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 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?

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

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

Cum

ulati

ve re

lati

ve fr

eque

ncy

(%)

Interpreting…

0

20

40

60

80

100

40 45 50 55 60 65 70Age at inauguration

Cum

ulati

ve re

lati

ve fr

eque

ncy

(%)

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….

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?

Estimate and interpret the 65th percentile of the distribution

Interpreting Cumulative Relative Frequency Graphs

47

11

65

58

Median Income for US and District of Columbia.

MedianIncome($1000s)

FrequencyRelativeFrequency

CumulativeFrequency

CumulativeRelativeFrequency

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

Graph it:

MedianIncome($1000s)

FrequencyRelativeFrequency

CumulativeFrequency

CumulativeRelativeFrequency

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:

0.0

0.2

0.4

0.6

0.8

1.0

35 40 45 50 55 60 65 70Median_Household_Income

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

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

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

Page 105 (1-15) odd