Test Norms and Basic Statistics

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Chapter 2

• Test Norms and Basic Statistics

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Types of Scores

• Raw score immediate result of an individuals
  responses to a test
• Normed score result of comparing an
  individuals raw score to those of individual in
  a norm group. Also called derived score or scale
  scores.
• Test norms are based on the elementary notions
  from descriptive statistics, so we will do a
  quick review.

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Why we need statistics

• To describe a set of data (descriptive
  statistics)
• To make inferences (inferential statistics)

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Levels of Variables

• Define construct Most general level. Give
  verbal descriptions and definitions of a
  variable.
• Measure variable this is the operational
  definition of the variable.
• Get raw data This is the result of the
  application of the measures

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Types of Scales

• Nominal (Names)
• Ordinal (Ranks /Order)
• Interval (Equal intervals)
• Ratio (Absolute zero, can form ratios)

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Organizing Raw Data

• Frequency distributions can be grouped or
  ungrouped
• X axis individual scores or intervals
• Y axis frequency

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Percentiles and Percentile Ranks

• Percentile is a point on a scale below which a
  specified percentage of the cases fall. Start
  with a given percentage, then find the raw score
  corresponding to this point. (e.g., what score
  corresponds to the 50th percentile?)
• Percentile rank tells the percentage of cases in
  the norm group falling below a given raw score.
  Start with the score, then find the percentage of
  cases falling below it. (e.g., what percentage
  of the class scored less than 75 on the test?)

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Calculating a percentile rank or percentile

• Pr (B/N) x 100
• Where
• Prpercentile rank
• Bnumber of scores below score of interest
• Ntotal number of scores
• Xscore of interest
• Remember to arrange data so best is on the top
  and worst is on the bottom

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Strengths and Weaknesses of Percentiles and
Percentile Ranks

• Strengths
• Easy to understand
• Easy to calculate

• Weaknesses
• Confused with percentage right score
• Inequality of units at various points on the
  scale. (Percentile ranks are bunched up in the
  middle of the distribution.)

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Measures of Central Tendency

• Mean
• Median 50th percentile, middle score when the
  scores are arranged from low to high
• Mode Most frequent score

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Measures of Variability

• Range distance from the lowest to the highest
  score
• Standard deviation average deviation around the
  mean.  

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Definitional formula for calculating the standard
deviationfor a population
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Computational Formula for Calculating the
Standard Deviation for a Sample
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Z-Scores
Mean of 0 Standard Deviation of 1 Tells how
far the score is from the mean and in what
direction
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Standard Normal Deviation

• Uses the normal distribution, i.e., a symmetrical
  binomial probability distribution
• Refer to the units on the X axis in Z score units
• The numbers under the normal curve are the
  proportions of cases we would expect to observe
  in each area.
• Proportions which begin at the bottom of the
  normal curve to the point of interest are
  percentiles or percentile ranks when multiplied
  by 100

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McCalls T scores

• Transformed z scores to have a mean of 50 and SD
  of 10.
• T10Z 50
• Used by MMPI and Strong Interest Inventory.

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Transforming z-scores to other standard scores
Once Z-scores have been obtained, they can be
transformed into any other set of scores with
the desired mean and standard deviation
New Score Z(new std. dev.) new mean
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Other standard scores

• Quartiles (distribution divided into 4 parts)
• Deciles (distribution divided into 10 parts)
• Stanines (distribution divided into 9 parts)
• They are non-linear transformations since they
  are derived by reference to percentile divisions

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Norms

• Norms refer to the performance of defined groups
  on particular tests.
• Norms are used to give information about
  performance relative to what has been observed in
  a standardization sample.
• Types of norms
• Percentile ranks
• Standard scores
• Developmental (age-related) Norms

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Developmental Norms

• Use either age equivalents or grade equivalents
• Are only meaningful in the range where the trait
  being measured is developing or growing with time
  in the relevant population
• MA (mental age) are determined by finding the
  typical or median score for examinees at
  successive age levels.
• Grade equivalents are developed by administering
  a test to students in different grade levels.
  The average or median performance at each grade
  are plotted and a curve fitted.

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Strengths and Weaknesses of Developmental Norms

• Strengths
• Easily understood, naturalness to their meaning.
• Provide basis for measuring growth across
  multi-level tests

• Weaknesses
• Applicable only to variables that show
  developmental pattern.
• Even variables that are developmental dont
  continue that way indefinitely
• SD not usually the same on different levels
  tests.
• same score may be obtained different ways,

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Issues in Norms Development

• Stability (so need large N) Look also at the
  size of the norm group for each population it
  represents (e.g., there may be a norm tables for
  5th grade girl, but at that level theres only 50
  in the norm group)
• Representativeness
• Separate norms for separate groups
• Tracking tendency to stay at about the same
  level relative to ones peers. (e.g., growth
  charts). Issue of children falling behind and
  the implications of that

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Norm- vs. Criterion-referenced Tests

• Norm referenced test compares each person with a
  norm.
• Criterion referenced test describes the specific
  things the test-taker can demonstrate.