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Standardized Scores (Z-Scores)

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Standardized Scores (Z-Scores) By: David Ruff Z-Score Defined The number of standard deviations a raw score (individual score) deviates from the mean Computing Z ... – PowerPoint PPT presentation

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Title: Standardized Scores (Z-Scores)


1
Standardized Scores(Z-Scores)
  • By David Ruff

2
Z-Score Defined
  • The number of standard deviations a raw score
    (individual score) deviates from the mean

3
Computing Z-Score
  • X - X
  • sx
  • where
  • Zx standardized score for a value of X
  • number of standard deviations a raw score
    (X-score)
  • deviates from the mean
  • X an interval/ratio variable
  • X the mean of X
  • sx the standard deviation of X

Zx

4
Direction of a Z-score
  • The sign of any Z-score indicates the direction
    of a score whether that observation fell above
    the mean (the positive direction) or below the
    mean (the negative direction)
  • If a raw score is below the mean, the z-score
    will be negative, and vice versa

5
Comparing variables with very different observed
units of measure
  • Example of comparing an SAT score to an ACT score
  • Marys ACT score is 26. Jasons SAT score is
    900. Who did better?
  • The mean SAT score is 1000 with a standard
    deviation of 100 SAT points. The mean ACT score
    is 22 with a standard deviation of 2 ACT points.

6
Lets find the z-scores
  • Jason 900-1000
  • 100
  • Mary 26-22
  • 2
  • From these findings, we gather that Jasons score
    is 1 standard deviation below the mean SAT score
    and Marys score is 2 standard deviations above
    the mean ACT score.
  • Therefore, Marys score is relatively better.

Zx

-1

Zx
2
7
Z-scores and the normal curve

SD
SD
SD
SD
SD
SD
SD
SD
SD
68
95
99
8
Interpreting the graph
  • For any normally distributed variable
  • 50 of the scores fall above the mean and 50
    fall below.
  • Approximately 68 of the scores fall within plus
    and minus 1 Z-score from the mean.
  • Approximately 95 of the scores fall within plus
    and minus 2 Z-scores from the mean.
  • 99.7 of the scores fall within plus and minus 3
    Z-scores from the mean.

9
Example
  • Suppose a student is applying to various law
    schools and wishes to gain an idea of what his
    GPA and LSAT scores will need to be in order to
    be admitted.
  • Assume the scores are normally distributed
  • The mean GPA is a 3.0 with a standard deviation
    of .2
  • The mean LSAT score is a 155 with a standard
    deviation of 7

10
GPA

SD
SD
SD
SD
SD
SD
SD
3.0
2.8
3.6
3.2
2.6
3.4
2.4
68
95
99
11
LSAT Scores

SD
SD
SD
SD
SD
SD
SD
155
148
176
162
141
169
134
68
95
99
12
What weve learned
  • The more positive a z-score is, the more
    competitive the applicants scores are.
  • The top 16 for GPA is from a 3.2 upwards for
    LSAT score, from 162 upwards.
  • The top 2.5 for GPA from a 3.2 upwards for LSAT
    score, from 169 upwards.
  • An LSAT score of 176 falls within the top 1, as
    does a GPA of 3.6.
  • Lesson the z-score is a great tool for analyzing
    the range within which a certain percentage of a
    populations scores will fall.

13
Conclusions
  • Z-score is defined as the number of standard
    deviations from the mean.
  • Z-score is useful in comparing variables with
    very different observed units of measure.
  • Z-score allows for precise predictions to be made
    of how many of a populations scores fall within
    a score range in a normal distribution.

14
Works Cited
  • Ritchey, Ferris. The Statistical
  • Imagination. New York McGraw- Hill, 2000.
  • Tushar Mehta Excel Page. lthttp//www.tushar- meht
    a.com/excel/charts/normal_dist ribution/gt
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