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 dents 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 class’s height
distribution if you converted each data value
from inches to centimeters. (2.54cm = 1 in)

How would this change of units affect the
measures of center, spread, and location
(percentile & z-score) that you calculated.
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
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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….
6
7
7
8
8
9

Mary, who earned a 74.

Two students who earned scores of 80.
7
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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…
When does it slow down?
Why?
100
Cumulative relative frequency (%)
Why does it get very steep
beginning at age 50?
80
60
What percent were
inaugurated before age 70?
40
20
What’s the IQR?
0
40
45
50 at inauguration
55 60 65
Age
70
Obama was 47….

Interpreting Cumulative Relative Frequency Graphs
11
47
58
Describing Location in a
Distribution
Use the graph from page 88 to answer
the following questions.
Was Barack Obama, who was
inaugurated at age 47, unusually
young?
65 and interpret the 65th
Estimate
percentile of the distribution
Median Income for US and District of Columbia.
Median
Income
($1000s)
Frequency
35 to < 40
1
40 to < 45
10
45 to < 50
14
50 to < 55
12
55 to < 60
5
60 to < 65
6
65 to < 70
3
Relative
Frequency
Cumulative
Frequency
Cumulative
Relative
Frequency
Graph it:
Median
Income
($1000s)
Frequency
Relative
Frequency
Cumulative
Frequency
Cumulative
Relative
Frequency
35 to < 40
1
1/51 = 0.020
1
1/51 = 0.020
40 to < 45
10
10/51 = 0.196
11
11/51 = 0.216
45 to < 50
14
14/51 = 0.275
25
25/51 = 0.490
50 to < 55
12
12/51 = 0.236
37
37/51 = 0.725
55 to < 60
5
5/51 = 0.098
42
42/51 = 0.824
60 to < 65
6
6/51 = 0.118
48
48/51 = 0.941
65 to < 70
3
3/51 = 0.059
51
51/51 = 1.000
Answer:
What is the relationship between
percentiles and quartiles?
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
Compare
Math: mean = 70
x = 62
s=6
English: mean = 80
x = 72
s=3
Be Careful!
Being better is relative to the situation.
What if I wanted to compare race times?
Homework
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Page 105 (1-15) odd