Find the following for the Weight
of Football Players
𝑥
Sample standard deviation
n=
Fun Coming Up!
3-3 Measures of Position
 Z-score
 Percentile
 Quartile
 Outlier
Bluman, Chapter 3
3
Measures of Position: Z-score

A z-score or standard score for a value
is obtained by following formulas:
X X
z
s

z
X 

A z-score represents the number of
standard deviations a value is above or
below the mean.
Bluman, Chapter 3
4
Rounding Rule for Z score

Round the Z score to 2 decimal places.
Chapter 3
Data Description
Section 3-3
Example 3-29
Page #142
Bluman, Chapter 3
6
Example 3-29: Test Scores
A student scored 65 on a calculus test that had a
mean of 50 and a standard deviation of 10; she
scored 30 on a history test with a mean of 25 and
a standard deviation of 5. Compare her relative
positions on the two tests.
X  X 65  50
z

 1.5 Calculus
s
10
X  X 30  25
z

 1.0 History
s
5
She has a higher relative position in the Calculus class.
Bluman, Chapter 3
7
Percentiles
Percentiles
divides the data into 100 equal groups.
Percentiles are symbolized
by P1, P2, P3, …,P99
Please take a look at data on page 135
of your text.
Formula
Percentile
formula:
Number of values below X   0.5 100%
Total # of values
Check this website out!
There are variation on the formula for
Percentiles. The website below does a
fantastic job of displaying and explaining
the variations.
 http://www.regentsprep.org/regents/mat
h/algebra/AD6/quartiles.htm

Definition 1: A percentile
is a measure that tells us
what percent of the total
frequency scored at or below
that measure. A percentile
rank is the percentage of
scores that fall at or below a
given score.
Formula:
To find the percentile rank of a score,
x, out of a set of n scores, where x is
included:
Where B
= number of scores
below x
E = number of scores
equal to x
n = number of scores
Definition 2: A percentile
is a measure that tells us
what percent of the total
frequency scored below that
measure. A percentile rank
is the percentage of scores
that fall below a given score.
About Percentile Ranks:
Percentile rank is a number between 0
and 100 indicating the percent of cases
falling at or below that score.
 Scores are divided into 100 equally sized
groups.
 Percentile ranks are usually written to the
nearest whole percent: 74.5% = 75% =
75th percentile.

About Percentile Ranks:




Scores are arranged in rank order from lowest to
highest
There is no 0 percentile rank-the lowest score is
the 1st percentile.
There is no 100th percentile- the highest score is
the 99th percentile.
You can’t perform the same mathematical
operations on percentile that you can on raw
scores. You can’t, for example, compute the
mean of percentile scores.
Example 3-31

The frequency distribution for the systolic
blood pressure readings (mm of mercury) of
200 randomly selected college students is
shown here. Construct a percentile graph
A
B
C
D
Class
Boundaries
Frequency
Cumulative
Freq
Cumulative
Percent
89.5-104.5
24
104.5-119.5
62
119.5-134.5
72
134.5-149.5
26
149.5-164.5
12
164.5-179.5
4
200
Measures of Position: Example of
a Percentile Graph
Bluman, Chapter 3
16
Chapter 3
Data Description
Section 3-3
Example 3-32
Page #147
Bluman, Chapter 3
17
Example 3-32
A teacher gives a 20-point test to 10
students. The scores are shown here.
18,15,12,6,8,2,3,5,20,10
 Find the percentile rank of a score of 12?
6?
Example 3-32: Test Scores
A teacher gives a 20-point test to 10 students.
Find the percentile rank of a score of 12.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10
Sort in ascending order.
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
6 values
# of values below X   0.5

Percentile 
100%
total # of values
6  0.5
A student whose score

100%
was 12 did better than
10
65% of the class.
 65%
Bluman, Chapter 3
19
Refer to the same data
the value of the 25th
percentile? 60th?
 Find
If the percentile is known,
then the formula is
n p
c
100
See page 149 for full explanation
Example 3-34: Test Scores
A teacher gives a 20-point test to 10 students. Find
the value corresponding to the 25th percentile.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10
Sort in ascending order.
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
n  p 10  25
c

 2.5  3
100
100
This is the location
of data. i.e. 3rd
value.
The value 5 corresponds to the 25th percentile.
Bluman, Chapter 3
21
Example 3-35: Test Scores
A teacher gives a 20-point test to 10 students. Find
the value corresponding to the 60th percentile.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10
Sort in ascending order.
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
𝑛𝑝
𝑐=
100
The answer is
between the 6th
and 7th data point.
10 × 60
=
=6
100
The value 11 corresponds to the 60th
percentile.
Bluman, Chapter 3
22
Measures of Position:
Deciles

Deciles separate the data set into 10
equal groups. D1=P10, D4=P40

Please Pages 150-151 for further
instructions.
Bluman, Chapter 3
23
Measures of Position:
Quartiles

Quartiles separate the data set into 4
equal groups. Q1=P25, Q2=MD, Q3=P75

Q2 = median(Low,High)
Q1 = median(Low,Q2)
Q3 = median(Q2,High)

The Interquartile Range, IQR = Q3 – Q1.
Bluman, Chapter 3
24
Chapter 3
Data Description
Section 3-3
Example 3-36
Page #150
Bluman, Chapter 3
25
Example 3-36: Quartiles
Find Q1, Q2, and Q3 for the data set.
15, 13, 6, 5, 12, 50, 22, 18
Sort in ascending order.
5, 6, 12, 13, 15, 18, 22, 50
6  12
Q1  median  Low, MD  
 9
2
13  15
Q 2  median  Low, High  
 14
2
18  22
Q3  median  MD, High  
 20
2
Bluman, Chapter 3
26
Measures of Position:
Outliers

An outlier is an extremely high or low
data value when compared with the rest of
the data values.

A data value less than Q1 – 1.5(IQR) or
greater than Q1 + 1.5(IQR) can be
considered an outlier.
Please refer to page 152 of your
test for the procedure of how to
identify the outliers.
Bluman, Chapter 3
27
Homework
Read section 3-3 and study the relevant
examples.
 Page 153
 #1-8 all, 11-31 odds
