measures of position - anderson school district five · national center for health statistics
TRANSCRIPT
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Measures of
Position
Percentiles
Z-scores
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0 min 30 min
The following represents my results when playing an online sudoku game…at www.websudoku.com.
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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.
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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!
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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!
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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)
How would this change of units affect the
measures of center, spread, and location
(percentile & z-score) that you calculated.
Converting from inches to centimeters will have NO
effect on shape.
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.
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National Center for Health
Statistics
Look at Clinical Growth Charts at
www.cdc.gov/nchs
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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.
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Test scores on last AP Test. Jenny made
an 86. How did she perform relative to her
classmates?
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.
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
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Find the percentiles for
the following students….
Mary, who earned a 74.
Two students who earned scores of 80.
6 7
7 2334
7 5777899
8 00123334
8 569
9 03 Percentileth16100
25
4
Percentileth4810025
12
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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%
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Cumulative Relative Frequency
Graph:
0
20
40
60
80
100
40 45 50 55 60 65 70Age at inauguration
Cu
mu
lati
ve r
ela
tive
fre
qu
en
cy (
%)
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Interpreting…
0
20
40
60
80
100
40 45 50 55 60 65 70Age at inauguration
Cu
mu
lati
ve r
ela
tive
fre
qu
en
cy (
%)
Why does it get very steep
beginning at age 50?
When does it slow down?
Why?
What percent were
inaugurated before age 70?
What’s the IQR?
Obama was 47….
Because most U.S. presidents were
inaugurated in their 50’s.
Slows at age 60 because most were
inaugurated in their 50’s.
100%
Roughly 63 – 53 = 10
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47
11
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.
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11
65
58
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.
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What is the relationship between
percentiles and quartiles?
Q1 = 25th Percentile
Q2 = Median = 50th Percentile
Q3 = 75th Percentile
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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.
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Discovery:
Mean = 90
St. dev = 10
Find z score for
80
95
73
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Z-Score Formula
mean
standard deviation
xz
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Compare…using z-score.
History Test
Mean = 92
St. Dev = 3
My Score = 95
Math Test
Mean = 80
St. Dev = 5
My Score = 90
13
9295
Historyz 2
5
8090
Mathz
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Compare
Math: mean = 70
x = 62
s = 6
English: mean = 80
x = 72
s = 3
33.16
7062
Mathz
67.23
8072
Englishz
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Be Careful!
Being better is relative to the situation.
What if I wanted to compare race times?
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Find the following percentiles.
X
Rel.
Freq C.F.
3 0.05 0.05
4 0.12 0.17
5 0.23 0.4
6 0.08 0.45
7 0.02 0.5
8 0.18 0.68
9 0.24 0.92
10 0.08 1
1. 40th percentile?
2. 17th percentile?
3. 70th percentile?
4. 25th percentile?
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3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
%
x
1. 40th Percentile?
2. 17th Percentile?
3. 70th Percentile?
4. 25th Percentile?
5
4
8.2
4.4
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Homework
Worksheet and
Textbook p. 105 (1 – 15) Odd