Chapter 6, part I: Educational Measurement

1
Chapter 6, part I Educational Measurement

• EDUC 502
• October 10, 2005

2
Definition of terms

• 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)

3
Purpose of measurement for research

• 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

4
Measurement scales

• Nominal - categories
• Race
• Gender
• Types of schools (e.g., public, private,
  parochial)
• Ordinal - ordered categories, but the degree of
  difference between the categories is not
  specified.
• Finishing position in a race
• Ranks in the military

5
Measurement scales

• 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
• Ratio - equal intervals and an absolute zero (0)
• Height
• Weight
• Time

6
Measurement scales

• 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.

7
Descriptive statistics

• 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

8
Types of descriptive statistics

• 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
• Example Scores on a test 20, 20, 30, 40, 50,
  60, 60, 70, 70, 70, 70, 90, 90, 100.

9
Shapes of distributions

• 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.

10
Central tendency - what is the typical score

• Mode the most frequently occurring score
• Median the score above and below which one-half
  of the scores occur
• Mean
• 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

11
Excerpt from How to Lie with Statistics

• 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

12
Excerpt from How to Lie with Statistics

• "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.

13
Excerpt from How to Lie with Statistics

• 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-and-out lie.

14
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)

15
Application

• Problem Seven 100 point tests were given during
  the Fall Semester. Erikas 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?
• Moral of the story Quantitative claims based on
  measures are never objective, although they often
  masquerade as such.

16
Variability - how different are the scores

• Range the difference between the highest and
  lowest scores
• Standard deviation (SD)
• The average distance of the scores from the mean
• Formula for calculating the SD of a population

17
Variability Measure Exercises

• Exercise 1 Calculate the range and standard
  deviation of Erikas 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 Erikas test scores out of
  the kindness of his heart. How would this impact
  the standard deviation? The range? Why?

18
Homework Exercises

• Textbook p. 147 (4, 5, 7, 9, 10)