Histograms and Descriptive Statistics Scoring GuideCRITERIANON.docx

Histograms and Descriptive Statistics Scoring Guide
CRITERIA
NON-PERFORMANCE
BASIC
PROFICIENT
DISTINGUISHED
Apply the appropriate SPSS procedures for creating histograms
to generate relevant output.
Does not provide SPSS output.
Provides SPSS output with errors.
Applies the appropriate SPSS procedures for creating
histograms to generate relevant output.
Analyzes the histogram output, demonstrating insight and
understanding of relevant data.
Interpret histogram results, including concepts of skew,
kurtosis, outliers, symmetry, and modality.
Does not provide an interpretation of histogram results.
Provides an interpretation of histogram results.
Interprets histogram results, including concepts of skew,
Evaluates histogram results, including concepts of skew,
Analyze the strengths and limitations of examining a
distribution of scores with a histogram.
Does not identify the strengths and limitations of examining a
Identifies the strengths and limitations of examining a
Analyzes the strengths and limitations of examining a
Evaluates the strengths and limitations of examining a
distribution of scores with a histogram. Demonstrates insight
and understanding of relevant data.
Apply the appropriate SPSS procedure for generating

descriptive statistics to generate relevant output.
Includes some, but not all, of the required output. Numerous
errors in SPSS output.
Applies the appropriate SPSS procedure for generating
descriptive statistics to generate relevant output.
Applies the appropriate SPSS procedure for generating
descriptive statistics to generate relevant output. Includes all
relevant output; no irrelevant output is included. No errors in
SPSS output.
Analyze meaningful versus meaningless variables reported in
descriptive statistics.
Does not identify meaningful versus meaningless variables
reported in descriptive statistics.
Identifies meaningful versus meaningless variables reported in
Analyzes meaningful versus meaningless variables reported in
Evaluates meaningful versus meaningless variables reported in
Interpret descriptive statistics for meaningful variables.
Does not identify meaningful variables.
Identifies meaningful variables.
Interprets descriptive statistics for meaningful variables.
Evaluates descriptive statistics for meaningful variables.
Apply the appropriate SPSS procedures for creating z scores
and descriptive statistics to generate relevant output.
Provides SPSS output with errors.
Applies the appropriate SPSS procedures for creating z scores
and descriptive statistics to generate relevant output.
Analyzes the z scores and descriptive statistics output,
demonstrating insight and understanding of relevant data.
Analyze the relevant data from the computation, interpretation,
and application of z scores.
Does not identify the relevant data or generate output.

Identifies the relevant data and generates output.
Analyzes the relevant data from the computation, interpretation,
and application of z scores.
Evaluates the relevant data from the computation,
interpretation, and application of z scores. Justifies the
meaningfulness of selected variables.
Analyze real-world application of Type I and Type II errors,
and the research decisions that influence the relative risk of
each.
Does not describe a real-world application of Type I and Type
II errors and the research decisions that influence the relative
risk of each.
Describes, but does not analyze, a real-world application of
Type I and Type II errors and the research decisions that
influence the relative risk of each.
Analyzes a real-world application of Type I and Type II errors
each.
Evaluates real-world application of Type I and Type II errors
each.
Apply the logic of null hypothesis testing to cases.
Does not apply the logic for null hypothesis testing.
Inconsistently applies the logic for null hypothesis testing.
Applies the logic of null hypothesis testing to cases.
Analyzes the logic of null hypothesis testing. Demonstrates
insight and understanding of relevant data to either reject or not
reject the null hypothesis.
Communicate in a manner that is scholarly, professional, and
consistent with expectations for members of the identified field
of study.
Does not communicate in a manner that is scholarly,
professional, and consistent with the expectations for members
in the identified field of study.
Inconsistently communicates in a manner that is scholarly,
professional, and consistent with the expectations for members

in the identified field of study.
Communicates in a manner that is scholarly, professional, and
consistent with the expectations for members in the identified
field of study.
Communicates in a manner that is professional, scholarly, and
consistent with expectations for members of the identified field
of study. Adheres to APA guidelines, and work is appropriate
for publication.
Running head: Z SCORE ASSESSMENT ANSWER TEMPLATE
1
UNIT 4 ASSIGNMENT 1 ANSWER TEMPLATE
5z-Score Assessment Answer TemplateStudent NameCapella
UniversityUnit 4 Assignment 1 Answer Template
The following assignment includes three sections consisting of:
1. z scores in SPSS.
2. Case studies of Type I and Type II errors.
3. Case studies of null hypothesis testing.
Additional notes:
· Answer in complete sentences.
· Follow APA rules for scholarly writing.
· Include a reference list if necessary.
· Save your answers and upload this template to the assignment
area for grading.
Section 1: z Scores in SPSS
A z score is typically analyzed when population mean (µ) and

population standard deviation (σ) are known. However, in SPSS,
we can still calculate z scores with the grades.sav data using the
sample mean (M) and sample standard deviation (s). To do this,
open grades.sav in SPSS. On the Analyze menu, point to
Descriptive Statistics, and then click Descriptives…
You will be calculating and interpreting z scores for the total
variable. In the Descriptives dialog box, move the total variable
into the Variable(s) box. Select the Save standardized values as
variables option and click OK.
SPSS provides descriptive statistics for total in the Output
window. SPSS also creates a new variable in the far right
column, labeled Ztotal, in the Data Editor area. Ztotal provides
a z score for each case on the total variable. You are now
prepared to answer the following Section 1 questions.Question 1
What is the sample mean (M) and sample standard deviation (s)
for total? You will use these values in Question 2 below.
[Answer here in complete sentences. Also insert the output
from SPSS here. Replace this prompt and the prompts below,
using as much space as necessary to answer questions.]
Question 2
A z score for this sample is calculated as [(X – M) ÷ s]. Locate
Case #53’s unstandardized total score (X) in the Data Editor. In
the formula below, replace X, M, s, and ? to show how the z
score in Ztotal is derived for Case #53.
(X – M ) ÷ s = ?Question 3
Run Descriptives… on Ztotal. What are the mean and standard
deviation of Ztotal? (Hint: “0E7” in SPSS is scientific notation
for 0). Are the mean and standard deviation what you would
expect? Justify your answer.

[Answer here in complete sentences. Also place the SPSS output
here.]
Question 4
Case number 6 has a Ztotal score of 1.22. What does a z value
of 1.22 represent?
[Answer here in complete sentences.]
Question 5
Identify the case with the lowest z score. Refer to Z Scores in
Suggested Resources. Interpret the percentile rank of this z
score rounded to whole numbers.
Question 6
Identify the case with the highest z score. Refer to Z Scores in
Suggested Resources. Interpret the percentile rank of this z
score rounded to whole numbers.
[Answer here in complete sentences.]Section 2: Cases Studies
of Type I and Type II Errors
Question 7
A jury must determine the guilt of a criminal defendant (not
guilty, guilty). Identify how the jury would make a correct
decision. Analyze how the jury would commit a Type I error
versus a Type II error.
[Answer here in complete sentences.]Question 8
An I/O psychologist asks employees to complete surveys
measuring job satisfaction and organizational citizenship
behavior. She intends to measure the strength of association

between these two variables. The researcher is concerned that
she will commit a Type I error. What research decision
influences the magnitude of risk of a Type I error in her study?
[Answer here in complete sentences]
Question 9
A clinical psychologist is studying the efficacy of a new drug
medication for depression. The study includes a placebo group
(no medication) versus a treatment group (new medication). He
then measures the differences in depressive symptoms across
the two groups.
What would a Type I error represent within the context of his
study? How can he reduce the risk of committing a Type I
error? How does this decision affect the risk of committing a
Type II error?
Section 3: Case Studies of Null Hypothesis TestingQuestion 10
You are running a series of statistical tests in SPSS using the
standard criterion for rejecting a null hypothesis. You obtain the
following p values.
Test 1 calculates group differences with a p value = .07.
Test 2 calculates the strength of association between two
variables with a p value = .50.
Test 3 calculates group differences with a p value = .001.
For each test below, state whether or not you reject the null
hypothesis. For each test, also explain what your decision
implies in terms of group differences (Test 1 and Test 3) and in
terms of the strength of association between two variables (Test
2).
Test 1 (group differences) =

Test 2 (strength of association) =
Test 3 (group differences) =
Question 11
A researcher calculates a statistical test and obtains a p value of
.86. He decides to reject the null hypothesis. Is this decision
correct, or has he committed a Type I or Type II error? Explain
your answer.
[Answer here in complete sentences]
Question 12
You are proposing a research study that you would like to
conduct while attending Capella University. During the
proposal, a committee member asks you to explain in your own
words what you meant by saying “p less than .05.” Provide an
explanation.
[Answer here in complete sentences]References
Provide references if necessary. This concludes Unit 4
Assignment 1. Save your answers and upload this template to
the assignment area.
Warner, R. M. (2013). Applied statistics: From bivariate
through multivariate techniques (2nd ed.). Thousand Oaks, CA:
Sage.
Print Copy/Export Output Instructions
SPSS output can be selectively copied and pasted into Word by
using the Copy command:
1. Click on the SPSS output in the Viewer window.
2. Right-click for options.
3. Click the Copy command.
4. Paste the output into a Microsoft Word document.
The Copy command will preserve the formatting of the SPSS
tables and charts when pasting into Microsoft Word.

An alternative method is to use the Export command:
1. Click on the SPSS output in the Viewer window.
2. Right-click for options.
3. Click the Export command.
4. Save the file as Word/RTF (.doc) to your computer.
5. Open the .doc file.
Data Set Instructions
The grades.sav file is a sample SPSS data set. The fictional data
represent a teacher’s recording of student demographics and
performance on quizzes and a final exam across three sections
of the course. Each section consists of about 35 students (N =
105).Software Installation
Make sure that IBM SPSS Statistics Standard GradPack is fully
licensed, installed on your computer, and running properly. It is
important that you have either the Standard or Premium version
of SPSS that includes the full range of statistics. Proper
software installation is required in order to complete your first
SPSS data assignment in Assessment 1.
Next, click grades.sav in the Assessment 1 Resources to
download the file to your computer.
· You will use grades.sav throughout the course.
The definition of variables in the grades.sav data set are found
in the Assessment 1 Context. Understanding these variable
definitions is necessary for interpreting SPSS output.
In Assessment 1, you will define values and scales of
measurement for all variables in your grades.sav file.
Verify the values and scales of measurement assigned in the
grades.sav file using information in the Data Set on page 2 of
this document.
Data Set
There are 21 variables in grades.sav,. Open your grades.sav file
and go to the Variable View tab. Make sure you have the

following values and scales of measurement assigned.
SPSS variable
Definition
Values
Scale of measurement
id
Student identification number
Nominal
lastname
Student last name
Nominal
firstname
Student first name
Nominal
gender
Student gender
1 = female; 2 = male
Nominal
ethnicity
Student ethnicity
1 = Native; 2 = Asian; 3 = Black;
4 = White; 5 = Hispanic
Nominal
year
Class rank
1 = freshman; 2 = sophomore;
3 = junior; 4 = senior
Scale
lowup
Lower or upper division
1 = lower; 2 = upper
Ordinal
section

Class section
Nominal
gpa
Previous grade point average
Scale
extcr
Did extra credit project?
1 = no; 2 = yes
Nominal
review
Attended review sessions?
1 = no; 2 = yes
Nominal
quiz1
Quiz 1: number of correct answers
Scale
quiz2
Scale
quiz3
Scale
quiz4
Scale
quiz5
Scale
final

Final exam: number of correct answers
Scale
total
Total number of points earned
Scale
percent
Final percent
Scale
grade
Final grade
Nominal
passfail
Passed or failed the course?
Nominal
2
Remove or Replace: Header Is Not Doc Title
Assessment 1 ContextTransitioning from Descriptive Statistics
to Inferential Statistics
In this assessment, we begin the transition from descriptive
statistics to inferential statistics which include correlation, t-
tests, and analysis of variance (ANOVA). This context
document includes information on key concepts related to
descriptive statistics, as well as concepts related to probability
and the logic of null hypothesis testing (NHT).Scales of

Measurement
In our initial quest to develop comprehension of statistical
analysis, it is first important to be sure the raw materials used
in the activity are understood. Statistical methods are methods
of analyzing data. What does that mean? In order to understand
statistics, we must first develop a basic vocabulary for
describing data, and recognize a system of names for different
categories and kinds of data. For the most part, the statement
means statistics provide various ways of answering specific
questions about data. It may serve us well to first back up and
make sure the fundamental units of statistical data are
understood.
In statistical analysis we make use of a concept called variables.
A variable is an abstract concept of a placeholder or a reserved
space. For instance, we may have a variable named GENDER.
Gender can have two possible values, male or female. The
concept of a variable is often easiest to understand in a concrete
sense by likening data to a typical table like those seen in
textbooks, or like a spreadsheet table found on computers. It is
a series of rows and columns which cross over each other. A
column in a table may be arranged such that the gender
identities of a group of people is recorded. We may have a list
of names in each of the rows of the table in the first column,
then a second column in the table which has the letters M or F
for each name. The title, or heading, of this column might read
"gender." The concept of a variable is like the column heading
where the gender is recorded. The column is the reserved space
for gender data, called a variable, and the column heading is the
name of the variable. Notice that the values of gender—male
and female—vary among different people in the rows. This is
why the entire column is called a variable. The values of the
variable vary among the rows.
Given this concept, be sure you understand that male and female

are NOT variables. The variable corresponds to the name of the
column where the information is recorded in the table. In the
case of this example, the name of the variable would probably
be GENDER—not male or female. Make sure you understand
this distinction so you do not use the term variable incorrectly,
as it will cause a great deal of confusion when trying to
communicate with others about statistics. In the case of the data
we are discussing here, it would make no sense whatsoever, and
would confuse whoever you were trying to communicate with, if
you said something like "the male variable," or that you are
working with two variables—male and female. That is not true,
and it will confuse whoever you say it to.
Consider that in the table where our list of people's names were
recorded along with their gender values, we now have two
columns. The first column contains names, the second column
contains the gender value for each person. Suppose there was a
third column in the table. In that column, we may decide to
write down each person's height in inches. At the top of the
column, we would place the title HEIGHT, and each row in the
table would have a number which was the height of the person
designated in the first column of each row. Notice that HEIGHT
is another variable. Notice it is a fundamentally different kind
of variable from GENDER, because it is a number which
corresponds to how tall the person is, while GENDER only has
two qualitative values—male and female. This points to another
important idea about data. There are several different types of
data, and you must be able to use a system of names which have
been developed to distinguish between different types of
variables. Understanding these categories of data is critical to
launching your effort to understand statistics, because if you do
not learn this system before beginning your exposure to
statistical analysis, you will be completely lost more times than
you are successful. It is worth your time to spend the requisite
amount of time and effort required to understand the concepts
we are about to discuss before you speed ahead to your

introduction to statistics in order to be finished. Of course, that
is only true if you want to understand the statistics. If you
would rather be completely lost, and have to call your instructor
later in the course and ask for extensions on your assignments
while you seek help, or if you look forward to asking for an
incomplete grade because you cannot comprehend anything you
are studying and you cannot do the assignments, then simply
browse through this concept of levels of measurement without
understanding it. You are guaranteed to suffer that fate.
There are two ways to understand what is important about the
different types of variables. The first is the most extensive and
formal, and the second is a rough approximation that will
usually serve your purposes.
First, an important concept in understanding descriptive
statistics is the four levels of measurement—nominal, ordinal,
interval, and ratio (Warner, 2013). These are sometimes called
"scales of measurement," "levels of data," or simply "data
levels." These scales of measurement, or kinds of variables, are
concepts which are needed to begin your study, because each
technique of analysis to which you are introduced is designed
for only specified kinds of variables, or levels of measurement.
In order to understand what kind of variables to use for the
analysis, and which kinds of variables are involved in different
parts of the analysis, you must be able to recognize the level of
measurement of a variable when you see its definition. There
are two important concepts involved in determining the level of
measurement for a variable. It is critical to know both the
characteristics of the variable's values (such as male or female,
or height in inches), but also what it is the values are being used
to measure or designate in the research project where the
variable is being used.
Consider that last statement again. We must know (1) the kinds
of values that are written in the column where the variable is

recorded, and (2) what those values correspond to in the people
they are being used to describe. Failure to consider both ideas
will result in misunderstanding levels of measurement. It is
instructive to begin by simply describing the four levels, and
then move on to some examples and explanations. First, please
remember that the four levels are a hierarchy. That is, they are
in order from lowest to highest, and each successive level has
all the qualities of the levels before it, plus some new added
category of information.
· Nominal data refers to numbers (or letters) arbitrarily assigned
to represent group membership, such as gender (male = 1;
female = 2). Nominal data is useful in comparing groups, but
they are meaningless in terms of measures of central tendency
and dispersion reviewed below. It does not matter whether the
values of the nominal variable are numbers or letters. If it is
numbers, they are mathematically meaningless. That is, 2 is not
"more" than 1.
· Ordinal data represents ranked data, such as coming in first,
second, or third in a marathon. However, ordinal data does not
tell us how much of a difference there is between
measurements. The first-place and second-place finishers could
finish 1 second apart, whereas the third-place finisher arrives 2
minutes later. Ordinal data lacks equal intervals, which again
prevents most mathematical interpretations, such as adding or
averaging the values.
· Interval data refers to data with equal intervals between data
points. This is the first level of measurement where the values
can have general mathematical meaning. An example is degrees
measured in Fahrenheit. The drawback for interval data is the
lack of a "true zero" value (freezing at 32 degrees Fahrenheit,
and 0 does not mean "no heat"). The most serious consequence
of a lack of a true zero is that one cannot reason using ratios – 4
is not necessarily twice as large as 2, and 5 is not necessarily

half as much as 10.
· Ratio data do have a true zero, such as heart rate, where "0"
represents a heart that is not beating. This level allows full
mathematical interpretation of the values, including ratios.
These four scales of measurement are routinely reviewed in
introductory statistics textbooks as the "classic" way of
differentiating measurements. However, the boundaries between
the measurement scales are fuzzy; for example, is intelligence
quotient (IQ) measured on the ordinal or interval scale?
Recently, researchers have argued for a simpler dichotomy in
terms of selecting an appropriate statistic. Most of the time,
being able to classify a variable into one of the two following
categories will serve the purposes needed:
· Categorical versus quantitative measures.
· A categorical variable is a nominal variable. It simply
categorizes things according to group membership (for example,
apple = 1, banana = 2, grape = 3).
· A quantitative measure represents a difference in magnitude of
something, such as a continuum of "low to high" statistics
anxiety. In contrast to categorical variables designated by
arbitrary values, a quantitative measure allows for a variety of
arithmetic operations, including =, <, >, +, -, x, and ÷.
Arithmetic operations generate a variety of descriptive statistics
discussed next.
Note that categorical variables generally correspond to the
nominal and ordinal levels of measurement in the previous
system, and quantitative variables typically correspond to the
interval and ratio levels of measurement. Quantitative variables,
or variables which are at the interval or ratio level of
measurement, are designated as scale variables in SPSS
software.

In order to determine the level of measurement of a variable,
one must consider the nature of the values as well as what those
values represent, or what the variable is measuring. For
instance, the same type of variable may have two levels of
measurement if the two versions are measuring different things
in a research project. The level of measurement refers to the
construct in the research project which is being measured—not
the values of the variable itself. A most interpretable example
would be a distinction between two different variables which
are exactly the same, except their operational definition in the
research project in which they are used is different. Consider a
score on a teacher-made test, where the score consists of the
number of correct answers out of 100 questions. The variable
can be defined in two ways:
1. An index which corresponds to the amount of knowledge the
test taker has in the topic area covered by the test.
2. The number of grade points credit earned on the test by the
test taker.
Notice that every person will have the same values on each of
these two variables. It is tempting to say there are no
differences. A closer look reveals these two variables are at two
different levels of measurement. First, when defined by the first
definition above—as the amount of knowledge—we know very
little about the meaning of the numbers involved. We do not
know if each question has the same amount of knowledge in it.
Also, we certainly cannot say that a score of zero means the
person has "no knowledge." Notice that this restricts the
variable to the ordinal level of measurement, because we cannot
even be sure that the necessary qualities for interval level
measurement are met (equal distances between points in terms
of amount of knowledge). We cannot say that a person that
scores 10 has twice as much knowledge as a person who scores
5, and we cannot say that the difference in knowledge between

two people who score 5 and 6 is the same as the difference
between two people who score 1 and 2. On the other hand, when
the measurement is defined by the second definition above, the
level of measurement is ratio. Clearly, all the requirements for
ratio level data are met. Zero means no grade points, and a
person who scores 6 has twice as many grade points as a person
who scores 3. The point of the preceding paragraph is that the
level of measurement of a variable depends on what it is being
used for—what is being measured in the research project.
Measures of Central Tendency and Dispersion
Descriptive statistics summarize a set of scores in terms of
central tendency (for example, mean, median, mode) and
dispersion (for example, range, variance, standard deviation).
As an example, consider a psychologist who measures 5
participants' heart rates in beats per minute: 62, 72, 74, 74, and
118.
· The simplest measure of central tendency is the mode. It is the
most frequent score within a distribution of scores (for example,
two scores of hr = 74). Technically, in a distribution of scores,
you can have two or more modes. An advantage of the mode is
that it can be applied to categorical data. It is also not sensitive
to extreme scores. This measure is suitable for all levels of
measurement.
· The medianis the geometric center of a distribution because of
how it is calculated. All scores are arranged in ascending order.
The score in the middle is the median. In the five heart rates
above, the middle score is a 74. If you have an even number of
scores, the average of the two middle scores is used. The
median also has the advantage of not being sensitive to extreme
scores. This measure is only suitable for data which are at the
ordinal level of measurement or above.
· The meanis probably what most people consider to be an

average score. In the example above, the mean heart rate is
(62+72+74+74+118) ÷ 5 = 80. Although the mean is more
sensitive to extreme scores (for example, 118) relative to the
mode and median, it can be more stable across samples, and it is
the best estimate of the population mean. It is also used in many
of the inferential statistics studied in this course, such as t-tests
and analysis of variance (ANOVA). This measure is suitable
only for interval and ratio level variable (quantitative
measures). It relies on math (sums and quotients) which are not
valid for ordinal measures.
· In contrast to measures of central tendency, measures of
dispersion summarize how far apart data are spread on a
distribution of scores. The rangeis a basic measure of dispersion
quantifying the distance between the lowest score and the
highest score in a distribution (for example, 118 – 62 = 56). A
deviance represents the difference between an individual score
and the mean. For example, the deviance for the first heart rate
score (62) is 62– 80, or - -18. By calculating the deviance for
each score above from a mean of 80, we arrive at -18, -8, -6, -6,
and +38. Summing all of the deviances equals 0, which is not a
very informative measure of dispersion. An alternative measure
of dispersion is the interquartile range (IQR), as well as the
semi-interquartile range (sIQR). If the values are placed in
order from lowest to highest, each value can be assigned a rank,
with the lowest rank being 1 which corresponds to the lowest
value in the data set. The highest value of the variable will have
a rank equal to the number of values in the data set. The ranks
can be used to identify two important scores in the data set.
First, the 25th percentile is the score which 25% of the scores
fall at or below. There is also a 75th percentile, which is the
score which 25% of the sample scores above. The IQR is the
difference between the scores at the 75th and 25th percentiles,
and the sIQR is half of that value. These measures are suitable
for variables at the ordinal level of measurement and above.
· A somewhat more informative measure of dispersion is sum of
squares (SS), which you will see again in the study of analysis

of variance (ANOVA). To get around the problem of summing
to zero, the sum of squares involves calculating the square of
each deviation and then summing those squares. In the example
above, SS = [(-18)2 + (-8)2 + (-6)2 + (-6)2 + (+38)2] = [(324) +
(64) + (36) + (36) + (1444)] = 1904. The problem with SS is
that it increases as data points increase (Field, 2009), and it still
is not a very informative measure of dispersion. This measure is
suitable only for interval and ratio levels of measurement.
· This problem is solved by next calculating the variance (s2),
which is the average distance between the mean and a particular
score (squared). Instead of dividing SS by 5 for the example
above, we divide by the degrees of freedom, N – 1, or 4. The
variance is therefore SS ÷ (N– 1), or 1904 ÷ 4 = 476. The
problem with interpreting variance is that it is the average
distance of "squared units" from the mean. What is, for
example, a "squared" heart rate score? This measure is suitable
only for interval and ratio levels of measurement.
· The final step is calculating the standard deviation(s), which
is simply calculated as the square root of the variance, or in our
example, √476 =21.82. The standard deviation represents the
average deviation of scores from the mean. In other words, the
average distance of heart rate scores to the mean is 21.82 beats
per minute. If the extreme score of 118 is replaced with a score
closer to the mean, such as 90, then s = 9.35. Thus, small
standard deviations (relative to the mean) represent a small
amount of dispersion; large standard deviations (relative to the
mean) represent a large amount of dispersion (Field, 2009). The
standard deviation is an important component of the normal
distribution. This measure is suitable only for interval and ratio
levels of measurement.
Notice that the various methods of expressing central tendency
and dispersion are suitable for different levels of measurement.
A brief summary of the types of measures used with different
levels of measurement are shown in the table below:

.
Typical Measures Central Tendency Dispersion
Levels of Measurement
Categorical Quantitative Nominal Ordinal Interval
Ratio Mode Median
Mean
k*
sIQR
Standard Deviation
Note: The number of categories or groups (often designated as
k) is the primary expression of variation for nominal level
data.Visual Inspection of a Distribution of Scores
An assumption of the statistical tests that you will study in this
course is that the scores for a dependent variable, like a range
of heart rate scores, are normally (or approximately normal) in
shape. The assumption is first checked by examining a
histogram of the distribution. This method is meaningful only
for quantitative variables—interval or ratio levels of
measurement. It makes no sense to create histograms of nominal
or ordinal (categorical) variables.
Departures from normality and symmetry are assessed in terms
of skew and kurtosis. Skewness is the tilt or extent a
distribution deviates from symmetry around the mean. A
distribution that is positively skewed has a longer tail extending
to the right (that is, the "positive" side of the distribution) A
distribution that is negatively skewed has a longer tail
extending to the left (that is, the "negative" side of the
distribution) In contrast to skewness, kurtosis is defined as the
peakedness of a distribution of scores.
The use of these terms is not limited to your description of a

distribution following a visual inspection. They are included in
your list of descriptive statistics and should be included when
analyzing your distribution of scores. Skew and kurtosis scores
of near zero indicate a shape that is symmetric or close to
normal respectively. Values of -1 to +1 are considered ideal,
whereas values ranging from -2 to +2 are considered acceptable
for psychometric purposes.Outliers
Outliers are defined as extreme scores on either the left of right
tail of a distribution, and they can influence the overall shape of
that distribution. There are a variety of methods for identifying
and adjusting for outliers. Outliers can be detected by
calculating z-scores or by inspection of a box plot. Once an
outlier is detected, the researcher must determine how to handle
it. The outlier may represent a data entry error that should be
corrected. The outlier may be a valid extreme score, and can
either be left alone, deleted, or transformed. Whatever decision
is made regarding an outlier, the researcher must be transparent
and justify his or her decision.
Probability and Hypothesis Testing
The interpretation of inferential statistics requires an
understanding of probability and the logic of null hypothesis
testing (NHT). The logic and interpretative skills addressed in
this assessment are vital to your success in interpreting data in
this course as well as to your success in advanced courses in
statistics.
It is useful to first consider some general ideas about hypothesis
testing to be sure terminology is clear and automatically
understood. Inaccuracies in vocabulary can present barriers
which will follow you through to the advanced concepts and
spoil your learning experience, so it is best to clear up common
errors before beginning to study the more complex ideas. An
hypothesis is simply a precise statement or prediction of a fact
we think is true. The first thing you should note is that we are
using two words here: hypothesis and hypotheses. The only
difference is the “is” and “es” in the last two letters. This is

simply the distinction between singular and plural. We may
examine a single hypothesis, or we may speak of many different
hypotheses. The first term rhymes with “this” and the second
rhymes with “these.” In fundamental statistics (and even in
most advanced methods), we encounter hypotheses of two types.
Confusion among these types is a common barrier in learning
statistics, so it will be valuable for you to study these ideas
carefully in order to provide and advance organizer for material
to come. The two types of hypotheses tests we will study are
hypotheses tests related to (1) group differences, and
hypotheses related to (2) association between variables. Each of
these types of hypotheses tests have two mutually exclusive and
conflicting statistical hypotheses, called the null hypothesis and
the alternative hypothesis. In each of these two general types of
hypotheses tests, the purpose of the statistical test is to show
that the null hypothesis has a low probability of being true. The
reason this is done is usually to support the alternative
hypothesis, because if the null hypothesis is false, the
alternative hypothesis must be true.Group Differences
An hypothesis of group differences is related to the question of
whether or not two or more groups have the same central
tendency (such as equal means). There are two possible
hypotheses in this category: The groups have equal means or the
groups do not have equal means. The first hypothesis, that the
means are equal, is called the null hypothesis. The hypothesis
that the groups do not have equal means is the alternative
hypothesis. This is true regardless of what the researcher is
interested in (whether the researcher wants the groups to be
equal or not). It is related to the mathematical structure of the
tests, and it is a constant in the statistical analysis. The purpose
of the statistical test is to demonstrate that the null hypothesis
has a very small probability of being true. If the null hypothesis
is not true, then the alternative hypothesis must be
true.Association Between Variables

The second type of is whether or not two variables are
association (related or correlated). When two variables are
associated, scores or values are analyzed in pairs. The
hypothesis is related to the idea that the values of the first
variable somehow predict the values of the second. The
relationship between the two variables is usually expressed by
some type of correlation coefficient. It is usually an index
which varies between -1 and +1. If the relationship between the
two variables is computed to be zero, then there is no
association between them. The value on the first variable has
nothing to do with the value on the second. If the correlation
coefficient is positive (between 0 and +1), it means those cases
with high values on the first variable tend to have high values
on the second. If the correlation coefficient is negative
(between -1 and 0), then it means high values on the first
variable tend to match up with low values on the second. The
magnitude or size of the correlation coefficient (its distance
from zero and closeness to -1 or +1) is the strength of the
relationship between the two variables. If the strength of the
relationship is zero, then there is no relationship – the value of
the first variable has nothing to do with the second.
Consider a table of data where students are listed in rows, and
variables are listed in columns, like a teacher’s gradebook.
There may be two variables, such as total number of absences in
one column and final test scores in another. Each case (each
student) has a value on both of these variables. We might expect
those students who have high numbers on absences would tend
to have lower numbers on the final exam. Note we would expect
the correlation between absences and exam scores to be
negative—those students who have many absences will tend to
have low exam scores. Statistical hypotheses are generally
related to questions of whether there is a relationship between
the two variables or not. In this type of statistical test, there are
two hypotheses: (1) the correlation between the two variables is
zero (the null hypothesis), or the correlation between the two

variable is not zero (the alternative hypothesis). This is true
regardless of what the researcher is interested in (regardless of
whether the research wants the variables to be related, or thinks
they are related, or not). The purpose of the statistical test is to
demonstrate that the null hypothesis has a very small
probability of being true. If the null hypothesis is not true, then
the alternative hypothesis must be true.The Standard Normal
Distribution and z-Scores
A student receives an intelligence quotient (IQ) score of 115 on
a standardized intelligence test. What is his or her percentile
rank? To calculate the percentile rank, you must understand the
logic and application of the standard normal distribution. The
advantage of the standard normal distribution is that the
proportional area under the curve is constant. This constancy
allows for the calculation of a percentile rank of an individual
score X for a given distribution of scores.
When a population mean (µ) and population standard deviation
(σ) are known, such as a distribution of scores for a
standardized intelligence test, the standard normal distribution
determines the percentile rank of a given X-score (for example,
50th percentile on IQ). Around 2/3rds of standard scores
(68.26%) fall within +/- 1 population standard deviation from
the population mean. Approximately 95 percent of standard
scores fall within +/- 2 population standard deviations.
Approximately 99 percent of standard scores fall within +/- 3
population standard deviations. Knowing this, we can begin the
process of determining a percentile rank from an individual
score.
Consider a standardized intelligence quotient (IQ) test where µ
= 100 and σ = 15. A standard score, or z-score, is calculated
with individual X scores rescaled to µ = 0 and σ = 1. The
formula for a z-score is [( X - µ) ÷ σ]. For example, an IQ score
of X = 100 would be rescaled to z = 0.00 [(100 – 100) ÷ 15 = 0].

A z-score of 0 means that the X-score is 0 standard deviations
above or below the mean. An IQ score of X = 85 would be
rescaled to z = -1.00 [(85 – 100) ÷ 15 = -1.00]. A z-score of -1
represents an IQ score 1 standard deviation below the mean. An
IQ score of 130 would be rescaled to z = 2.00 [(130 – 100) ÷ 15
= 2.00). A z-score of 2 represents an IQ score 2 standard
deviations above the mean. In short, a negative z-score falls to
the left of the population mean, whereas a positive z-score falls
to the right of the population mean.
Once a given z-score is calculated for a given X-score, its
percentile rank can be determined. The proportion of area under
the curve can also be calculated.
An important z-score is +/- 1.96, where 95 percent of scores fall
under the area of the curve, whereas 2.5 percent fall to the left
and 2.5 percent fall to the right (2.5% + 2.5% = 5%). A z score
beyond these cutoffs is typically considered to be "extreme"
(Warner, 2013). In addition, we will see below that most
inferential statistics set "statistical significance" to obtained
probability values of less than 5 percent.
Hypothesis Testing
Probability is crucial for hypothesis testing. In hypothesis
testing, you want to know the likelihood that your results
occurred by chance. As was discussed in the introduction to
hypothesis testing, we are generally concerned with the
probability that a null hypothesis is true (recall that a null
hypothesis is an hypothesis that two group means are equal, or
that two variables have no relationship). Whenever two groups
have equal means in a population, taking random samples of
those two groups and computing their means, then examining
the difference between those means, will rarely result in the
means having exactly zero difference. Because the samples are
selected randomly and do not contain all members of the
population, there will tend to be some small or negligible

difference in means for almost any two samples. These
differences are due to chance alone in the random selection
process. In the same way, whenever two variables are
completely unrelated in a population, random selection of a
small sample will rarely result in the two variable having
exactly zero correlation in those samples.
There will be some small amount of correlation between the two
variables in most random samples by chance alone. No matter
how unlikely, there is always the possibility that your results
have occurred by chance, even if that probability is less than 1
in 20 (5%). However, you are likely to feel more confident in
your inferences if the probability that your results occurred by
chance is less than 5 percent compared to, say, 50 percent. Most
psychologists find it reasonable to designate less than a 5
percent chance as a cutoff point for determining statistical
significance. This cutoff point is referred to as the alpha level.
An alpha level is set to determine when a researcher will reject
or fail to reject a null hypothesis (discussed next). The alpha
level is set before data are analyzed to avoid "fishing" for
statistical significance. In high-stakes research (for example,
testing a new cancer drug), researchers may want to be even
more conservative in designating an alpha level, such as less
than 1 in 100 (1%) that the results are due to chance.Null and
Alternative Hypotheses
The null hypothesis (H0) refers to the difference between a
group mean and a given population parameter, such as a
population mean of 100 on IQ, or to the difference between two
group means. Imagine that we ask two groups of students to
complete a standardized IQ test, and then we calculate the mean
IQ score for each group. We observe that the mean IQ for Group
A is 100 (MA = 100), whereas the mean IQ for group B is 115
(MB = 115). Is a mean difference of 15 IQ points statistically
significant or just due to chance? The null hypothesis predicts
that H0: MA = MB. That is, the null hypothesis predicts "no"

difference between groups. Remember that "null" also means
"zero," so we could also state the null hypothesis as H0: MA –
MB = 0. When comparing groups, in general, the null
hypothesis predicts that group means will not differ. When
testing the strength of a relationship between two variables,
such as the correlation between IQ scores and grade point
average (GPA), in general, the null hypothesis is that the
relationship (expressed as a single correlation coefficient)
between Variable x and Variable y is zero.
By contrast, the alternative hypothesis (H1)does predict a
difference between two groups, or in the case of relationships,
that two variables are significantly related. An alternative
hypothesis can be directional (for example, H1: Group A has a
higher mean score than Group B) or nondirectional (H1: Group
A and Group B will differ). In hypothesis testing, you either
reject or fail to reject the null hypothesis. Note that this is not
stating, "accept the null hypothesis as true." By default, if you
reject the null hypothesis, you accept the alternative hypothesis
as true (because the hypotheses are mutually exclusive).
However, if you do not reject the null hypothesis, you cannot
accept the null hypothesis as true. You have simply failed to
find statistical justification to reject the null hypothesis, and the
null hypothesis was assumed to begin with. There was no
evidence for it. The test was simply done to show it was false in
order to support the allegation that the alternative hypothesis
was true.Type I and Type II Errors
If you commit a Type Ierror, this means that you have
incorrectly rejected a true null hypothesis. You have incorrectly
concluded that there is a significant difference between groups,
or a significant relationship, where no such difference or
relationship actually exists. Type I errors have real-world
significance, such as concluding that an expensive new cancer
drug works when actually it does not work, costing money and
potentially endangering lives. Keep in mind that you will
probably never know whether the null hypothesis is "true" or
not, as we can only determine that our data fail to reject it.

If you commit a Type IIerror, this means that you have NOT
rejected a false null hypothesis when you should have rejected
it. You have incorrectly concluded that no differences or no
relationships exist when they actually do exist. Type II errors
also have real-world significance, such as concluding that a new
cancer drug does not work when it actually does work and could
save lives.
Your alpha level will affect the likelihood of making a Type I
or a Type II error. If your alpha level is small (for example, .01,
less than 1 in 100 chance), you are less likely to reject the null
hypothesis. So, you are less likely to commit a Type I error.
However, you are more likely to commit a Type II error.
You can decrease the chances of committing a Type II error by
increasing the alpha level (for example, .10, less than 1 in 10
chance). However, you are now more likely to commit a Type I
error. Since the chances of committing Type I and Type II
errors are inversely proportional, you will have to decide which
type of error is more serious. You need to assess the risk
associated with each type of error. Your research questions will
help in this decision. In standard psychological research, alpha
level is set to .05 (that is, a 1 in 20 chance of committing a
Type I error). An alpha level of .05 is used throughout the
remainder of this course.Probability Values and the Null
Hypothesis
The statistic used to determine whether or not to reject a null
hypothesis is referred to as the calculated probability value or p
value, denoted p. When you run an inferential statistic in SPSS,
it will provide you with a p value for that statistic (SPSS labels
p values as “Sig.”). If the test statistic has a probability value of
less than 1 in 20 (.05), we can say "p < .05, the null hypothesis
is rejected." Keep in mind in the coming weeks that we are
looking for values less than .05 to reject the null hypothesis.
This may seem counterintuitive at first, because usually we
assume that "bigger is better." In the case of null hypothesis
testing, the opposite is the case—if we expect to reject a null
hypothesis, remember that, for p values, "smaller is better." Any

p value less than .05, such as .02, .01, or .001, means that we
reject the null hypothesis. Any p value greater than .05, such as
.15, .33, or .78, means that we do not reject the null hypothesis.
Make sure you understand this point, as it is a common area of
confusion among statistics learners.
Based on your understanding of the null hypothesis, the
alternative hypothesis, the alpha level, and the p value, you can
begin to make statements about your research results. If your
results fall within the rejection region, you can claim that they
are "statistically significant," and you reject the null hypothesis.
In other words, you will conclude that groups do differ in some
way or that two variables are significantly related. If the results
do not fall within the rejection region, you cannot make this
claim. Your data fail to reject the null hypothesis. In other
words, you cannot conclude that the groups differ in some way,
or that the two variables are related.
This assessment covers the terminology and concepts behind
hypothesis testing, which prepares you for the remaining
assessments in this course. The statistical tests include:
· Correlations.
· t-Tests.
· Analysis of Variance.
ReferencesFarris, J. R. (2012). Organizational commitment and
job satisfaction: A quantitative investigation of the
relationships between affective, continuance, and normative
constructs (Doctoral dissertation, Capella University). Available
from ProQuest Dissertations and Theses database. (UMI No.
3527687)Field, A. (2009). Discovering statistics using SPSS
(3rd ed.). Thousand Oaks, CA: Sage Publications.George, D., &
Mallery, P. (2014). IBM SPSS statistics 21 step by step: A
simple guide and reference (13th ed.). Upper Saddle River, NJ:

Pearson.Lane, D. M. (2013). HyperStat online statistics
textbook. Retrieved from
http://davidmlane.com/hyperstat/index.htmlOnweugbuzie, A. J.
(1999). Statistics anxiety among African American graduate
students: An affective filter? Journal of Black Psychology, 25,
189–209.Pan, W., & Tang, M. (2004). Examining the
effectiveness of innovative instructional methods on reducing
statistics anxiety for graduate students in the social sciences.
Journal of Instructional Psychology, 31, 149–159.
Scoville, D. M. (2012). Hope and coping strategies among
individuals with chronic neuropathic pain (Doctoral
dissertation, Capella University). Available from ProQuest
Dissertations and Theses database. (UMI No. 3518448)
Stewart, A. M. (2012). An examination of bullying from the
perspectives of public and private high school children
(Doctoral dissertation, Capella University). Available from
ProQuest Dissertations and Theses database. (UMI No.
3505818)
Warner, R. M. (2013). Applied statistics: From bivariate
through multivariate techniques (2nd ed.). Thousand Oaks, CA:
Sage.
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