Zscores

1
Z-scores

• Standardizing Scores Distributions

2
From now on...
3
What are z-scores?

• z-scores are standardized scores that tell where
  a raw score (X) is located in an entire
  distribution in terms of standard deviation (SD)
  units.

4
What are z-scores?
5
Purpose of z-scores

• Z-scores enable you to
• determine the relative standing of a raw score
  with a distribution.
• compare scores that come from different
  distributions.

6
Purpose of z-scores

• Z-scores enable you to
• standardize a distribution to have a mean of 0
  and a standard deviation of 1, or any mean and
  standard deviation you specify.
• determine probability values associated with a
  range of raw scores.

7
Computing z-scores

• Z-score formula

8
Components of z-score

• 1. Sign tells you if the score is above
• (z gt0), below (z lt 0), or at (z 0) the
• mean.
• 2. Magnitude tells you how far away the
• score is from the mean in standard
• deviation units.

9
Z-Score Transformations Finding location of a
raw score within a distribution
10
Converting z-score to raw score

• Apply z-score formula in reverse

11
Z-score TransformationsComparing Scores from
different distributions

• For which class would you expect a
• higher grade?
• Bio test X 56, µ 48
• Psy test X 60, µ 50

12
Z-score TransformationsComparing Scores from
different distributions

• To compare scores, need to put them on a common
  metric.
• To do this, raw scores within each distribution
  are transformed to z-scores.

13
Z-score TransformationsComparing Scores from
different distributions

• Bio test X56
• If µ 48 and s 4
• Psy test X60
• If µ 50 and s 10

14
Comparing Exam ScoresRole of the mean and
variability
15
Extreme Scores
16
Transforming Distributions of Scores
17
Standardizing a Distribution to Have a Mean of 0
and SD1
18
Transforming Raw ScoresDesignating your own µ
and s

• Sam got a 64 on his achievement test (µ 57, s
  14).
• To make score more digestible, you decide to
  standardize the original distribution to have a µ
  50 and s 10.
• After so doing, what is Sams new raw score?

19
Transforming Raw ScoresDesignating your own µ
and s

• Convert raw score to z score using parameters
  from original distribution
• Calculate new raw score, substituting in the new
  µ and s

20
Transforming Raw ScoresDesignating your own µ
and s

• Given X64, first calculate Sams z-score, using
  the original parameter values µ 57, s 14)
• Second, calculate Sams new raw score on the new
  distribution, substituting in the new parameter
  values
• µ 50 and s 10

21
Transforming Raw Scores to Z-scores

• Transforming a set of raw scores to z-scores
• Does not change the shape of the distribution.
• Does not change the location of individual scores
  within the distribution.