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Zscores and the Empirical Rule
For moundshaped data, we can apply Zscores to the Empirical Rule to help us create guidelines for
identifying “unusual” observations.
For a given data set, the mean Zscore is 0, while the standard deviation is 1. If the data is mound
shaped, then:
•
68% of our observations will have Zscores between 1 and 1
•
95% of our observations will have Zscores between 2 and 2
•
99% of our observations will have Zscores between 3 and 3
EXAMPLE: Return to our GMAT scores (
μ
= 663 points,
σ
= 37 points). For the student who had a
GMAT score of 719 points, is this an exceptionally high score?
What about a student who had a GMAT test score of 800 points?
If a student had a test score of 600 points, would this be considered a really bad score?
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Calculating a Zscore determines how many standard deviations any given observation is away from it’s
mean. Since Zscores are unitless, we can use them to compare observations from two data sets.
EXAMPLE: Students who want to get into medical school are required to write the Medical College
Admissions Test (MCAT). According to an October 2016 article in Maclean’s magazine, the average
MCAT score for students entering University of Toronto’s medical program in fall 2015 was 11.03 points.
Let’s assume MCAT scores have a mound shape distribution, with a standard deviation of 0.75 points.
()
Which student performed better on their respective test: the student who wrote the GMAT and ob
tained a score of 719, or a student who wrote the MCAT and obtained a score of 12.53?
PERCENTILES:
describe the location of an observation relative to the other observations in the data
set, when
the data is listed in ascending order
.
•
•
EXAMPLES: A student who scores 736 points on the GMAT is in the 95
th
percentile of all students
who wrote the GMAT.
Quartiles
and
deciles
are specific percentile values. Quartiles divide the data into 4 quarters, while
deciles divide the data into 10 sections:
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Interquartile Range:
IQR
=
Q
3

Q
1
.
There is no single defined method for calculating a percentile. Di
↵
erent procedures may result in slightly
di
↵
erent values for the percentiles.
One way to calculate a percentile/quartile value is:
To find the
p
th
percentile:
•
Take
n
⇤
p
100
•
If the number calculates to a
decimal
value, round up.
The
p
th
percentile is then this ordered
value in the data set.
•
If the number calculates to a
whole
number, the
p
th
percentile is then the
mean
of this ordered
value and the one above it from our data set.