Describing Location in a Distribution Section 2.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Lesson 2.1.1

Starter Problem Draw a geometric figure that starts at (0,0) and goes in straight line segments to (0, .5), (.5, .5), (.5, 1.5), (1, 1.5), (1, 0), (0,0) in order. Find the area A of the figure. Lesson 2.1.1

Today’s Objectives Percentiles: The Pth Percentile Standardized value: Z-score Density curves Activity! “where do I stand” Lesson 2.1.1

WE DON’T USE THIS DEFINITION Percentiles One way to describe the location in a distribution is to tell what percent falls below a value. That is called the percentile. The pth percentile of a distribution is the value with p percent of the observations less than it. NOTE: There IS an alternative definition that includes “less than or equal to” making it possible to be in the 100th percentile. WE DON’T USE THIS DEFINITION

Percentiles

Percentile example Here are all the scores of all 25 students in Mr. Baca’s Algebra class on their first test: The red score is Jill’s 1)Make a Stemplot 2) How many scores are below 79 81 80 77 73 83 74 93 78 75 67 89 84 82 72 86 90 85 Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jill is at the 84th percentile in the class’s test score distribution. 6 7 7 2334 7 5777899 8 00123334 8 569 9 03

Another example? Yes? No? “ages of U.S. Presidents” Then lets look to check your understanding

YES Was Barack Obama, who was inaugurated at age 47, unusually young? Interpreting Cumulative Relative Frequency Graphs Was Barack Obama, who was inaugurated at age 47, unusually young? Estimate and interpret the 65th percentile of the distribution YES 65 11 58 47

NO Mark received a score report detailing his performance on a statewide test. On the math section, Mark earned a raw score of 39, which placed him at the 68th percentile. This means that Mark did better than about 39% of the students who took the test. Mark did worse than about 39% of the students who took the test. Mark did better than about 68% of the students who took the test. Mark did worse than about 68% of students who took the test. Mark got fewer than half of the questions correct on this test.

Remember Standard Deviation? When we looked at the standard normal curve we looked specifically at 1, 2, and 3 SD away from the mean…what if we wanted to find a value that falls between 1,2 and 3 SD??

Z-score If x is an observation from a distribution that has a known mean and standard deviation, the standardized value (z) of x is: 𝑧 𝑠𝑐𝑜𝑟𝑒= 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛−𝑚𝑒𝑎𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 **Z scores DON’T have units!** *The Z score tells us how many standard deviations from the mean an observation falls, and in what direction. *

Example! If Jill scored an 86 on the test, and I tell you the mean of the data is 80, with a standard deviation of 6.07, find her z-score value. 𝑥 = 86 µ= 80 𝜎= 6.07 𝑧= 86−80 6.07 = 6 6.07 =0.9885

Try it! Show all your work, write your final answer in a sentence If Katie scored an 93 on the test, and I tell you the mean of the data is 80, with a standard deviation of 6.07. If Norman scored an 72 on the test, and I tell you the mean of the data is 80, with a standard deviation of 6.07. Show all your work, write your final answer in a sentence

Good Job!

Break! - 5 Minutes

Density Curves A histogram can be approximated by a smooth curve. The curve retains the shape, center & spread of the original distribution. Lesson 2.1.1

Density Curves If the area of a histogram is 1, then the area under the curve must also be 1. This is called a DENSITY CURVE. Density curves can be ANY shape so long as the area is 1 and the entire curve is above the horizontal axis.

The median of a density curve If the median is the value with one half the observations below it and one half above, then 50% lies to the left, and 50% lies to the right. Likewise If the mean is normally distributed, then the mean will be the same point as the median….but…. Lesson 2.1.1

The Mean of a Density Curve What if the graph is skewed? Remember we said that if the graph is skewed left or right the mean gets pulled. Its not resistant.

Stand Up! Activity! Arrange yourself from TALLEST to SHORTEST! There are a total of ______ AP Stats students. Count how many students are shorter than you. ______ What is your percentile for this class of height?

Today’s Objectives Percentiles: The Pth Percentile Standardized value: Z-score Density curves Activity! “where do I stand” Lesson 2.1.1

1) Start Chapter 2 Reading Guide Homework 1) Start Chapter 2 Reading Guide 2) 2.1 Homework Worksheet Lesson 2.1.1

Test Results! 2A Grade: Amount: Marginal % ……A......……….3..……….10% …….B……………7……....23% 66% Passed …….C…………..10.……....33% …….D…………...5……....16% …….F…..............5……….16% 32% Failed Mean: % Max: % Min: % No Outliers

Test Results! 5B Grade: Amount: Marginal % ……A......……....4.……….33% …….B…………..1……..... 8% 75% Passed …….C…………..4……....33% …….D…………..3……....25% …….F…..............0……….0% 25% Failed Mean: % Max: % Min: % No Outliers

Tracking AP Stats 2016-2017 (WHS) Ch. 1 Test Ch. 2 Test Ch. 3 Test B 7/1 C10/4 D5/3 F5/0

14/15 VS 15/16 VS 16/17 14/15 15/16 16/17 Chapter 1 Test A -5 A -7 B-5 D-2 D-10 D-8 F-1 F-9 F-5 14/15 15/16 Chapter 2 Test A - B- C- D- F- 14/15 15/16 Chapter 3 Test A - B- C- D- F- 14/15 15/16 Chapter 4 Test A - B- C- D- F-