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-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 variableX= the mean of X

sx= the standard deviation of X

Zx =¯

¯

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

Comparing variables with very different observed units of measure

Example of comparing an SAT score to an ACT score Mary’s ACT score is 26. Jason’s 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.

Let’s find the z-scores

Jason: 900-1000 100Mary: 26-22

2 From these findings, we gather that Jason’s

score is 1 standard deviation below the mean SAT score and Mary’s score is 2 standard deviations above the mean ACT score.

Therefore, Mary’s score is relatively better.

Zx =

Zx =

=

=

-1

+2

Z-scores and the normal curve

SD SDSD SDSD SDSD SD SD

68%95%99%

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.

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

GPA

SD SD SDSD SDSD SD

68%95%99%

3.02.8 3.63.22.6 3.42.4

LSAT Scores

SD SD SDSD SDSD SD

68%95%99%

155148 176162141 169134

What we’ve learned The more positive a z-score is, the more

competitive the applicant’s 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 population’s scores will fall.

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 population’s scores fall within a score range in a normal distribution.

Works Cited Ritchey, Ferris. The Statistical

Imagination. New York: McGraw-Hill, 2000.

Tushar Mehta Excel Page. <http://www.tushar-mehta.com/excel/charts/normal_dist ribution/>