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 Her score was greater than 7 2334 21 of the 25 observations. 7 5777899 Since 21 of the 25, or 84%, of the scores are below 8 00123334 hers, Jenny is at the 84th 8 569 percentile in the class’s test 9 03 score distribution. 6 7 Find the percentiles for 7 2334 the following students…. 7 5777899 8 00123334  Mary, who earned a 74. 8 569 4 9 03 100 16th Percentile 25

 Two students who earned scores of 80.

12 100  48th Percentile 25 Cumulative Relative Frequency Table:

Age of First 44 Presidents When They Were Inaugurated

Age Frequency Relative Cumulative Cumulative frequency frequency 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:

100

80

60

40

20

Cumulative relativeCumulative frequency (%) 0

40 45 Age50 at inauguration55 60 65 70 Why does it get very steep beginning at age 50? Interpreting… Because most U.S. presidents were inaugurated in their 50’s. When does it slow down? 100 Why? 80 Slows at age 60 because most were inaugurated in their 50’s. 60 What percent were

40 inaugurated before age 70?

20 100%

Cumulative relativeCumulative frequency (%) 0 What’s the IQR? Roughly 63 – 53 = 10 40 45 Age50 at inauguration55 60 65 70

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 90  80 z  1 z   2 History 3 Math 5 Compare

Math: mean = 70 62  70 z   1.33 x = 62 Math 6 s = 6

English: mean = 80 72 80 x = 72 z   2.67 English 3 s = 3 Be Careful!

Being better is relative to the situation.

What if I wanted to compare race times? Find the following percentiles.

Rel. 1. 40th percentile? X Freq C.F. 3 0.05 0.05 th 4 0.12 0.17 2. 17 percentile? 5 0.23 0.4 6 0.08 0.45 3. 70th percentile? 7 0.02 0.5 8 0.18 0.68

9 0.24 0.92 4. 25th percentile? 10 0.08 1 th 10 1. 40 Percentile?

9 5

8 2. 17th Percentile? 7 4 6

% 5 3. 70th Percentile?

4 8.2 3

th 2 4. 25 Percentile?

1 4.4 0 3 4 5 6 7 8 9 10

x Homework

 Worksheet and Textbook p. 105 (1 – 15) Odd