Scaling and Zscores

1
Scaling and Z-scores
2
What are scaled scores?

• Scores that are adjusted through applying
• the same constant to all scores in the
  distribution.
• The constant is either added, subtracted,
  multiplied, or divided to all scores. The mean
  and standard deviation may change, but their
  relative position remains the same.

3
Why are scaled scores needed?

• There are many reasons to scale scores.
• For example
• Tests may be unusually difficult or easy and
  need to be adjusted for a comparison to another
  test.
• Tests may be measured on different scales and
  need to have a common scale.
• Data can be transformed into something much more
  meaningful

4

• Raw scores are often meaningless without more
  info
• E.g., SAT and ACT
• E.g., The Big Lebowski received a rating of 8.1
  out of 10 from IMDB.com
• Shawshank Redemption 9.1
• E.g., The Big Lebowski received an 84 rating on
  Rottentomatoes.com
• Must know the values of the scale
• What is it measuring

5

• All scores are relative
• In psychology, the distribution of the scores and
  relative rank are more important than the raw
  data.
• Must be able to compare individual scores to some
  standard
• Compare to others scores
• Often, the data must be transformed to make sense
  of it, or to adjust the scores as needed.

6
Adding or Subtracting a Constant

• Adding or subtracting a constant changes the mean
  the amount of the constant and the standard
  deviation remains the same.

Original Score Constant
New Score 5 5 10 4 5 9
3 5 8 Sum 12

Sum 27 Mean 4 5
Mean 9

7

• Shifts the distribution, but the distribution
  itself remains the same.

?
8
Multiplying or Dividing a Constant

• Multiplying or dividing a constant changes the
  mean and the standard deviation both. The spread
  of the scores change.
• Can use more elaborate transforms
• Common in experiments to construct variables that
  are transformations of raw data
• E.g., statistical curve
• E.g., Stroop experiment
• Size of Stroop effect
• Size incongruent words congruent words /
  congruent words

9
Example
Original Score Constant
New Score 5 x 5 25 4 x 5
20 3 x
5 15 Sum 12
Sum 60 Mean 4
x 5 Mean 20
S 1 x 5 S 5

Spread changes
10
Standard Scores (z Scores)

• Transforms scores to a distribution that has a
  mean of 0 and standard deviation of 1. (z
  distribution)

Use the transformation rules Subtract the
constant (the mean of the distribution. Divide
the constant (the standard deviation of the
distribution).
11
Compute a z Score
Example The original distribution has a mean
of 10 and standard deviation of 2. Raw score
20.
12

• Key Z-scores are transformed into standard
  deviation units.
• E.g., z 1 ? 1 standard deviation above the
  mean.
• Z-scores give tremendous amounts of information
• The higher the score, the further away from the
  mean
• The sign represents the position relative to the
  mean
• z ? greater than the mean
• -z ? less than the mean

13
Example
History Exam Prof A Mean 70 Standard
Deviation 10
History Exam Prof B Mean 65 Standard
Deviation 5
Mary was in Prof As class and her score was 80
on the exam. Minnie was in Prof Bs class and
her score was 75 on the exam. Did Mary or
Minnie do better on the exam?
14
Compute a z score to compare.

• Mary

• Minnie

By converting both scores to a distribution with
a mean of zero and a standard deviation of 1, the
scores can be compared. Result
Minnie had a higher score than Mary.
15
IMDB information

• Mean 6.2
• Median 6.4
• Standard deviation 1.4
• Big Lebowski scored an 8.1
• What is the Z-score?

16
(No Transcript)
17
Example

• Calculate z scores on an IQ test if the mean is
  100 and the standard deviation is 15
• Brian 130
• Jan 72
• Jim 100

18

• Converting z-scores to raw scores
• Transformation can be reversed.
• X ZS M
• On an IQ test if the mean is 100 and the standard
  deviation is 15
• Jody z 3
• Jeremiah z 0
• Zach z - 2.5

19

• Z-scores allow you to compare apples to oranges.
  Puts two or more different scales on the same
  metric.
• Comparison between tests
• Compare z-scores
• Ex. Jerry makes a 1200 on the SAT, Terry makes a
  30 on the ACT. Who scored better
• SAT mean1000, sd 150
• ACT mean 21, sd 3

20

• In 1960, the mean baseball salary was 50,000
  with a standard deviation of 10,000. Today, the
  mean salary is 2,000,000, with a standard
  deviation of 500,000.  In 1960, Clete Boyer, the
  third baseman for the New York Yankees, made
  30,000.  What would he earn at today's salaries?