Scales, Transformations,

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• Scales, Transformations, Norms

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• Norms
• Norm-Referenced Test one of the most useful
  ways of describing a persons performance on a
  test is to compare his/her test score to the test
  scores of some other persons or group of people.
• Norms are average scores computed for a large
  representative sample of the population.
• The arithmetic average (mean) is used to judge
  whether a score on the scale above or below the
  average relative to the population of interest.
• a representative sample is required to ensure
  meaningful comparisons are made.

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• Norms (cont.)
• No single population can be regarded as the
  normative group.
• a representative sample is required to ensure
  meaningful comparisons are made.
• When norms are collect from the test performance
  of groups of people these reference groups are
  labeled normative or standardized samples.

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Norms (cont.)

• The normative sample selected as the normative
  group, depends on the research question in
  particular.
• It is necessary that the normative sample
  selected be representative of the examinee and of
  the research question to be answered, in order
  for meaningful comparisons to be made.
• For example tests measuring attitudes towards
  federalism having norm groups consisting of only
  students in the province of Quebec might be very
  useful for interpretation regionally in Quebec,
  however their generalizability in other parts of
  the country (Yukon, Toronto, Ontario) would be
  suspect.

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Sample Groups
Although the three terms below are used
interchangeably, they are different.
Standardized Sample - is the group of individuals
on whom the test is standardized in terms of
scoring procedures, administration procedures,
and developing the tests norms. (e.g., sample
used in technical manual)
Normative Sample - can refer to any group from
which norms are gathered. Norms collected after
test is published
Reference Group - any group of people against
which test scores are compared. (e.g., a
designated group such as students in 3090.03 or
World Champions)
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• Types of Norms
• Norms can be Developed
• Locally
• Regionally
• Nationally
• Normative Data Can be Expressed By
• Percentile Ranks
• Age Norms
• Grade Norms

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• Local Norms
• Test users may wish to evaluate scores on the
  basis of reference groups drawn from specific
  geographic or institutional setting.
• For Example
• Norms can be created employees of a particular
  company or the students of a certain university.
• Regional National norms examine much broader
  groups.

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• Subgroup Norms
• When large samples are gathered to represent
  broadly defined populations, norms can be
  reported in aggregate or can be separated into
  subgroup norms.
• Provided that subgroups are of sufficient size
  and fairly representative of their categories,
  they can be formed in terms of
• - Age
• - Sex
• - Occupation
• - Education Level
• Or any other variable that may have a significant
  impact on test scores or yield comparisons of
  interest.
• .

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• Percentile Ranks
• The most common form of norms and is the
  simplest method of presenting test data for
  comparative purposes.
• The percentile rank represents the percentage of
  the norm group that earned a raw score less than
  or equal to the score of that particular
  individual.
• For example, a score at the 50th percentile
  indicates that the individual did as well or
  better on the test than 50 of the norm group.
• When a test score is compared to several
  different norm groups, percentile ranks may
  change.
• For example, a percentile rank on a mathematical
  reasoning test may be lower when comparing it to
  math grade students, than music students.

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• Age Norms
• Method of describing scores in terms of the
  average or typical age of the respondents
  achieving a specific test score.
• Age norms can be developed for any
  characteristic that changes systematically with
  age.
• In establishing age norms, we need to obtain a
  representative sample at each of several ages and
  measure the particular age related characteristic
  in each of these samples.
• It is important to remember that there is
  considerable variability within the same age,
  which means that some children at one age will
  perform similar to children at other ages.

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• Grade Norms
• Most commonly used in school settings.

• Similar to age norms except the baseline is
  grade level rather than age.

• It is important to remember that there is
  considerable variability within individuals of
  different grade, which means that some children
  in one grade will perform similar to or below
  children in other grades.

• One needs to be extremely careful when
  interpreting grade norms not to fall into the
  trap of saying that, just because a child obtains
  a certain grade-equivalent on a particular test,
  he/she is the same grade in all areas.

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Evaluating Suitability of a Normative Sample

• How large is the normative sample?
• When was the sample gathered?
• Where was the sample gathered?
• How were individuals identified and selected?

• What was the composition of the normative
  sample?
• - age, sex, ethnicity, education level,
  socioeconomic status

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Caution When Interpreting Norms

• Norms are not based on samples that adequately
  represent the type of population to which the
  examinees scores are compared.

• Normative data can become outdated very quickly.

• The size of the sample taken.

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Setting Standards/Cutoffs

• Rather than finding out how you stand compared
  to others, it might be useful to compare your
  performance on a test to some external standard.

For Example - if most people in class get an F on
a test and you get a D, your performance in
comparison to the normative group is good.
However, overall your score is not good.
Criterion-Referenced Tests - assesses your
performance against some set of standards. (e.g.,
school tests, Olympics)
Cutoff Scores - 1 SD?, 2 SD?
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Raw Scores Raw scores are computed for
instruments using Likert scales (interval or
ordinal) by assigning scores to responses and
totaling the scores of the items. - For
positively phrase items, e.g., I think things
will turn out right 5Always, 4Often,
3Sometimes, 2Seldom, 1Never - For positively
phrase items, e.g., I think things will turn
out right 1Always, 2Often, 3Sometimes,
4Seldom, 5Never The raw score would be the
sum of the scores for pertinent items. The
problem with raw scores are that they are fairly
meaningless without some sort of benchmark with
which to make a comparison (e.g., What would a
raw score of 30 on an Optimism scale mean?)
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• Transformations
• Raw scores (i.e., simplest counts of behaviour
  sampled by a measuring procedure) do not always
  provide useful information.
• It is often necessary to reexpress, or transform
  raw scores into some more informative scale.
• The simplest form of transformation is changing
  raw scores to percentages.
• For Example
• If a student answers 35 questions out of 50
  correctly on a test, that students score could
  be reexpressed as a score of 70.

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• Linear Transformations
• Changes the units of measurement, while leaving
  the interrelationship unaltered.
• An advantage of this procedure is that the
  normally distributed scores of tests with
  different means and score ranges can be
  meaningfully compared and averaged.
• Most familiar linear transformation is the z
  score.

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Standard Scores Standard scores allow each
obtained score to be compared to the same
reference value. In order to facilitate
comparison between obtained scores and the scores
of other individuals (i.e., the normative
sample), as well as comparison among the various
scales and instruments. Standard scores are
calculated from raw scores such that each scale
and subscale will have the same mean (or average)
score and standard deviation. For example, IQ
scores are transformed so that the average score
is 100, with a SD of 15.
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• Z Scores
• A z-score tells how many standard deviations
  someone is above or below the mean. Simply put,
  the mean of the distribution is given the z value
  of zero (0) and is standard deviation is counted
  by ones.
• A z-score of -1.4 indicates that someone is 1.4
  standard deviations below the mean. Someone who
  is in that position would have done as well or
  better than 8 of the students who took the test.
  
• To calculate a z-score, subtract the mean from
  the raw score and divide that answer by the
  standard deviation. (i.e., raw score 15, mean
  10, standard deviation 4. Therefore 15 minus 10
  equals 5. 5 divided by 4 equals 1.25. Thus the
  z-score is 1.25.)
• Z scores have negative values, which can be
  difficult to interpret to test users. How can you
  explain an examinee that his z score is -1.5? For
  this reason it is often convenient to perform a
  linear transformation on z-scores to convert them
  to values that are easier to record or explain.
  The general form of such a transformation is

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T Scores

• T-Scores (or standardized scores) are a
  conversion (transformation) of raw individual
  scores into a standard form, where the conversion
  is made without knowledge of the population's
  mean and standard deviation.
• The scale has a mean set at 50 and a standard
  deviation at 10.
• T 50 l0 x z score
• An advantage of using a T-Scores is that none of
  the scores are negative.

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• Area Transformations
• Area transformations do more than simply put
  scores on a new and more convenient scale -- it
  changes the point of reference.
• Area transformations adjust the mean and
  standard deviation of the distribution into
  convenient units.
• Advantages of area transformations are obvious.
  Out of the infinite number of possible empirical
  distributions of test scores, the normal
  distribution is most frequently assumed and
  approximated. It is also most frequently studied,
  in considerably greater detail than other
  possible test score distributions.
• Normalization thus allows the application of
  knowledge concerning properties of standard
  normal distribution toward the interpretation of
  the obtained scores.

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• Normal Distribution Curve
• Many human variables fall on a normal or close
  to normal curve including IQ, height, weight,
  lifespan, and shoe size.
• Theoretically, the normal curve is bell shaped
  with the highest point at its center. The curve
  is perfectly symmetrical, with no skewness (i.e.,
  where symmetry is absent). If you fold it in half
  at the mean, both sides are exactly the same.
• From the center, the curve tapers on both sides
  approaching the X axis. However, it never touches
  the X axis. In theory, the distribution of the
  normal curve ranges from negative infinity to
  positive infinity.
• Because of this, we can estimate how many people
  will compare on specific variables. This is done
  by knowing the mean and standard deviation.

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Normal Distribution The bell-shaped curve has
the following properties 1. bilaterally
symmetrical (right and left halves are mirror
images) 3. the limits of the curve are plus and
minus infinity, so the tails of the curve will
never quite touch the baseline4. about 68 of
the total area of the curve lies between one
standard deviation below the mean and one
standard deviation above the mean 5. about 95 of
the total area of the curve lies between two
standard deviations below the mean and two
standard deviations above the mean 6. about
99.8 of the total area of the curve lies between
three standard deviations below the mean and
three standard deviations above the mean.
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• Skewness
• Skewness is the nature and extent to which
  symmetry is absent.
• Positive Skewness - when relatively few of the
  scores fall at the high end of the distribution.
• For Example - positively skewed examination
  results may indicate that a test was too
  difficult.
• Negative Skewness - when relatively few of the
  scores fall at the low end of the distribution.
• For Example - negatively skewed examination
  results may indicate that a test was too easy.

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Standard Deviations The standard deviation
represents the average distance each score is
from the mean. Use of Standard Deviations with
Norms Knowing the average of a population
allows for a determination as to whether a
particular respondent scored above or below that
average, but does not indicate how much above or
below average the score falls. Standard Deviation
plays a role in this. Scores within 1 SD of
average are pretty much in the middle cluster of
the population. Scores between 1 2 SDs from the
average are moderately above or below the average
, and scores 2 SDs from the average are markedly
for above or below the average.