Measures of
Position
Percentiles
Z-scores
The following represents my results
when playing an online sudoku
game…at www.websudoku.com.
0 min
30 min
Introduction


A student gets a test back with a score of 78 on
it.
A 10th-grader scores 46 on the PSAT Writing
test
Isolated numbers don’t always provide enough
information…what we want to know is where we
stand.
Where Do I Stand?
Let’s make a dotplot of our heights from 58
to 78 inches.
 How many people in the class have
heights less than you?
 What percent of the heights in the class
have heights less than yours?

This is your percentile in the distribution of
heights!
Finishing….

Calculate the mean and standard deviation.

Where does your height fall in relation to the
mean: Above or Below?

How many standard deviations above or below
the mean is it?
This is the z-score for your height!
Let’s discuss

What would happen to the shape of the
class’s height distribution if you converted
each data value from inches to centimeters.
(2.54cm = 1 in)
Converting from inches to centimeters will have NO
effect on shape.

How would this change of units affect the
measures of center, spread, and location
(percentile & z-score) that you calculated.
It will multiply the center and spread by 2.54.
Converting the class heights to z-scores and percentiles will
not change the shape of the distribution. It will change the
mean to 0 and the standard deviation to 1.
National Center for Health
Statistics

Look at Clinical Growth Charts at
www.cdc.gov/nchs
Percentiles

Value such that r% of the observations in
the data set fall at or below that value.

If you are at the 75th percentile, then 75%
of the students had heights less than
yours.
Test scores on last AP Test. Jenny made
an 86. How did she perform relative to her
classmates?
6
7
7
8
8
9
7
2334
5777899
00123334
569
03
Her score was greater than
21 of the 25 observations.
Since 21 of the 25, or 84%,
of the scores are below
hers, Jenny is at the 84th
percentile in the class’s test
score distribution.
Find the percentiles for
the following students….

Mary, who earned a 74.
4
100  16th Percentile
25

6
7
7
8
8
9
Two students who earned scores of 80.
12
100  48th Percentile
25
7
2334
5777899
00123334
569
03
Cumulative Relative Frequency Table:
Age of First 44 Presidents When They Were Inaugurated
Age
Frequency
Relative
frequency
Cumulative
frequency
Cumulative
relative frequency
40-44
2
2/44 = 4.5%
2
2/44 =
4.5%
45-49
7
7/44 = 15.9%
9
9/44 = 20.5%
50-54
13
13/44 = 29.5%
22
22/44 = 50.0%
55-59
12
12/44 = 34%
34
34/44 = 77.3%
60-64
7
7/44 = 15.9%
41
41/44 = 93.2%
65-69
3
3/44 = 6.8%
44
44/44 = 100%
Cumulative Relative Frequency
Graph:
Cumulative relative frequency (%)
100
80
60
40
20
0
40
45
50 at inauguration
55 60 65
Age
70
Interpreting…
Because most U.S. presidents were
inaugurated in their 50’s.
When does it slow down?
Why?
100
Cumulative relative frequency (%)
Why does it get very steep
beginning at age 50?
Slows at age 60 because most were
inaugurated in their 50’s.
80
60
40
What percent were
inaugurated before age 70?
20
100%
What’s the IQR?
0
40
45
50 at inauguration
55 60 65
Age
70
Roughly 63 – 53 = 10
Obama was 47…. 
Was Barack Obama, who was
inaugurated at age 47, unusually
young?
He was inaugurated at the 11th percentile for age  This means that he was
younger than 89% of all U.S. presidents.
11
47
Estimate and interpret the 65th
percentile of the distribution.
This means that about 65% of all U.S. presidents were younger than 58
when they took office.
65
11
58
What is the relationship between
percentiles and quartiles?
Q1 = 25th Percentile
Q2 = Median = 50th Percentile
Q3 = 75th Percentile
Z-Score – (standardized score)
It represents the number of deviations
from the mean.
 If it’s positive, then it’s above the mean.
 If it’s negative, then it’s below the mean.
 It standardized measurements since it’s in
terms of st. deviation.

Discovery:
Mean = 90
St. dev = 10
Find z score for
80
95
73
Z-Score Formula
x  mean
z
standard deviation
Compare…using z-score.
History Test
Math Test
Mean = 92
Mean = 80
St. Dev = 3
St. Dev = 5
My Score = 95
My Score = 90
95  92
z History 
1
3
90  80
zMath 
2
5
Compare
Math: mean = 70
x = 62
s=6
English: mean = 80
x = 72
s=3
62  70
zMath 
 1.33
6
72  80
z English 
 2.67
3
Be Careful!
Being better is relative to the situation.
What if I wanted to compare race times?
Find the following percentiles.
X
3
4
5
6
7
8
9
10
Rel.
Freq
0.05
0.12
0.23
0.08
0.02
0.18
0.24
0.08
1. 40th percentile?
C.F.
0.05
0.17
0.4
0.45
0.5
0.68
0.92
1
2. 17th percentile?
3. 70th percentile?
4. 25th percentile?
1. 40th Percentile?
10
5
9
8
2. 17th Percentile?
7
4
6
%
3. 70th Percentile?
5
4
8.2
3
4. 25th Percentile?
2
1
4.4
0
3
4
5
6
7
x
8
9
10
Homework

Worksheet and
Textbook p. 105 (1 – 15) Odd