What is a Number?
What is a number?
Names and symbols are arbitrary.
What is a number?
Names and symbols are arbitrary.
Four…. IV …. 4….
What is a number?
Names and symbols are arbitrary.
Numbers that are not numbers….
0
http://en.wikipedia.org/wiki/0_%28number%29
Numbers that are not numbers…
Some make the world go around.
e
What is a number?
Names and symbols are arbitrary.
Measurement:
“Rules for assigning numbers to objects
(or concepts) to represent quantities of attributes.”
What is a number?
Names and symbols are arbitrary.
Measurement:
What is a number?
Names and symbols are arbitrary.
But to be a true number scale the symbols
must follow some logical
and
systematic arrangement.
What is a number?
Names and symbols are arbitrary.
Measurement:
“Standardized process of assigning symbols to
objects according to certain prespecified and
nondegenerating rules.”
Degenerating!
160
Is it possible to have an IQ of 160? But what does it mean?
What is a number?
Names and symbols are
arbitrary.
Measurement:
What is a number?
Names and symbols are arbitrary.
Measurement:
“An object is never measured… only the
object’s attributes.”
Object characteristics.
are measured,
not objects,
So… what are these people rating?
What are they assigning numbers to?
What is a number?
Scales:
“A scale is the continuum upon which
measurements are located.”
Zero degrees
centigrade….
So then what is this…..
What is a number?
Scales:
Likert Scale
Is a statement (not a question)
followed by five categories of
agreement.
What is a number?
Scales:
Likert Scale
Ice cream is good for breakfast.
1. Strongly disagree
2. Disagree
3. Neither agree nor disagree
4. Agree
5. Strongly agree
What is a number?
Scales:
Likert Scale
What is a number?
Scales:
Likert Scale
What is a number?
Scales: Likert
What is a number?
Scales:
What is a number?
Scales:
Semantic scales:
Typically: Opposite adjectives
separated by 7 selection points.
What is a number?
Scales:
Semantic
scales:
Semantic
scales:
Hybrid
Scales:
But complex concepts in business
may not be easily measured.
What is a number?
So…..
Harvard professor S.S. Stevens
created numerical scales to measure
difficult concepts.
S. S. Stevens
1906 - 1973
Steven’s original paper in Science, 103(2684), June 7, 1946.
Steven’s Scales:
Ratio
Steven’s Scales:
1. Nominal Scales
Steven’s Scales:
1. Nominal Scales
a. Name
Steven’s Scales:
1. Nominal Scales
a. Name
b. Classify
Steven’s Scales:
1. Nominal Scales
a. Name
b. Classify
c. Categorize
Steven’s Scales:
1. Nominal Scales
a. Name
b. Classify
c. Categorize
Steven’s Scales:
1. Nominal Scales
2. Ordinal Scales
Steven’s Scales:
1. Nominal Scales
2. Ordinal Scales
Does everything a nominal scales does.
Ranks objects or concepts by some
characteristic.
http://www.usatoday.com/sports/ncaab/sagarin/
Steven’s Scales:
1. Nominal Scales
2. Ordinal Scales
3. Interval scales
Steven’s Scales:
1. Nominal Scales
2. Ordinal Scales
3. Interval scales
Does everything an
ordinal scale does.
Interval is now
meaningful.
Steven’s Scales:
1.
2.
3.
4.
Nominal Scales
Ordinal Scales
Interval scales
Ratio scales
Steven’s Scales:
1.
2.
3.
4.
Nominal Scales
Ordinal Scales
Interval scales
Ratio scales
Has all the characteristics of all other scales,
but it also has meaningful ratios. It has a true
zero.
Good source:
http://www.fao.org/docrep/W3241E/w3241e04.htm
Steven’s Scales:
1.
2.
3.
4.
Nominal Scales
Ordinal Scales
Interval scales
Ratio scales
X = f(x)
X = kx + c
X = kx
Which scale to use?
1. Amount of information needed
Which scale to use?
1. Amount of information needed
Each higher scale carries more information
than the one before it.
Which scale to use?
1. Amount of information needed
2. Characteristics of stimulus or concept
Which scale to use?
1. Amount of information needed
2. Characteristics of stimulus or concept
3. Application context
Which scale to use?
1.
2.
3.
4.
Amount of information needed
Characteristics of stimulus or concept
Application context
Capacity of scale
Which scale to use?
1.
2.
3.
4.
5.
Amount of information needed
Characteristics of stimulus or concept
Application context
Capacity of scale
Post-measurement analysis
Which scale to use?
1.
2.
3.
4.
5.
Amount of information needed
Characteristics of stimulus or concept
Application context
Capacity of scale
Post-measurement analysis
Statistics are designed for specific types
of scales. Using the wrong scale will give
answers that are nonsense.
Measurement characteristics:
Measurement characteristics:
Y = x(true) + x(sy-error) + x(random)
Measurement characteristics:
Y = x(true) + x(sy-error) + x(random)
Systematic error can be eliminated.
Measurement characteristics:
Y = x(true) + x(sy-error) + x(random)
Random error cannot be eliminated.
Measurement characteristics:
Y = x(true) + x(sy-error) + x(random)
If a sample is taken to estimate an answer:
another form of error is added……
Measurement characteristics:
This is called a
Sampling Error
Y = x(true) + x(sy-error) + x(random) + x(sampling error)
If you take a sample… you will create a sampling error!
You and a friend (in the same class) take the
same exam at the same time and
get different grades.
WHY?
Take a piece of paper…
write down five different reasons
why these two friends taking the same class
would get different grades...
What then did the grade actually measure?
Write down a definition of a “grade.”
If you suggested that a “grade” is a measurement
of what a student knows, how many “grades” would
you suggest need to be taken in order to be confident that
the student actually knows what the grades indicate
that they know?
Measurement characteristics:
Validity
Before validity can be established, it is necessary to
show that measurements have reliability.
A measurement can be reliable without being
valid, but it cannot be judged to be valid without
reliability.
Measurement characteristics:
Reliability
Measurement characteristics:
Reliability
1. Stability
Measurement characteristics:
Reliability
1. Stability
a. Test-retest
b. Equivalent forms
Measurement characteristics:
Reliability
1. Stability
a. Test-retest
b. Equivalent forms
2. Equivalence
Measurement characteristics:
Reliability
1. Stability
a. Ttest-retest
b. Equivalent forms
2. Equivalence
a. Kuder-Richardson
b. Cronbach’s Alpha
Measurement characteristics:
Reliability
1. Stability
a. Test-retest
b. Equivalent forms
2. Equivalence
a. Kuder-Richardson
b. Cronbach’s Alpha
Lee Cronbach
Measurement characteristics:
Reliability
1. Stability
a. Test-retest
b. Equivalent forms
2. Equivalence
a. Kuder-Richardson
b. Cronbach’s Alpha
Learn, Effective, & Like the instructor
Measurement characteristics:
Reliability
1. Stability
a. Test-retest
b. Equivalent forms
2. Equivalence
a. Kuder-Richardson
b. Cronbach’s Alpha
3. Inter-rater Consistency
a. Krippendorff’s Alpha
Klaus Krippendorff
1932 -
Measurement characteristics:
If a measurement is reliable, it
may be valid:
But there are many ways that
a measurement could be valid or invalid.
Measurement characteristics:
Validity
1. Face validity
Measurement characteristics:
Validity
1. Face
2. Content
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
a. Convergent
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
a. Convergent
b. Divergent
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
a. Convergent
b. Divergent
c. Discriminant
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
a. Convergent
b. Divergent
c. Discriminant
d. Nomological
Measurement characteristics:
Validity
1. Face
2. Content
3. Criteria
a. Concurrent
b. Predictive
4. Construct
5. Utilitarian (?)
A measurement may satisfy a utilitarian goal
independently of any validity of the actual measurement.