
TRENDS OF MAJOR, AGE AND TYPE OF
DISABILITIES FOR STUDENTS RECEIVING SERVICES
Cortiella, C., & Horowitz, S. H. (2014). The state of learning
disabilities: Facts, trends and emerging issues. New York:
National center for learning disabilities, 2-45.
Statement of the Problem
The statement of the problem that faculty and
administrators of MSU Billings required an analysis and
description of the trends of major, age and type of
disabilities for students receiving services via Disability
Support Services from 2000 to 2014.

The earliest study that identifies students with disabilities
in selected public colleges and Universities.
The type of Research Methods
The type of research is qualitative to analyze and describe the
trends of age and type of major disabilities for students
receiving disability support services between the years 2000 and
2014.
The data collection strategy
Interview with every student after acceptance for services based
on mental or medical health documentation of a disability.
The data Analysis Methods
There are two analytical techniques that were employed in the
data analysis. For instance, descriptive graphs and data
offered an overall perception of the trends from 2000 to
2014. The second technique to data analysis to data
analysis employed is descriptive trend analysis basing on
events to carry out the analysis of data to understand
what occurred with disabilities basing on political,
economic and social forces during the period.
The strengths of the study included the fact that interviews were
used as they are best suited in qualitative surveys, but
interviewing students at a particular year could not give the
trends on the subject matter unless complemented by other ways
of data collection such as literature review; which is one of the
weaknesses.
Summary
Based on the data analysis, there was an increase in all type of
disabilities categories. However, there was an exception, of
the number of college students with movement impairments
that reduced during the study period.
Statement that support my proposed article
Descriptions in this particular study incorporated various
types of disabilities of students so as to offer insight

into several people with severe weakness rather than people
with less severe circumstances.
EDUCATION CURRICULUM AND NOT EQUITABLE
CURRICULUM
Ennis, C. D. (2017). Creating a culturally relevant curriculum
for disengaged girls. Sport, Education and Society, 4(1), 31-49.
Ensure that each student has access to education curriculum
and not equitable curriculum via placement in education
classrooms is regarded as a problem of social justice.
The people with Disabilities Education Act pushes the
urge for inclusive instruction. The triumph of laws on the
mindset and expertise teachers show in the classroom.
Type of research methods
The type of research method used is qualitative research to
gauge the satisfaction levels of students with disabilities with
the curriculum based on their experiences..
Data collection strategy
Participants were inquired to give their experiences in
response to the questions asked in the interview.
Data Analysis Strategy
A digital audio recorder was employed to record the
interview so as to attain accuracy. The digital recording
for every interview was stored in a file of audio
computer and transcribed. The data collected was analyzed
via descriptive coding.
The strengths included the fact that data coding was used in
data analysis of results obtained from interviews tape recorded
as it ensured categorization of data. One of the weaknesses
encompass failing to use questionnaires to complement the
interviews as some aspects might have been lost due to fear for
confidentiality and persona security or stigma that could have

otherwise been availed in questionnaires.
Summary
The findings of the study indicate that there exists no one
size fitting the models of students with learning
disabilities. Those involved in the study demonstrated the
idea of services continuum be readily available to meet
each student’s needs. The study results give
information to be utilized by special education
administrator’s, personnel , dismissal, review and admission
committees as they try to find the needs of each student
such as students with disabilities.
Statement that support my Research Proposal
The research limited qualitative data with education
administrators and educational personnel. Future study can
incorporate perspectives from other important stakeholders
including general education students, parents and teachers.
Yell, M. L. (2015). The law and special education.
Merrill/Prentice-Hall, Inc., 200 Old Tappan Road, Old Tappan,
NJ 07675.
The individuals with Disabilities Act and No Child Left
Act ensures that education programs of teacher prepare
qualified teachers for schools.
Teacher educators have the role of engaging in a
program evaluation to identify their preparation programs
aspects that might assist transition of novice teachers
to full time teaching and might affect the retention of
qualified educators.
Type of research methods
The type of research method used is mixed method research.
Data Collection Strategy
The data collection strategy that was used Path wise
observations and survey instrument. They gave the faculty

with several sources of data for improving decision-
making.
The Data Analysis Strategy
Formative/ screening processes to enhance individual
teaching performance of student and program improvement.
Summary
The mixed methods data offered the areas for improvement in
data collection. Whereas reflections are informative,
guided reflections might be important to identify areas
where candidates need help. Faculty will focus on the
reflections of candidates on identified areas in the data and
literature obtained from candidates.
The use of mixed methods was a strength as qualitative research
gave a better reflection of the performance and opinion on the
important roles played by novice teachers whereas quantitative
availed numerical value performance of the teachers thereby
giving the researcher a wide field of view of the subject under
study. The weakness was the fact that the data collection
strategy as it poses a threat of privacy violation.
Statement that support my Research Proposal
Mixed-methods strategies are suitable as accrediting agencies
need data from multiple forms and multiple forms.
AMERICAN WITH DISABILITIES
Price, M. (2015). Mad at school: Rhetorics of mental disability
and academic life. University of Michigan Press.
The problem to be studied
The purpose of the study to discuss the discourse of practices
and context in the literature about higher education practitioners
and researchers since passing the American with disabilities
The presence of articles were 20 out of 2300 published
matched the search. The article covered the representation of
the students, how the students are perceived as well as what the
articles state about them. The article is focused on need and
experiences, academic achievement, programs and services and

attitudes of the faculty and peer
The type of research that was conducted
Qualitative research was used to approach the theoretical
aspects of the interests and needs of students with disabilities
which are either absence or visible; Areas of disabilities:
Speech, mental development, learning, hearing, general health,
orthopedics, psychiatric health, and vision.
The Data Collection Strategy.
The data collection strategy that was used was a survey.
The Data Collection tools.
70% of the articles about college students with disabilities in
top two peer reviewed journals published in the 1990s in
comparison to 10% published in 2014. 3/5 studies
quantitative data. The method used was surveyed. It compared
students without disabilities and students with disabilities.
The main strength of the survey was that it used literature
review that covered both qualitative and quantitative methods,
increasing the relevance and validity of the results of the
survey. Using literature review in data collection could also be
a weakness as it does not reflect the real-time updated issues on
the ground.
Summary
This study takes into considerations the article about students
with disabilities. The articles were published in top two journals
on higher education. One percent of the two journals discussed
students with disabilities. The percentage is disproportionate in
regard that 10 percent of students enrolled currently in higher
learning have disabilities.
A statement that supports my proposed Study.
Limited data collection and research has been carried out
recently, leaving scarce resources

References
Cortiella, C., & Horowitz, S. H. (2014). The state of learning
disabilities: Facts, trends and emerging issues. New York:
National center for learning disabilities, 2-45.
Ennis, C. D. (2017). Creating a culturally relevant curriculum
for disengaged girls. Sport, Education and Society, 4(1), 31-49.
Fuller*, M., Bradley, A., & Healey, M. (2014). Incorporating
disabled students within an inclusive higher education
environment. Disability & Society, 19(5), 455-468.
Price, M. (2015). Mad at school: Rhetorics of mental disability
and academic life. University of Michigan Press.
Skiba, R. J., Simmons, A. B., (2008). Achieving equity in
special education: History, status, and current
challenges. Exceptional Children, 74(3), 264-288.
Yell, M. L. (2015). The law and special education.
Merrill/Prentice-Hall, Inc., 200 Old Tappan Road, Old Tappan,
NJ 07675.
Assignment 2: RA: Annotated Bibliography
In your final paper for this course, you will need to write a
Methods section that is about 3–4 pages long where you will
assess and evaluate the methods and analysis of your proposed
research.
In preparation for this particular section, answer the following
questions thoroughly and provide justification/support. The
more complete and detailed your answers for these questions,
the better prepared you are to successfully write your final
paper:
· What is the problem being addressed by your research study?
· State the refined research question and hypothesis (null and
alternative).
· What are your independent and dependent variables? What are
their operational definitions?
· Who will be included in your sample (i.e., inclusion and
exclusion characteristics)?

· How many participants will you have in your sample?
· How will you recruit your sample?
· Identify the type of measurement instrument to be used to
collect the raw numeric data to be statistically analyzed and the
type of measurement data the instrument produces.
· What issues will you cover in the informed consent?
· If there is potential risk or harm, how will you ensure the
safety of all participants?
· Name any possible threats to validity and steps that can be
taken to minimize these threats.
· What type of parametric or nonparametric inferential
statistical process (correlation, difference, or effect) will you
use in your proposed research? Why is this statistical test the
best fit?
· State an acceptable behavioral research alpha level you would
use to fail to accept or fail to reject the stated null hypothesis
and explain your choice.
This paper may be written in question-and-answer format rather
than a flowing paper. Write your response in a 3- to 4-page
Microsoft Word document.
All written assignments and responses should follow APA rules
for attributing sources.
Submission Details:
· By the due date assigned, save your document as
M4_A2_Lastname_Firstname.doc and submit it to
the Submissions Area .
Assignment 2 Grading Criteria
Maximum Points
Stated the problem being addressed.
8
Stated the refined research question and hypothesis (null and
alternative).
6
Stated the independent and dependent variables and provided
the operational definitions.
12

Discussed sample characteristics and size.
8
Discussed a sample recruitment strategy.
6
Identified the type of measurement instrument to be used and
the type of measurement data the instrument produces.
8
Discussed the informed consent and potential risk and
protection factors.
12
Named the possible threats to validity and steps that can be
taken to minimize these threats.
12
Discussed the type of parametric or nonparametric inferential
statistical process that will be used and why it is a best fit.
8
Stated an acceptable behavioral research alpha level for
analyzing the data.
4
Wrote in a clear, concise, and organized manner; demonstrated
ethical scholarship in accurate representation and attribution of
sources; displayed accurate spelling, grammar, and punctuation.
16
Total:
100
Computing Descriptive Statistics
© 2014 Argosy University

of 5
Research and Evaluation Design
©2014 Argosy University
2 Computing Descriptive Statistics
Computing Descriptive Statistics: “Ever Wonder What Secrets
They Hold?” The Mean, Mode, Median, Variability, and
Standard Deviation
Introduction
Before gaining an appreciation for the value of descriptive
statistics in behavioral science
environments, one must first become familiar with the type of
measurement data these statistical
processes use. Knowing the types of measurement data will aid
the decision maker in making
sure that the chosen statistical method will, indeed, produce the
results needed and expected.
Using the wrong type of measurement data with a selected
statistic tool will result in erroneous
results, errors, and ineffective decision making.
Measurement, or numerical, data is divided into four types:
nominal, ordinal, interval, and ratio.
The businessperson, because of administering questionnaires,
taking polls, conducting surveys,

administering tests, and counting events, products, and a host of
other numerical data
instrumentations, garners all the numerical values associated
with these four types.
Nominal Data
Nominal data is the simplest of all four forms of numerical data.
The mathematical values are
assigned to that which is being assessed simply by arbitrarily
assigning numerical values to a
characteristic, event, occasion, or phenomenon. For example, a
human resources (HR) manager
wishes to determine the differences in leadership styles between
managers who are at different
geographical regions. To compute the differences, the HR
manager might assign the following
values: 1 = West, 2 = Midwest, 3 = North, and so on. The
numerical values are not descriptive of
anything other than the location and are not indicative of
quantity.
Ordinal Data
In terms of ordinal data, the variables contained within the
measurement instrument are ranked in
order of importance. For example, a product-marketing
specialist might be interested in how a
consumer group would respond to a new product. To garner the
information, the questionnaire
administered to a group of consumers would include questions
scaled as follows: 1 = Not Likely, 2
= Somewhat Likely, 3 = Likely, 4 = More Than Likely, and 5 =
Most Likely. This creates a scale
rank order from Not Likely to Most Likely with respect to
acceptance of the new consumer

product.
Interval Data
Oftentimes, in addition to being ordered, the differences (or
intervals) between two adjacent
measurement values on a measurement scale are identical. For
example, the differences in age
between managers 25 years of age and 30 years of age are the
same as the differences in age
between managers who are 40 years of age and 45 years of age.
That is to say, when each
interval represents the same increment of that which is being
measured, the measure is referred
to as an interval measurement or interval mathematical value.
Page 3 of 5
Ratio Data
Some measurements, in addition to having an equal interval
value, also have an absolute zero
value. In this instance, zero represents the absence of the
variable being measured. With an
absolute zero value, for example, either you have some money
or you do not have any money. In

the money scenario, adding the interval quality to ratio data
would mean that you have no money,
$1.00 to $5.00, $6.00 to $10.00, and so on. What is being stated
here is that ratio data is
quantitative as it tells us the amount of the variable being
measured. Consider other examples of
ration data such as the percentage of votes received by a
candidate, the gross national product
per capita, the current American crime rate, and the number of
finished consumer products
manufactured per day per person—all of these are examples of
ratio data.
Why Measurement Data Matters
To the behavioral scientist, when conducting research, the level
of measurement is important
because the higher the level of measurement of a variable, the
more powerful the statistical
techniques that can be employed to analyze the data. Take, for
example, the voters’ choice
wherein the nominal variables in the 2004 presidential race
were Bush, Kerry, Nader, etc. One
can count the number of votes each candidate received as well
as calculate the percentage each
candidate received. One can also calculate joint frequencies and
percentages by region and by
gender. One can also calculate the relationship between region
and vote and whether the
relationship occurred by chance. Unfortunately, using nominal
measurement data does not permit
one to use advanced methods of statistics. In the example
presented above, even when we
assign numbers to each candidate, we cannot very well
determine that Bush plus Kerry equals
Nader or that Nader divided by Kerry is half way between Bush

and Kerry, and so on. In an
attempt to calculate the situation, only the mentioned higher
level statistical techniques are
required.
If one uses a technique that assumes a higher level of
measurement than is appropriate for the
data, there is a risk of getting meaningless results and answers.
At the same time, if one uses a
technique that fails to take advantage of a higher level of
measurement, important things are often
overlooked about the data collected.
Computing and Using the Mean, Median, Mode, Frequency
Distribution, and Standard
Deviation
Regardless of the type of measurement data garnered from a
data collection set, the mean,
mode, median, frequency distribution, and standard deviation
can be calculated. However, simply
because mathematical calculations can be made does not imply
the legitimacy of their use. The
remainder of this section will deal with how one computes and
uses measures of central tendency
in business situations.
Mean
Mean is the arithmetic average of a group of measurement
values. To be meaningful, the
resulting mathematical value must be based on at least an
ordinal set of data or above. Interval
and ratio are also appropriate for calculating the mean. The
resulting mean of a group of data will
only describe the data in general descriptive terms. The mean

alone cannot be used to draw
conclusions and make inferences about a population being
studied.
Page 4 of 5
Formula:
N (n) = Number of participants in a study, or numeric values
Σ = Sum of all numeric values
X = Raw score, or individual measurement score
= The mean
Page 5 of 5

Example
Consider a production manager who wants to determine whether
the 11 employees of the second
shift of line employees are producing more baby strollers than
the 11 first-shift employees. The
second-shift employees produce the following number of baby
strollers: 5, 7, 8, 5, 7, 7, 9, 9, 5, 5,
5. The first-shift employees produce the following number of
units: 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8.
Solution
:
Second shift total units (ΣΧ) = 72
Second shift total number of employees (n) = 11
Mean = 72/11 = 6.55 units produced
First shift units (ΣΧ) = 69

First shift total number of employees (n) = 11
Mean = 69/11 = 6.27 units produced
Conclusion
What appears to be happening is that the second shift of line
employees produces more baby
strollers than the first shift of line employees.
Caution: No other conclusions can be drawn from the mean. To
determine whether production
differences are significant, a higher level statistical process
must be used. We can only “describe”
what has happened here and cannot draw conclusions or make
inferences as to why or how
much.
Median
Median is broadly defined as the middle value of a set of
measurement values. Just like medians
divide roads down the middle, so does the median in statistics
in that the median is simply the
middle number For highly skewed distributions, the median is a

better measure of central
tendency than the mean as extreme outliners or measurement
values do not affect it.
Formula:
No statistical formula is needed. To calculate the median of a
group of measurement scores,
simply find the
midpoint of the distribution by arranging the scores in
ascending order—from low to high.
Example
Second-shift stroller production 5, 7, 8, 5, 7, 7, 9, 9, 5, 5, 5
Ordered Values –5, 5, 5, 5, 5, 7, 7,
7, 8, 9, 9
Median = 7
Page 6 of 5

First-shift stroller production 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8
Ordered values –2, 4, 6, 6, 6, 7, 7, 7, 7, 8, 9
Median = 7
Note: As the median values are equal, the mean is a better
choice to describe the measurement
data.
Mode
Mode is defined as the most frequent value in a measurement
data set and is of limited value.
Example
Second-shift stroller production: 5, 7, 8, 5, 7, 7, 9, 9, 5, 5, 5

Mode = 5
First-shift stroller production: 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8
Mode = 7
Variability
Furthermore, the central tendency is a summary measure of the
overall quantity of a
measurement data set. Variability (or dispersion) measures the
amount of spread in a
measurement data set. Variability is generally measured using
three criteria: range, variance, and
standard deviation.
Example
Range
The difference between the largest and the smallest value in the
data set is calculated by

subtracting the smallest value from the largest measurement
value. Although the range is a crude
measure of variability, it is easy to calculate and useful as an
outline description of a data set.
Page 7 of 5
Example
Second-shift stroller production: 5, 7, 8, 5, 7, 7, 9, 9, 5, 5, 5
Range = 9 – 5 = 4
First-shift stroller production: 7, 7, 7, 7, 6, 6, 9, 2, 4, 6, 8

Range = 9 – 2 = 7
Variance
Variance is a deviation. It is a measure of by how much each
point frequency distribution lies
above or below the mean for the entire data set:
Note: If you add all the deviation scores for a measurement data
set together, you will
automatically get the mean for that data set.
In order to define the amount of deviation of a data set from the
mean, calculate the mean of all
the deviation scores, i.e., the variance.
Formula:
Standard Deviation
In statistics, the standard deviation represents the measure of
the spread of a set of
measurement values from the mean of the data set. Putting it

another way, the standard deviation
can be defined as the average amount by which scores in a
distribution differ from the mean while
ignoring the sign of the difference, i.e., the plus or minus value.
Further, standard deviations are
only good when referring to single data or measurement values,
i.e., finding out when a single
score falls with reference to being above or below the mean.
To find out the standard deviation of a data set, you must
perform the following steps:
1. Calculate the mean of all the scores.
2. Find the deviation of each score from the mean.
3. Square each deviation.
4. Calculate the average of each deviation.
5. Calculate the square root of the average deviation.

of 5
Formula:
Example
Consider, for example, a real estate manager who wants to
determine where his or her
department employees are placed with respect to the average
number of real estate closings they
accomplish per month.
The monthly closings are as follows: 8, 25, 7, 5, 8, 3, 10, 12, 9
1. First, calculate the mean and determine N.
2. Remember, the mean is the sum of scores divided by N,
where N is the number of scores.

3. Therefore, the mean = (8+25+7+5+8+3+10+12+9) / 9 or 9.67
4. Then, calculate the standard deviation, n, as illustrated
below.
Squared
Score Mean Deviation* Deviation
8 9.67 –1.67 2.79
25 9.67 +15.33 235.01
7 9.67 –2.67 7.13
5 9.67 –4.67 21.81
8 9.67 –1.67 2.79
3 9.67 –6.67 44.49
10 9.67 +.33 .11
12 9.67 +2.33 5.43

9 9.67 –.67 .45
(*deviation from mean = score – mean)
Sum of squared deviation = 320.01
Standard Deviation = Square root (sum of squared deviations /
(N – 1))
Page 9 of 5
= Square root (320.01/ (9 – 1))
= Square root (40)
= 6.32

Conclusion
Real estate closings for the month vary ±6.32 closings above or
below the mean of a 9.67 closing
average. Caution must be exercised here as no conclusion can
be drawn as to whether this is an
acceptable range of real estate closing activity. In other words,
you cannot draw a conclusion
about whether the closings are profitable, not profitable, or
represent an industry average. Further
statistical data analysis would have to be conducted to draw any
such conclusion.
Page 10 of 5
The Color of Data: Visually Reported Data for Descriptive
Statistical Presentations

The Pros and Cons of Visually Reported Data
For the behavioral scientist, whether in business, forensics,
sociology, or anthropology, a host of
other related facts such as profit reporting, result forecasting,
time sequences, overtime hours,
personality test scores, anxiety scores, and IQ are often reported
visually. The reason for using
visual presentations is not only because “a picture is worth a
thousand words” but also because
visually presented data is not generally bogged down with
useless facts and figures. That is to
say, visual presentations usually present the most salient or
robust features of an event,
occurrence, phenomenon, situation, problem, or condition.
Visually presented information can
also be considered the road map to what has happened or lies
ahead. Further, visually reported
data are not only colorful but also easy to construct, and one
does not have to be a statistician to
create them.
Unfortunately, visually reported data results in the behavioral
science arena have two primary

drawbacks, namely, the presenter and the lack of data
sophistication. For the most part, visually
presented data are based on percentages, raw numeric data
scores, and frequencies. Rarely is
visually presented material based on true value statistics based
on inferential statistical findings.
The reason is that the values for inferential statistics are not
amenable to graphs and charts, and
their values speak mathematically for themselves. That is to
say, inferential statistical values are
mathematical values that must be interpreted and are not
subjected to graphic presentation.
A Prelude to Statistical Data Analysis: Raw Data’s Wardrobe in
the Form of Bar Graphs,
Pie Charts, Line Graphs, Stem and Leaf Displays, and Box and
Whisper Plots
Introduction
Regardless of the type of chart or graph you use to illuminate or
express a concept or idea, they
all have one thing in common, namely, to communicate via
picture what is being studied or what
is happening. A behavioral scientist, or any other person who

wants to show what is taking place,
makes use of bar graphs, pie charts, line graphs, leaf displays,
and box and whisper plots.
Unfortunately, however, those who are not well-grounded and
informed about statistical
processes oftentimes rely on these pictorial presentations to
draw conclusions or make inferences
about what is being studied and evaluated. When this happens in
the behavioral science arena,
wrong decisions are often made about important matters.
Nonetheless, graphs and charts are an
important step in achieving what statistical processes will
eventually resolve.
Bar Graph
Bar graphs allow and encourage a great deal of poetic license to
those who design or make use
of them. The reason is that the one designing the bar graph
determines what scale is to be used.
This means that the information can be presented in a
misleading way. For example, by using a
smaller scale (for example, having each half inch of the height
of a bar represent 10 widgets
versus 50 widgets produced), one can exaggerate the truth,

make production differences look
more dramatic, or even exaggerate values. On the other hand, by
using a larger scale (for
example having each half inch of a bar represent 50 widgets
versus 10 widgets produced), a
person can downplay differences, make the end results look less
dramatic than they actually are
or even make small differences appear to be nonexistent. When
evaluating a bar graph, one
should do the following:
Page 11 of 5
Make sure that the bars that divide up values are equal in width
for an equitable comparison.

Make sure that there is an appropriate representation of all
information being presented.
Knowing that the information being presented might not be a
fair representation of all
information, be prepared to dig deeper and use more advanced
statistical processes.
Example
Suppose, for example, we want to pictorially present, via a bar
graph, information on how much
money is spent on transportation by individuals of different
income levels. The first step is to
gather the information from a representative sample (sampling
will be discussed later in the
course) and determine how much money each participant spends
on transportation in a year. The
second step is to define income level. The third step is to
determine the horizontal axis and the
vertical axis. When you are seeking the “how” of something,
always remember that it becomes
the vertical axis and the “category” becomes the horizontal axis.

For this particular example, the
bar graph might possibly look like the following:
To construct a bar graph, go to your Windows task bar, click on
Insert, click on Chart, and enter
your information when prompted.
Pie Charts
Similar to bar graphs, pie charts, or circle graphs, are a pictorial
means using which the individual
can present information in a simple, nonstatistical form.
Further, like bar graphs, pie charts are
usually employed to compare percentages of the same whole.
Unlike bar graphs, pie charts do
not use a set of axes to plot information or data points. Pie
charts are display percentages and
are used to compare different parts of the same whole. With pie
charts, it is important to
remember how they are sometimes misrepresented in business
situations, namely leaving out
parts of the whole and not defining what the whole really is.
When a part of the whole is omitted,

then it increases the percentage values of the remaining parts
that are displayed. When the whole
Page 12 of 5
is not well-defined, the reader is unclear about what the parts
represent. Again, a pie is a
graphical representation of how many individual parts
contribute to the total.
Example
Take, for example, a human resource manager who is interested
in finding out how three different
departments in a business situation waste time on the Internet
on a given day when they should
be engaged in company business. The human resource person

would collect data through a time
study process and determine the number of times each employee
in each department logged on
and off the Internet for personal business. The times would be
collected and added together, and
each department’s time would be converted to percentages.
Going further, the human resource
manager reported that, cumulatively, the employees of
Department 1 spent a total of five hours a
day on the Internet, those of Department 2 spent two hours a
day, and those of Department 3
spent six hours. The pie chart would look like the following:

Running Head DESCRIPTIVE STATISTICS COMPUTING .docx

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