For this assignment, you will identify a specific organizational problem that could be addressed through statistical applications, and you will create a business case (justification for why your problem is important and should be prioritized above other projects requiring resources) to support the need for the analysis. For example, you might want to explore how a working team could be more efficient in their productivity or how your company could generate incremental revenue through better product design and/or advertising campaigns. As such, you would want to explain the problem, why it is important, and how it could be addressed through the use of statistical applications. You can use the dataset provided for this assignment and all subsequent assignments, or you may use your own dataset. Whichever dataset you use, it should be used throughout the course given that the assignments build upon prior assignments.
Your business case should consist of the following components:
Length: 6 pages, not including title or reference pages
References: Include a minimum of 5 scholarly resources not more than 5 years old.
The completed assignment should demonstrate thoughtful consideration of the ideas and concepts presented in the course by providing new thoughts and insights relating directly to this topic. The content should reflect scholarly writing and current APA 7th edition standards. Include a plagiarism report.
| Growth Opportunity Scoring Definitions | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Evaluation Criteria | Higher | Attractiveness | Medium Attractiveness / Fit (3 Points) |
Lower Attractiveness / Fit (1 Point) |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Revenue Potential | 3 Year revenue potential of $1, | 0 | 3 Year revenue potential of $999,999 – $400,000 | 3 Year revenue potential of $399,999 or less | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Pretax Potential | More than 40% | Between | 30% | Less than 30% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Strategic Alignment | Fits a key strategic growth initiative / lever and it fits our culture / business model | Fits a strategic growth initiative / lever | Unclear fit with current business strategies | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Client Need | Unmet need validated by potential customers; unmet need with customer request for service | Unmet need identified and confirmed (not with customer); met need with customer openess to service | Unmet need may exist but has not been confirmed; met need with customer not intersted in service | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Customers | Targets customer inside domain of interest, and decision maker is in a function we are very familiar with | Targets customer inside our domain of interest and the decision maker is unfamiliar with us | Targets customer outside our domain of interest | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Time to Revenue | Less than 6 months to initial revenue | 7- 18 months to initial revenue | Greater than 18 months to initial revenue | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Investment Required | Minor (0 – | 10% | Moderate (10-20% of revenue potential) | Significant (>20% revenue potential) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Progressive | Cutting Edge – Viewed as progressive by the target customer | Leading Edge – Viewed as “second” to the market but considered progressive | Standard – Effective and proven but not progressive | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Ability to Execute / Business Fit | Capabilities – Process | Does not require any significant additions to, or enhancement of, our existing processes | Requires enhancement of existing processes, but does not require new processes | Depends on process that do not exist in the business today | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Capabilities – Technology | Does not require any significant additions or upgrades to current tools | Requires substantial upgrades to existing tools, but no new tools | Requires new technology tools | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Capabilities – Skillsets | Only requires existing leadership, management, and operational skillsets | Requires new skillsets / talent from a leadership/management or an operational perspective (not both) | Requires the addition or new skillsets / talent from both a leadership/management and an operational perspective | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Competitors | Competitive set is limited or does not exist (less than 2) | Competitive set is moderate (2-6) | Competitive set is is very robust for our currents offering(s) (7+) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Pricing Model | Pricing terms and mechanics are consistent with current offerings and familiar to the target customer set | Pricing terms and mechanics are different from current offerings or unfamiliar to the target customer set (not both) | Pricing terms and mechanics are different from current offerings and will be unfamiliar to the target customer set |
| Growth Opportunity Scoring Sheet | ||||||||||||||||||||||||||||||||||
| Score Confidence | ||||||||||||||||||||||||||||||||||
| Growth Opportunity Name: | ||||||||||||||||||||||||||||||||||
| Instructions: For each of the evaluation criteria listed, please provide a score in the ‘Score’ column based on the criteria provided in the ‘Scoring Definitions’ tab | ||||||||||||||||||||||||||||||||||
| as well as a brief rationale for why you entered each score | ||||||||||||||||||||||||||||||||||
| Weight | Score (1,3,5) |
Weighted Score | Rationale for Score | Score (10/6/2) |
||||||||||||||||||||||||||||||
| Economic Fit / Attractiveness | 0.0 | |||||||||||||||||||||||||||||||||
| 5% | ||||||||||||||||||||||||||||||||||
| Total | 70% | |||||||||||||||||||||||||||||||||
| Total Score | 100% |
>Master Scoring Summary
0)
0)
0
4
5
0
80 38 28 75 50 12 65 52 28 80 65 48 22 48 22 60 50 28 75 52 28 26 58 28 90 28 54 28 95 54 28 50 26 100 26 80 58 28 100 [CELLRANGE] 38 44 52 44 60 38 50 50 52 48 48 48 50 52 58 42 58 54 54 54 50 46 58 22 14 28 10 18 28 12 12 28 26 22 22 28 28 26 24 28 28 28 28 26 26 28 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Economic Fit/Attractiveness Ability to Execute/Business Fit
ID
Initiative Name
Score
Economic Fit/ Attractiveness (
7
Ability To Execute / Business Fit (
3
Confidence Rating
1
Initiative 1
3
8
22
9
2
Initiative 2
4
14
5
3
Initiative 3
52
28
80
4
Initiative 4
44
10
75
5
Initiative 5
6
18
6
Initiative 6
7
Initiative 7
50
12
65
8
Initiative 8
9
Initiative 9
10
Initiative 10
48
26
11
Initiative 11
60
12
Initiative 12
13
Initiative 13
14
Initiative 14
70
15
Initiative 15
58
85
16
Initiative 16
42
24
90
17
Initiative 17
18
Initiative 18
54
95
19
Initiative 19
20
Initiative 20
100
21
Initiative 21
22
Initiative 22
46
23
Initiative 23
24
25
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]
[CELLRANGE]TIM-
1_Video_
_Data
| Date | Visits | VisitTime | TotalTime | Advertising | |||||||||||||||||||||||||||||||||||||||||||||||
| Friday | Police | Yes | |||||||||||||||||||||||||||||||||||||||||||||||||
| Saturday | 0.7 | 6 | |||||||||||||||||||||||||||||||||||||||||||||||||
| Sunday | |||||||||||||||||||||||||||||||||||||||||||||||||||
| Monday | No | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Tuesday | |||||||||||||||||||||||||||||||||||||||||||||||||||
| Wednesday | |||||||||||||||||||||||||||||||||||||||||||||||||||
| Thursday | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1. | 3 | 7.9 | 5 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 2.9 | 8 | 14.9 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 2.4 | 16.83 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 0.82 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.93 | 15.45 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.33 | 3.99 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.68 | Theif | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 0.67 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.88 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 7.97 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.41 | 4.22 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 10 | 2.85 | 28.45 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 4.44 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.23 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.15 | 12.89 | ||||||||||||||||||||||||||||||||||||||||||||||||||
Evidence Based Library and Information Practice 2007, 2:1
32
Evidence Based Library and Information Practice
Feature Article
A Statistical Primer: Understanding Descriptive and Inferential Statistics
Gillian Byrne
Information Services Librarian
Queen Elizabeth II Library
Memorial University of Newfoundland
St. John’s, NL , Canada
Email: gbyrne@mun.ca
Received: 13 December 2006 Accepted: 08 February 2007
© 2007 Byrne. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
As libraries and librarians move more towards evidence‐based decision making, the data
being generated in libraries is growing. Understanding the basics of statistical analysis is
crucial for evidence‐based practice (EBP), in order to correctly design and analyze research
as well as to evaluate the research of others. This article covers the fundamentals of
descriptive and inferential statistics, from hypothesis construction to sampling to common
statistical techniques including chi‐square, correlation, and analysis of variance (ANOVA).
Introduction
Much of the research done by librarians,
from bibliometrics to surveys to usability
testing, requires the measurement of certain
factors. This measurement results in
numbers, or data, being collected, which
must then be analyzed using quantitative
research methods. A basic understanding of
statistical techniques is essential to properly
designing research, as well as accurately
evaluating the research of others.
This paper will introduce basic statistical
principles, such as hypothesis construction
and sampling, as well as descriptive and
inferential statistical techniques. Descriptive
statistics describe, or summarize, data, while
inferential statistics use methods to infer
conclusions about a population from a
sample.
In order to illustrate the techniques being
http://creativecommons.org/licenses/by/2.0
Evidence Based Library and Information Practice 2007, 2:1
33
Great Job Lousy Job
If you accept the job Have a great experience Waste time & effort
If you decline the job Waste an opportunity Avoid wasting time & effort
Figure 1. Illustration of Type I & II errors.
described here, an example of a fictional
article will be used. Entitled Perceptions of
Evidence‐Based Practice: A Survey of Canadian
Librarians, this article uses various
quantitative methods to determine how
Canadian librarians feel about Evidence‐
based Practice (EBP). It is important to note
that this article, and the statistics derived
from it, is entirely fictional.
Hypothesis
Hypotheses can be defined as “untested
statements that specify a relationship
between two or more variables” (Nardi 36).
In social sciences research, hypotheses are
often phrased as research questions. In plain
language, hypotheses are statements of
what you want to prove (or disprove) in
your study. Many hypotheses can be
constructed for a single research study, as
you can see from the example in Fig. 1.
In research, two hypotheses are constructed
for each research question. The first is the
null hypothesis. The null hypothesis
(represented as H0) assumes no relationship
between variables; thus it is usually phrased
as “this has no affect on this”. The
alternative hypothesis (represented as H1) is
simply stating the opposite, that “this has an
affect on this.” The null hypothesis is
generally the one constructed for scientific
research.
Type I & II Errors
Anytime you make a decision in life, there is
a possibility of two things going wrong.
Take the example of a job offer. If you
decide to take the job and it turned out to be
lousy, you would have wasted a lot of time
and energy. However, if you decided to pass
on the job and it was great, you would have
wasted an opportunity. It’s best illustrated
by a two by two box (Fig. 1).
It is obvious that, despite thorough research
about the position (speaking to people that
work there, interview process, etc.), it is
possible to come to the wrong conclusion
about the job. The same possibility occurs in
research. If your research concludes that
there is a relationship between variables
when in fact there is no relationship (i.e.,
you’ve incorrectly assumed the alterative
hypothesis is proven), this is a Type I error.
If your research concludes that there is no
relationship between the variables when in
fact there is (i.e., you’ve incorrectly assumed
the null hypothesis is proven), this is a Type
II error. Another way to think of Type I & II
errors is as false positives and false
negatives. Type I error is a false positive,
like concluding the job is great when it’s
lousy. A Type II error is a false negative;
concluding the job is lousy when it’s great.
Type I errors are considered by researchers
to be more dangerous. This is because
concluding there is a relationship between
variables when there is not can lead to more
extreme consequences. A drug trial
illustrates this well. Concluding falsely that
a drug can help could lead to the drug being
put on the market without being beneficial
to the public. A Type II error would lead to
a promising drug being left off the market,
Evidence Based Library and Information Practice 2007, 2:1
34
which while serious, isn’t considered as dire.
To help remember this, think of the
conservative nature of science. Inaction (and
possibly more testing) is less dangerous
than action. Thus, disproving the null
hypothesis, which supposes no relationship,
is preferred to proving the alternative
hypnosis.
There are many safety features built in to
research methodology which help minimize
the possibility of committing both errors,
including sampling techniques and
statistical significance, both of which you
will learn about later.
Dependent and Independent Variables
Understanding hypotheses help you
determine which variables are dependent
and which are independent (why this is
important will be revealed a bit later).
Essentially it works like this: the dependent
variable (DV) is what you are measuring,
while the independent variable (IV) is the
cause, or predictor, of what is being
measured.
In experimental research (research done in
controlled conditions like a lab), there is
usually only one hypothesis, and
determining the variables are relatively
simple. For example, in drug trials, the
dosage is the independent variable (what
the researcher is manipulating) while the
effects are dependent variables (what the
researcher is measuring).
In non‐experimental research (research
which takes place in the ‘real world’, such as
survey research), determining your
dependent variable(s) is less straightforward.
The same variable can be considered
independent for one hypothesis while
dependent for another. An example – you
might hypothesize that hours spent in the
library (independent variable) are a
predictor of grade point average (dependent
variable). You might also hypothesize that
major (independent variable) affects how
much time students spend in the library
(dependent variable). Thus, your hypothesis
construction dictates your dependent and
independent variables.
A final variable to be aware of in
quantitative research is the confounding
variable (CV). Also know as lurking
variables, a confounding variable is an
unacknowledged factor in an experiment
which might affect the relationship between
the other variables. The classic example of a
confounding example affecting an
assumption of a relationship is that murder
rates and ice cream purchased are highly
correlated (when murder rates go up, so
does the purchase of ice cream?). What is
the relationship? There isn’t one; both
variables are affected by a third,
unacknowledged variable: hot weather.
Population, Samples & Sampling
Although it is possible to study an entire
population (censuses are examples of this),
in research samples are normally drawn
from the population to make experiments
feasible. The results of the study are then
generalized to the population. Obviously, it
is important to choose your sample wisely!
Population
This might seem obvious, but the first step is
to carefully determine the characteristics of
the population about which you wish to
learn. For example, if your research
involves your university, it is worthwhile to
investigate the basic demographic features
of the institution; i.e., what is the percentage
of undergraduate students vs. graduate
students? Males vs. females? If you think
these are groups you would like to compare
in your study, you must ensure they are
properly represented in your sample.
Sampling Techniques
Probability Sampling
Evidence Based Library and Information Practice 2007, 2:1
35
Probability sampling means that each
member of the population has an equal
chance of being selected for the survey.
There are several flavors of probability
sampling; the common characteristic being
that in order to perform probability
sampling you must be able to identify all
members of your population
Random sampling is the most basic form of
probability sampling. It involves identifying
every member of a population (often by
assigning each a number), and then
selecting sample subjects by randomly
choosing numbers. This is often done by
computer programs.
Stratified random sampling ensures the
sample matches the population on
characteristics important to a study. Using
the example of a university, you might
separate your population into graduate
students and undergraduate students, and
then randomly sample each group
separately. This will ensure that if your
university has 70% undergraduates and 30%
graduates, your sample will have a similar
ratio.
Cluster sampling is used when a population
is spread over a large geographic region.
For example, if you are studying librarians
who work at public libraries in Canada, you
might randomly sample 50 libraries, and
then randomly sample the librarians within
those libraries.
Non‐probability Sampling
Simply put, this is any sampling technique
that does not involve random sampling.
Often samples are not random because in
some research it is easier to perform
convenience sampling (surveying those who
volunteer, for example). Also, sometimes the
population from which the sample is to be
taken cannot be easily identified. A
common strategy employed by libraries is to
use patron records to derive random
samples. This is probability sampling only if
the population is library users; if the
population is an entire institution or city, it
is no longer random. With non‐probability
samples, you can only generalize to those
who participated, not to a population.
Sample Size
Sample size is also extremely important to
be able to accurately generalize to a
population. Generally, the bigger the sample,
the better. The Central Limit Theorem states
that the larger the sample, the more likely
the distribution of the means will be normal,
and therefore population characteristics can
more accurately be predicted. Some other
things to keep in mind:
• If you want to compare groups with
each other (for example, majors),
you will need at least 5 subjects in
each group to do many statistical
analyses.
• Poor response rate can severely
compromise a study, if surveys are
involved. Depending on the
distribution method, response rate
can be as low as 10% (ideally you
want a response rate over 70%)
(Weisberg 119).Ensure your sample
size is large enough to still provide
accurate results with a poor
response rate.
There is no magic formula to determine the
proper sample size – it depends on the
complexity of your research, how
homogenous the population is, and time
and human resources you have available to
compile and analyze data.
Descriptive Statistics
Once you have performed your research
and gathered data, you need to perform
Evidence Based Library and Information Practice 2007, 2:1
36
Table 1. Examples of hypotheses.
data analysis. Choosing the appropriate
statistical method for the data is crucial. The
bad news is, this means you have to know a
whole lot about your data – is it nominal,
ordinal or ratio? Is it normally distributed?
Let’s start from the very beginning.
A clear understanding of librarians’ perceptions of EBP is necessary to inform the development of
systems to support EBP in librarianship.
The following research questions were posed:
1. What are the perceptions of librarians of EBP?
2. Does institution type the librarian works at affect perception?
3. Does length of service of the librarian affect perception?
What are the hypotheses?
There are three being provided. Here is a rephrasing of number 3:
H0 = “Length of service of librarians has no affect on the perception of EBP”
H1 = “Length of service of librarians affects the perception of EBP”
What are the Type I & II error possibilities?
The real situation (in the population)
H0 is true H1 is true
No error
Type II error
Result of
Research
(from sample):
H0 is proven (length
of service doesn’t
affect perception)
H1 is proven (length
of service does affect
perception)
Type I error
No error
What are the dependent and independent variables?
The researchers are attempting to determine whether length of service can predict perception of EBP,
or to rephrase, is perception of EBP dependant on length of service. Therefore:
Dependent variables: perception of EBP
Independent variable: length of service
Evidence Based Library and Information Practice 2007, 2:1
37
Levels of Measurement
Nominal variables are measured at the most
basic level. They are discrete levels of
measurement where a number represents a
category (i.e., 1 = male; 2 = female), but these
numbers do not imply order and
mathematical calculations cannot be
performed on them. You could just as easily
say, 1 = male and 36,000 = female ‐ this
doesn’t mean that females are 35, 999 times
bigger or better than males! Nominal
variables are of the least use statistically.
Ordinal variables are also discrete categories,
but there is an order to the categories; they
increase and decrease at regular intervals. A
good example is a Likert scale: 1 = very
poor; 2 = poor; 3 = average, etc. In this
example, you can state 1 is ‘less’ or ‘smaller’
or ‘worse’ than 2. The disadvantage of
ordinal variables is that you cannot measure
in between the values. You do not know
how much worse 1 is than 2.
Ratio (sometimes known as scale,
continuous or interval) variables are the
most robust, statistically, of variable types.
Ratio variables have natural order, and the
distance between the points in the same.
Think of pounds on a scale. You know that
Table 2. Examples of sampling.
The sampling frame was the database of all librarians (defined as those who hold an MLS)
who were members of the Canadian Library Association in March 2005. A total of 5,683
librarians were on the list. The list was divided up by type of library worked at (academic,
public, school, special, and other / not stated). A proportional random sample of
210
was then
selected. This ensured that even at a return rate of 40% a final sample size of 150 would be
achieved.
Is this a random sample?
On first glance, yes. However, this is only a true random sample if all librarians in Canada
belonged to the Canadian Library Association. The design of this study means that the results
can only be generalized to Canadian Library Association members, not to Canadian librarians.
What sampling technique is used?
This survey used stratified random sampling to ensure that all types of librarians would be
represented, as illustrated in the chart below. Please remember that all values in this table are
for demonstration purposes and do not accurately reflect reality.
Academic
Librarians
Public
Librarians
School
Librarians
Special
Librarians
Other /
Not Stated
Totals
Real
Proportion
1136 (20%) 2273 (40%) 568 (10%) 582
(15%)
582 (15%) 5683
Sample Size 42
(20%)
84
(40%)
21
(10%)
31
(15%)
31
(15%)
210
Evidence Based Library and Information Practice 2007, 2:1
38
100 is lighter than 101. You also know that
101 is 1 pound heavier than 100. Finally the
scale is continuous; it is possible to weigh
100.58 pounds. The power of the ratio
variable is important to keep in mind for
your study. For example, rather than asking
subjects to tick off an age category in a box,
you can ask them to fill in their age. This
gives you the freedom to keep it as a ratio
variable, or to round the ages up into
appropriate ordinal values.
Measures of Central Tendency
The theory of normal distribution tells us
that, if you tested an entire population, the
result (parameter) would look like a bell
curve, with the majority of values grouped
in the middle. A good example of this
would be scores on test.
Table 3. Examples of variables.
Selection of variables used in the study
Variable Name Variable Label Values
TYPE
Type of library worked at 1 = academic, 2 = public…
LENGTH
Length of service
INCOME
Income of respondent 1 = under 30,000, 2 = 31,000‐
40,000…
AGE Age of respondent
EBP_AWARE Answer to the question I
have heard of EBP
1 = yes, 2 = no
EBP_SCORE Score on the EBP
Perceptions Test
What level of measurement is TYPE?
TYPE is a nominal measurement. The numbers represent types of libraries, but no
mathematical calculations can be performed on them. EBP_AWARE is also a nominal
measurement.
What level of measurement is LENGTH?
Because there are no values set for LENGTH it is a ratio variable. Each librarian’s length of
service will be entered in years. EBP_SCORE and AGE are also ratio variables.
What level of measurement is INCOME?
INCOME is an ordinal variable. It has numbers representing categories, but there is a clear
ranking. Librarians in category one earn less money than librarians in category two.
Evidence Based Library and Information Practice 2007, 2:1
39
Figure 2. Normal distribution of a bell curve.
However, when moving from parameters to
statistics, there is the probability that the
results will not reflect the population, and
thus not be normally distributed. Measures
of Central Tendency provide you with
information about how your results are
grouped. There are three measures, and
which one to use depends on what level of
measurement the variable is.
Mean (represented by M or μ) is the most
commonly referred to measure of central
tendency. It is the average measure, where
each value is added, and then the sum
divided by the number of cases. However,
it should be quite clear that the mean cannot
be used with nominal and ordinal variables.
Imagine again a Likert scale. The mean
value might be 2.36, but what does that tell
you? That the average respondent falls
somewhere closer to “I found this difficult”
than “I have no opinion”?
Median (represented by Mdn) is the measure
commonly used with ordinal data. The
median is the halfway point of the data. To
calculate simply order your values from
lowest to highest and see at what value half
the data is below, and half is above. The
median is also an extremely valuable
measure for ratio data when there are
outliers (think how the average income
variable would be skewed in a town with
one multimillionaire). This is because
median is not affected by how far away
from the middle values are, just the quantity
of them. The median for 2, 2, 3, 4, 4 is 3; the
median for 2, 2, 3, 4, 10 is also 3.
Mode is often used with nominal data
(though it can also be calculated for other
variable types). It is simply the most
frequently occurring value in a dataset. An
example of when this would be an
appropriate measure is for major. The
average major makes no sense, nor does the
halfway point major, but the most
frequently occurring major does.
Measures of Spread
Measures of central tendency reveal much
about data, but not the whole story. You
also need to know how the values are
spread across the spectrum. Measures of
spread will tell us whether the values are
clustered around the mean or more spread
out. Think of test scores; one group might
all score 70, while another group’s score
might range from 60‐90. In this case, it is
possible that the mean, median and mode
would be the same, but we can see the
distribution is quite different. There are
three main statistical methods for
determining spread.
Range is the most basic measure; it is
calculated simply by subtracting the lowest
score from the highest score. However, this
is not the most accurate method as the range
can be skewed by outlier values (a very high
or very low score).
Evidence Based Library and Information Practice 2007, 2:1
40
Interquartile range is less likely to be
distorted by outliers, as it is calculated by
ordering the sample from highest to lowest,
then dividing the sample into four equal
quarters (percentiles). The median is then
calculated for each quartile. Subtracting the
median of the first quartile from the third
quartile obtains the interquartile range.
Standard deviation (represented by SD or σ)
is the most sophisticated measure of spread,
and a widely used statistical concept.
Statistical software will easily calculate
standard deviation, so the formula will not
be covered here. Because standard
deviation relies on calculations of the mean
it can only be used with continuous
variables. A standard deviation score of 0
indicates that there is no variation of values.
The higher the standard deviation, the
larger the spread.
Bivariate Analysis
At heart of all research is an interest in
determining relationships between variables.
Table 4. An example of measures of central tendency and measures of spread.
Characteristics of the variable AGE
Age of Respondent
N
210
Mean 44.05
Median 43
Mode 33
Std Deviation 12.77
Range 38
Percentiles 25 33
50 43
75 56.50
What does this tell us about the central tendencies of the data?
The average age of librarian respondent to this survey is 44.05. Half of the librarians were
over 43, while other half were under 43. The most commonly occurring age was 33.
What does this tell about the spread of the data?
We can tell something about spread simply by looking at the difference between mean,
median and mode. The fact that the mean is slightly higher than the median and much higher
than the mode indicates that there are some older respondents skewing the data.
The range indicates that there are 38 years between oldest and youngest respondent. This
large value could be due to the outliers at the upper end of the scale. However, the large
standard deviation also indicates a wide spread of values. This is not surprising, as logically
in any profession, there is likely to be a wide variety of ages.
Evidence Based Library and Information Practice 2007, 2:1
41
There are many statistical methods for
exploring those relationships, which ones to
choose are often dependent on the type of
variables with which you are working
(nominal, ordinal or ratio). It is also
important to understand statistical
significance (the extent to which the
relationship can be generalized to the
population) and effect size (the strength of
the relationship) with bivariate analysis
techniques.
Statistical Significance
Comprehending inferential statistics
requires a clear understanding of what is
meant by statistical significance. For
something to be statistically significant, it is
unlikely to have occurred by chance
(remember that every time you are dealing
with a sample you are taking the chance that
your results will not reflect the population).
Another way of putting it is that significance
tests denote how large the possibility is that
you are committing a Type I error.
Significance tests are affected by the
strength of relationship between variables
and the size of the sample. Common levels
of significance (represented by alpha, or α)
are 5%, 1% and 0.1%; if α =.01, you are
stating that there is a one in one thousand
chance this happened by coincidence.
Cross Tabulation
What is a cross tabulation?
Essentially a cross tabulation (cross tab) is a
table in which each cell represents a unique
combination of values. This allows you to
visually analyze whether one variable’s
distribution is contingent on another’s.
When would you use a cross tabulation?
Cross tabulations can be used to show
relationships between two nominal
variables, nominal and ordinal variables, or
two ordinal variables. It can be used with
ratio data, as long as the variable has a
limited number of values. Limitations of the
cross tabulation
Cross tabulations provide you with a visual
view of comparative data, but because they
display simple values and percentages,
there is no way to gauge whether any
differences in the distribution are
statistically significant.
Chi‐Square
What is a chi‐square?
A chi‐square is a test which looks at each
cell in a cross tabulation and measures the
difference between what was observed and
what would be expected in the general
population. It is used to evaluate whether
there is a relationship between the values in
the rows and columns of a cross tab, and the
likeliness that any differences can be put
down to chance.
When would you use a chi‐square?
Chi‐square is one of the most important
statistics when you are assessing the
relationship between ordinal and/or
nominal measures.
Are there limitations of using chi‐square?
Chi‐square cannot be used if any cell has an
expected frequency of zero, or a negative
integer. It can be affected by low
frequencies in cells; if many of your cells
have a frequency of less than 5, the chi‐
square test might be compromised.
How do I know if the relationship is
statistically significant?
The chi‐square test provides a significance
value called a p‐value. The p‐value is
compared to α, which can be set at different
levels. If α = .05, then a p score less than .05
indicates statistical significant differences, a
p score greater than .05 means that there is
no statistical difference.
Evidence Based Library and Information Practice 2007, 2:1
42
Table 5. Example of cross tabulation.
T‐test
What is a t‐test?
A t‐test compares the means between two
values. It tests whether any differences in
the means are statistically significant or can
be explained by chance.
When do you use a t‐test?
T‐tests are normally used when comparing
differences between two groups (i.e.,
undergraduates versus graduates) or in a
before and after situation (student
achievement before versus after library
instruction). A t‐test involves means,
therefore the dependent variable (the
variable you are attempting to measure)
must be a ratio variable. The independent
variable is nominal or ordinal.
Limitations of the t‐test
A t‐test can only be used to analyze the
means of two groups. For more than two
groups, use ANOVA.
How do I know if the relationship is
statistically significant?
Cross tabulation of type of library and I have heard the
term evidence‐based practice
Yes No Total
Academic Library
Count 30 12 42
Percentage 71.42% 28.58% 100%
Public Library
Count 54 30 84
Percentage 64.28% 35.72% 100%
School Library
Count 9 12 21
Percentage 42.86% 57.14% 100%
Special Library
Count 22 10 31
Percentage 70.96% 29.04% 100%
Other/Not Stated
Count 20 11 31
Percentage 64.51% 35.49% 100%
Total Count 100 110 210
What does this table tell us?
This table allows us to see the numbers of librarians who have heard of the term Evidence‐
based Practice broken down by type of library worked at. As you can see, there are some
differences between the groups; a smaller percentage of school librarians have heard of EBP
(42.86%, N = 9) than other type of librarians. There is no indication from this table, however, if
that difference is statistically significant.
Evidence Based Library and Information Practice 2007, 2:1
43
Like the chi‐square test, the t‐test provides a
significance value called a p‐value, and is
presented the same way.
Correlation Coefficients
What are correlation coefficients?
Correlation coefficients measure the
strength of association between two
variables, and reveal whether the correlation
is negative or positive. A negative
relationship means that when one variable
increases the other decreases (e.g., drinking
alcohol and reaction time). A positive
relationship means that when one variable
increases so does the other (e.g.,study time
and test scores). Correlation scores range
from ‐1 (strong negative correlation) to 1
(strong positive correlation). The closer the
figure is to zero, the weaker the association,
regardless whether it is a negative or
positive integer.
Table 6. Example of chi‐square.
A chi‐square statistic was then performed to determine if type of library worked at affected whether
librarians had heard the term evidence‐based practice. As you can see by the table below, p>.05,
therefore there is no statistical difference in distribution of awareness of EBP based on the type of
library worked at.
Value Df Sig.
Chi‐Square 16.955 4 .990
Why use a chi‐square?
A chi‐square is the statistic being used here because the relationship between two ordinal variables
(type of library worked at and awareness of the term EBP) is being explored.
What does value mean?
It is simply the mathematical calculation of the chi‐square. It is used to then derive the p‐value, or
significance.
What does df mean?
Df stands for degrees of freedom. Degrees of freedom is the number of values that can vary in the
estimation of a parameter. It is calculated for the chi‐square statistic by looking at the cross
tabulation and multiplying the number of rows minus one by the number of columns minus one (r‐
1) x (c‐1). In this case, if we look back to Fig. 4, we can see that we have a two by five table. Thus, (2‐
1) x (5‐1) = 4.
What does sig. mean?
Sig. stands for significance level, or p‐level. In this case p = .990. As this number is larger than .05,
the null hypothesis is proven. There is no statistically relationship between type of library and
awareness of EBP, despite the differences in percentages we saw in Table 5.
Evidence Based Library and Information Practice 2007, 2:1
44
When should you use correlation
coefficients?
Correlation coefficients should be used
whenever you want to test the strength of a
relationship. There are many tests to
measure correlation; which one to use
depends on what variables you are
examining. A few are listed below:
Nominal variables: Phi, Cramer’s V, Lambda,
Goodman and Kruskal’s Tau
Ordinal variables: Gamma, Sommers D,
Spearman’s Rho
Ratio variables: Pearson r
Limitations of correlation coefficients
Correlation does not indicate causality.
Simply because there is a relationship
between two variables does not mean that
one causes the other. Keep in mind
correlation only looks at the relationship
between two variables; there many be others
affecting the relationship (remember the
confounding variable!). Correlation
coefficients can also be skewed by outlier
values.
How do I know if the relationship is
statistically significant?
Correlation scores range from ‐1 (strong
negative correlation) to 1 (strong positive
correlation). The closer the figure is to zero,
the weaker the association, regardless of
whether it is a negative or positive integer.
Analysis of Variance (ANOVA)
What is ANOVA?
Like the t‐test, ANOVA compares means,
but can be used to compare more than two
groups. ANOVA looks at the differences
between categories to see if they are larger
or smaller than those within categories.
When should you use ANOVA?
The dependent variable in ANOVA must be
ratio. The independent variable can be
Table 7. Example of a t‐test.
An independent samples t‐test was performed to determine if there was a statistical difference
between genders on the Evidence‐based Practice test. As the table below illustrates, there was
a significant difference in performance between males and females, t (19)=‐.398 p<.05
Value df Sig.
T‐test ‐.398 19 .049
Why use a t‐test?
A t‐test is used for these variables because we are comparing the mean of one variable (EPB
Test Score, a ratio variable) between 2 groups (sex, a nominal variable). An independent
samples t‐test is used here because the groups being compared are mutually exclusive ‐ male
and female.
How is the t‐test interpreted?
The t‐test value, degrees of freedom, and significance values can be interpreted in precisely
the same way as the chi‐square in Fig. 5. The significance value of .049 is less that .05,
therefore it can be stated that the null hypothesis is disproved; there is a statistical significant
difference between the performance of male librarians and the performance of female
librarians on the EBP Perceptions Test.
Evidence Based Library and Information Practice 2007, 2:1
45
Table 8. Example of a Pearson r correlation.
nominal or ordinal, but most be composed
of mutually exclusive groups
Limitations of ANOVA
ANOVA measures whether there are
significant differences between three or
more groups, but it does not illustrate where
the significance lies – there could be
differences between all groups or only two.
There are tests called post hoc comparisons
which can be performed to determine where
significance lies, however.
How do I know if the relationship is
statistically significant?
An ANOVA uses an f‐test to determine if
there is a difference between the means of
groups. The f‐test can be used to calculate a
p‐score, which is analyzed in the same way
as chi‐squares and t‐tests.
Statistical Significance and Effect Size
Measures
Significance tests have a couple of
weaknesses. One is the fairly arbitrary
value at which statistical significance is said
to have occurred. Why is α = .051 not a
significant finding while α = .049 is? The
second disadvantage is that significance
tests do not give an indication of the
strength of a relationship, merely that it
exists. A smaller significance value could be
the result of a larger sample rather than a
strong relationship. This is where effect
sizes come in. Effect sizes are tests which
gauge the strength of a relationship. There
are many different effect size indices; which
to use depends on the statistical test being
performed.
Multivariate Analysis
Any in‐depth discussion of multivariate
analysis is beyond the scope of a paper
entitled “Statistical Primer”; however, here
is a brief introduction.
Multivariate analysis looks at the
relationship between more than two
variables, for example length of service and
type of librarian might together be
predictors of perception of EBP. Using
bivariate statistical methods, it is not
possible to see the relationship between two
independent variables as well as their effect
on the dependent variable. There are several
multivariate statistical methods. Here are
two of the most common.
A Pearson r correlation was performed to determine if there was a relationship between age
and score on the EBP test instrument. The correlation revealed that the two were significantly
related, r=+.638, n=210, p<.05.
Why was a Pearson r correlation performed?
A Pearson r was done because both variables involved, Age and EBP Perceptions Test score,
are ratio variables.
What does the r value tell us?
The r is correlation score. Remember that correlation scores range from +/‐1 to 0. Therefore, a
score of +.638 reveals that there is a strong positive correlation between age and EBP score.
The fact that it is positive means that when one variable increases so does the other – the older
the librarian, the higher they scored on the EBP test instrument.
Evidence Based Library and Information Practice 2007, 2:1
46
Table 9. Example of ANOVA.
Statistical Test Effect Size Measure Comments
Chi‐square phi Phi tests return a value between zero (no
relationship) and one (perfect relationship).
T‐test Cohen’s d
Cohen’s d results are interpreted as 0.2 being a
small effect, 0.5 a medium and 0.8 a large effect
size. (Cohen 157)
ANOVA Eta squared Eta square values range between zero and one,
and can be interpreted like phi and Cohen’s d.
Table 10. Statistical tests and effect size measures.
Multivariate analysis of variance (MANOVA)
is an ANOVA which analyses several
dependent variables. It can be interpreted
in much the same way as ANOVA tests.
MANOVA has advantages over doing
multiple ANOVA tests, including reducing
the potential for Type I errors (concluding
that there is a relationship when there is not).
Conversely, MANOVA tests can also reveal
relationships not apparent in ANOVA tests.
Multiple linear regression examines “the
relationship between one ‘effect’ variable,
called the dependent or outcome variable,
and one or more predictors, also called
independent variables” (Muijs 168). It is
designed to work with continuous variables,
though there are different techniques
available for analyzing other variable types.
While performing and analyzing regressions
are complicated, they are valuable tools for
examining the relationship between many
variables. It is important to note that, like
other inferential statistical techniques,
values are created that provide the statistical
significance of the relationships.
For the EBP Test Instrument Score, the analysis of variance (ANOVA) revealed that there was
not a significant difference in performance F (3, 47)=3.43, p<.05 between types of librarians.
The critical value (.245) for the scores was obtained the F distribution table using dfbetween=4
and dfwithin=16.
Why was an ANOVA performed?
An ANOVA was the appropriate statistical technique because the dependent variable (EBP
Test score) is continuous, while the independent variable (type of library worked at) is
nominal and composed of several groups.
What does this tell us?
The F test score was calculated at 3.43. This score was used in conjunction with the degrees of
freedom (because we are comparing several groups, there are two degrees of freedom scores,
one for between the groups (4) and one for within the groups (16) to calculate the p‐score. P
= .245, which is greater than .05. Therefore there is no difference in performance on the test
based on the type of library worked at.
Evidence Based Library and Information Practice 2007, 2:1
47
Conclusion
This paper is not intended to produce
statistical experts. Rather, it is a guide to
understanding the basic principles and
techniques common in library and related
research. Most statistical software packages,
such as SPSS or SAS, will effortlessly
perform statistics, so it is far more important
that as a researcher you know a) how to
select an appropriate sample; b) know what
statistical technique is appropriate in which
situations; and c) be able to interpret results
correctly. There are a few things you can do
to make yourself more comfortable with
statistics. One is to purchase a basic
quantitative methods textbook. Look for one
that comes with a CD of sample data sets.
Running through the exercises in the
textbook will provide you with valuable
practice in performing and analyzing
statistics. There are several textbooks
available in the library field, although any
social science quantitative methods texts
would be useful. The second thing you can
do is to read the research literature in your
field. If you know the topic well, it is easier
to evaluate and interpret results.
Works cited
Cohen, J. “A Power Primer.” Psychological
Bulletin 112 (1992): 155‐159.
Muijs, Daniel. Doing Quantitative Research in
Education with SPSS. London: SAGE,
2004.
Nardi, Peter M. Doing Survey Research: A
Guide to Quantitative Methods. Boston:
Allyn and Bacon, 2003.
Weisberg, Herbert F., Jon A. Krosnick, and
Bruce D. Bowen. An Introduction to
Survey Research, Polling, and Data
Analysis. 3rd ed. Thousand Oaks, Calif.:
Sage Publications, 1996.
