AP Statistics – 2.1 Notes Name: _____________________ Describing Location in a Distribution

1. Find and interpret the percentile of an individual value within a distribution of data. Percentile: The Pth Percentile of a distribution is the value with p percent the observations less than it.

Human Dot Plot: Independent Practice:

a) What percentile is your height? Interpret what this means.

2. Find and Interpret the standardized score (z-score) of an individual value within a distribution of data.

Standardized Score (z-score) - tells the number of standard deviations an observation is away from the mean, and the direction (above or below). b) Calculate the mean and standard deviation of the distribution of the class heights (on Calc). c) What is the z-score for your height? Can you use this information to figure out the formula for

calculating a z-score?

Formula for a z-score

Z =

d) Interpret your height’s z-score.

key

HeightsinInches

mefii i

60 62 64 66 as io is 24 26 78

myheightis the 75thPercentile because it is larger

21502 0.75 than 75 of theclassheights

5 66.2 5 3.24

z 2066.22 1.173.24

Sx

2 1.17 Myheight is 1.17standard deviations above themean

AP Statistics – 2.1 Notes Name: _____________________ Describing Location in a Distribution

Comparing Z-Scores • We can also use z-scores to compare the positions of individuals in different distributions. Example Problem: In the 2018/19 English Premier League season Mohamed Salah won the Golden Boot Award for scoring the most goals. The Liverpool star scored 22 goals and lead the team to win the UEFA Champions League. Eden Hazard also had a successful season and led the league in assists with total of 15. Which of these two accomplishments was more impressive? The mean number of goals scored among the top 50 scorers was �̅�𝐺 = 11.28 with a standard deviation of 𝑆𝐺 = 4.33 and the mean number of assists among the top 50 assists was �̅�𝐴 = 6.98 with a standard deviation of 𝑆𝐴 =2.43

3. Describe the effect of adding, subtracting, multiplying, or dividing by a constant on the shape,

center, and spread of a distribution.

A Simple Example Effects of Adding (or Subtracting) a Constant Effects of Multiplying (or Dividing) by a Constant

Adding the same number a to (subtracting a from) each observation: • adds a to (subtracts a from) measures of center and location (mean, median, quartiles,

percentiles), but • Does not change the shape of the distribution or measures of Spread (range, IQR, standard

deviation).

Multiplying (or dividing) each observation by the same number b: • multiplies (divides) measures of Center and Location (mean, median, quartiles, percentiles)

by b • multiplies (divides) measures of Spread (range, IQR, standard deviation) by b, but • does not change the Shape of the distribution

2211.282 2.48 Hazardwas moreimpressive bcZs 4.33 his Z score was larger

15 6.98 3.302.43

X iI l l l l l l l l 1 Transformation Mean Range2 4 6 8 10

X 3 2X 12 i i I i ti i l

2 4 6 8 10 X 2 5 2X 2 i

I l l l l l l l l l 2X 6 42 4 6 8 102 2 i

I l l l l l l l l l 2 2 8 42 4 6 8 10

AP Statistics – 2.1 Notes Name: _____________________ Describing Location in a Distribution

Using the class heights in inches, fill out the summary statistics table below.

Summary Statistics for Heights in Inches Min Q1 Med Q3 Max Mean Standard Dev.

Adding or Subtracting a Constant Suppose we re-measured each student’s height in the class, and required everyone to wear 2-inch heels (yes even the guys). How would this change the distribution of heights? Measures of center, spread, and location (percentiles and z-scores)?

Summary Statistics for Heights + 2 Min Q1 Med Q3 Max Mean Standard Dev.

Multiplying or Dividing by a Constant Suppose we wanted to distribute the class heights in Centimeters instead of inches. How would this change the distribution of heights? Measures center, spread, and location (percentiles and z-scores)?

Summary Statistics for Height X 2.54 Min Q1 Med Q3 Max Mean Standard Dev.

61 63.5 66.5 69.5 71 66.2 3.24

63 65.5 68.5 71.5 73 68.2 3.24

154.94 161.29 168.91 176.53 180.34 168.148 8.224

AP Statistics – 2.1 Notes Name: _____________________ Describing Location in a Distribution

4. Estimate percentiles and individual values using a cumulative relative frequency graph. A cumulative relative frequency graph displays the cumulative relative frequency of each class of a frequency distribution.

Interpret the Graph:

Practice: a) Was Obama, who was inaugurated at 47, unusually young?

b) Estimate and interpret the 65th Percentile of the distribution.

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%

0

20

40

60

80

100

40 45 50 55 60 65 70

Cum

ulat

ive

rela

tive

freq

uenc

y (%

)

Age at inauguration

Thegraphgrowsgradually at firstbecause afewpresidents were inauguratedin their40s It steepens at 50 bc 65

most were inaugurated in their 50sIt becomes lesssteep after 60

symmetry skewedlefty skewedRighti

Age47 puts him at about the 11th PercentileHewasyoung butnotunusuallyyoung

An ageof 58 is approximately the 65thpercentile

AP Statistics – 2.1 Notes Name: _____________________ Describing Location in a Distribution

Independent Practice: 1. Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test:

a) Jenny earned an 86, calculate her percentile.

b) Norman scored a 72, calculate his percentile.

c) Calculate the percentile for the two students who scored an 80.

d) Calculate and interpret the standardized score for Katie, who scored a 93.

e) Calculate and interpret the z-score for Norman, who scored a 72. 2. Here are a graph and table of summary statistics for a sample of 30 test scores. The maximum

possible score was 50 points.

a) Suppose the teacher was nice and added 5 points to each score. How would this change the

summary statistics? Complete the table below

b) Suppose the teacher wanted to convert the original test scores to percent. Since the test was

out of 50 points, he should multiply each score by 2 to make them out of 100 points. How would this change the original distribution of scores? Complete the table below

Score10 15 20 25 30 35 40 45 50

Collection 1 Dot Plot

Score

Score_Plus510 15 20 25 30 35 40 45 50

Collection 1 Dot Plot

Score

Scorex210 20 30 40 50 60 70 80 90 100

Collection 1 Dot Plot

6 7

7 2334

7 5777899

8 00123334

8 569

9 03

2 0.84 j 84thPercentile

215 0.04 4thpercentile

LF 0.48 48thPercentile

z 93 soKate'sresultis 2.14 standarddeviations

6.02 2.14 abovethemean forHistest

z 72 soNorman'sscoreis 1.32standarddeviations below

1.32 Hemean of 80

3040.88.17 17 37 42 4653 9 363071.6 16.3424 6474 8296 18 72