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
Zx = 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
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 zscore 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:
Zx = 900-1000 = -1
100
Mary:
Zx = 26-22 = +2
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.
Z-scores and the normal curve

SD
SD
SD
SD
SD
68%
95%
99%
SD
SD
SD
SD
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
2.4
SD
2.6
SD
2.8
SD
3.0
68%
95%
99%
SD
3.2
SD
3.4
SD
3.6
LSAT Scores

SD
134
SD
141
SD
148
SD
155
68%
95%
99%
SD
162
SD
169
SD
176
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: McGrawHill, 2000.
Tushar Mehta Excel Page.
<http://www.tusharmehta.com/excel/charts/normal_dist
ribution/>