Transcript Chapter 6, part I: Foundations of Educational Measurement

Chapter 6, part I:
Educational
Measurement
EDUC 502
October 10, 2005
Definition of terms
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Measurement: assignment of numbers to
differentiate values of a variable
Evaluation: procedures for collecting
information and using it to make decisions for
which some value is placed on the results
Assessment - multiple meanings
Measurement of a variable
 Evaluation
 Diagnosis of individual difficulties
 Procedures to gather information on student
performance (formative)
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Purpose of measurement for
research
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Obtain information about the variables being
studied
Provide a standard format for recording
observations, performances, or other responses
of subjects
Provide for a quantitative summary of the
results from many subjects
Measurement scales
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Nominal - categories
Race
 Gender
 Types of schools (e.g., public, private, parochial)
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Ordinal - ordered categories, but the degree of
difference between the categories is not
specified.
Finishing position in a race
 Ranks in the military
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Measurement scales
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Interval - equal intervals between numbers on
the scale – one score can be compared directly
to another in terms of the amount of
difference.
Classroom Test scores
 Some Achievement Test Levels
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Ratio - equal intervals and an absolute zero (0)
Height
 Weight
 Time
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Measurement scales
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The line between interval and ordinal scales is
not always clear-cut.
Example: Is a Likert scale that runs from 1-5 an
example of an interval scale or an ordinal scale?
The difference matters here, because some will
argue that it makes sense to calculate an average
value on a Likert scale, while others argue that it
does not.
Descriptive statistics
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Definition of terms
Statistics: procedures that summarize and analyze
quantitative data
 Descriptive statistics: statistical procedures that
summarize a set of numbers in terms of central
tendency, variation, or relationships
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Types of descriptive statistics
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Frequency distributions: an organization of the
data set indicating the number of times (i.e.,
frequency) each score was present
Types of presentations
Frequency table
 Frequency polygon
 Histogram
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Example: Scores on a test: 20, 20, 30, 40, 50, 60,
60, 70, 70, 70, 70, 90, 90, 100.
Shapes of distributions
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Symmetric - a set of scores that are equally
distributed around a middle score.
Positively skewed - a set of scores
characterized by a large number of low scores
and a small number of high scores.
Negatively skewed - a set of scores
characterized by a large number of high scores
and a small number of low scores.
See Figure 6.2 in Chapter 6 of the text.
Central tendency - what is the typical
score
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Mode: the most frequently occurring score
Median: the score above and below which onehalf of the scores occur
Mean
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The arithmetic average of all scores
Statistical properties make it very useful
Concerns related to outlying scores
Determine each of these for the earlier data
set
Excerpt from “How to Lie with
Statistics”
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In a classic book entitled "How to Lie with
Statistics," George Huff makes the point that
individuals will often choose the average (mean,
median, or mode) that best supports their
argument. Here is an excerpt from the beginning
of his chapter that is entitled "The Well-Chosen
Average". Keep in mind this book was written in
1955, when it was a big deal to make $15,000 a
year:
Excerpt from “How to Lie with
Statistics”
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"You, I trust, are not a snob, and I certainly am not in
the real-estate business. But let's say that you are and I
am and that you are looking for property to buy along a
road that is not far from the California valley in which I
live.
Having sized you up, I take pains to tell you that the
average income in this neighborhood is some $15,000 a
year. Maybe that clinches your interest in living here;
anyway, you buy and that handsome figure sticks in
your mind. More than likely, since we have agreed that
for the purposes of the moment you are a bit of a
snob, you toss it in casually when telling your friends
about where you live.
Excerpt from “How to Lie with
Statistics”
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A year or so later we meet again. As a member of some
taxpayer's committee I am circulating a petition to keep the tax
rate down or assessments down or bus fare down. My plea is that
we cannot afford the increase: After all, the average income in
this neighborhood is only $3,500 a year. Perhaps you go along
with me and my committee in this-you're not only a snob, you're
stingy too-but you can't help being surprised to hear that measly
$3,500. Am I lying now, or was I lying last year?
You can't pin it on me either time. That is the essential
beauty of lying with statistics. Both those figures are legitimate
averages, legally arrived at. Both represent the same data, the
same people, the same incomes. All the same it is obvious that at
least one of them must be so misleading as to rival an out-andout lie.
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Excerpt from “How to Lie with
Statistics”
My trick was to use a different kind of average each time, the
word "average" having a very loose meaning. It is a trick
commonly used, sometimes in innocence but often in guilt, by
fellows wishing to influence public opinion or sell advertising
space. When you are told that something is an average you still
don't know very much about it unless you can find out which of
the common kinds of average it is - mean, median, or mode.
The $15,000 figure I used when I wanted a big one is a
mean, the arithmetic average of the incomes of all the families in
the neighborhood. You get it by adding up all the incomes and
dividing by the number there are. The smaller figure is the
median, and so it tells you that half the families in question have
more than $3,500 a year and half have less. I might also have
used the mode, which is the most frequently met-with figure in a
series. If in this neighborhood there are more families with
incomes of $5,000 a year than with any other amount, $5,000 a
year is the modal income" (pp. 27-29)
Application
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Problem: Seven 100 point tests were given
during the Fall Semester. Erika’s scores on the
tests were: 76, 82, 82, 79, 85, 25, 83. If her grade
for the semester is based completely on these
tests, what grade should she receive?
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Moral of the story: Quantitative claims based on
measures are never objective, although they
often masquerade as such.
n
(X
i 1
i
 X )2
N
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Variability - how different are the
scores
Range: the difference
between the highest
and lowest scores
Standard deviation
(SD):
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The average distance of
the scores from the
mean
Formula for calculating
the SD of a population:
n
(X
i 1
i
 X)
N
2
Variability Measure Exercises
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Exercise 1: Calculate the range and standard
deviation of Erika’s test scores. Which measure
more accurately describes the variation in her
scores?
Exercise 2: Suppose the instructor decided to
add 5 points to each of Erika’s test scores out of
the kindness of his heart. How would this
impact the standard deviation? The range? Why?
Homework Exercises
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Textbook p. 147 (4, 5, 7, 9, 10)