Measures of

Position

Percentiles

Z-scores

0 min 30 min

The following represents my results when playing an online sudoku game…at www.websudoku.com.

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)

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.

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?

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

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

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:

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 (

%)

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

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.

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.

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

mean

standard deviation

xz

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

Compare

Math: mean = 70

x = 62

s = 6

English: mean = 80

x = 72

s = 3

33.16

7062

Mathz

67.23

8072

Englishz

Be Careful!

Being better is relative to the situation.

What if I wanted to compare race times?

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?

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

Homework

Worksheet and

Textbook p. 105 (1 – 15) Odd