Worked problems for standard scores, percentiles, and stanines

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Worked problems for standard scores, percentiles,
and stanines

• The problems are given in pairs. The first slide
  presents the problem, the second slide gives the
  answer.

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Item 1

• Rosemary and Angela are new 10th-grade students
  to Lincoln High School. They have just moved in
  to the area from different states, Rosemary from
  Virginia and Angela from Texas. Both states have
  statewide standardized testing programs. The
  Lincoln High counselor is trying to place the
  students in appropriate classes. Right now he is
  trying to place the two students in math and
  English.
• He notes that Rosemarys standard scores on her
  previous states math and English tests were .25
  and 1.2, respectively. Angelas scores on her
  previous states math and English tests were .75
  and .95, respectively.
• Which of the following statements is true?
• A. Angela should be placed in higher level math
  and English Classes.
• B. The students should be placed in the same
  level math and English classes.
• C. Both should be placed in higher than average
  level classes in math and English.
• D. Not enough information is given to determine
  placements.

3

• The answer is D Not enough information is given.
• The two girls standardized test scores were
  obtained from different norm groups and are,
  hence, not comparable. In order to be able to
  compare two individuals norm-referenced scores
  (standard scores, percentiles, grade equivalent
  scores, etc.) the scores have to be compared to
  the same norm group. Rosemarys scores are from a
  test normed in Virginia Angelas from a test
  normed in Texas. It is possible that an
  above-average performance on, say, a math test in
  Texas would be equivalent to a below-average
  performance on a similar test in Virginia.

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Item 2

• Recall that Rosemarys standard scores on her
  previous states math and English tests were .25
  and 1.2, respectively, and that Angelas scores
  on her previous states math and English tests
  were .75 and .95, respectively.
• Which of the following interpretations can most
  easily be supported?
• Rosemary and Angela both perform better in
  English than in Math.
• Only Rosemary is clearly better at English than
  at Math.
• C. While Rosemary is clearly better than Angela
  in English, Angela is better in math.

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• The correct answer is B Only Rosemary is clearly
  better at English than at math.
• The difference between Rosemarys standard scores
  in math and English (.25 and 1.2) correspond,
  roughly, to a difference in percentiles of 51
  tile for math and the 90th tile for English.
  The difference in Angelas standard scores for
  math and English, on the other hand, correspond,
  again roughly, to a difference between the the
  79th tile and the 83ed tile.
• Choice C is incorrect for reasons given for the
  previous item. The girls test scores come from
  test normed on different populations and, hence,
  are not comparable.

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Item 3

• Tom, Dick, and Harry reported their scores on the
  same science test to their mothers. Each reported
  his score as 70.
• In Harrys case, his score was actually 70 of
  the items correct.
• Toms score really referred to the 70th
  percentile.
• Dick was actually reporting the number of items
  he got correct.
• The test, which contained 75 items, had a mean of
  60 and a standard deviation of 10. Who, if any,
  had the best performance on the test?
• A. Tom C. Harry
• B. Dick D. Cant tell

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• The correct answer is B. Dick, who scored 70 of
  75 items correct did best on the test. His score
  is clearly better than Harrys. Harry, who got
  70 of the items correct only got 52 or 53 items
  correct.
• The only thing we have to check is whether Dick
  scored above or below Toms 70th percentile. To
  check this we first have to calculate his
  standard score. A simple calculation reveals that
  Dicks standard score is 1. He scored one
  standard deviations above the mean. Since one
  standard deviation above the mean equates to the
  84th percentile, Dick clearly outscored Tom.

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Item 4

• Picabo came in at a speed of 100 mph on the
  downhill. Tommy, on a bad day, came in at the
  same speed. The average female speed on the
  downhill is 80 mph with a SD of 5. The average
  male speed on the downhill is 110 with a SD of
  10. Just how much better than Tommy is Picabo?

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• Tommy is coming in a the 16th percentile. He
  needs to stop that partying.
• Picabo is at the 99percentile. She needs to
  watch those steroids.
• Picabo is definitely better than Tommy. Whoa!
  Actually, they are the samethey both skied at
  the same speed, 100 mph.
• Only when compared to the norms for their gender
  do we see a difference in their performance.
• In this case, however, there is an absolute
  standard of comparison. This is a case of
  criterion-referenced measurement.