Standardized Scores ZScores

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Standardized Scores(Z-Scores)

• By David Ruff

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Z-Score Defined

• The number of standard deviations a raw score
  (individual score) deviates from the mean

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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

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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

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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.

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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
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Z-scores and the normal curve

SD
SD
SD
SD
SD
SD
SD
SD
SD
68
95
99
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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.

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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

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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
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LSAT Scores

SD
SD
SD
SD
SD
SD
SD
155
148
176
162
141
169
134
68
95
99
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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.

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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.

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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