Reminders. Homework scores will be up by tomorrow morning. Please me and the TAs with any grading questions by tomorrow at 5pm

Transcription

1 Reminders Homework scores will be up by tomorrow morning Please me and the TAs with any grading questions by tomorrow at 5pm 1

2 Chapter 12: Describing Distributions with Numbers Aaron Zimmerman STAT Summer 2014 Department of Statistics University of Washington - Seattle 2

3 UW football team wins Year # Wins Motivating Examples UW Coach Salaries (2014) Coach Salary ($) Chris Peterson Pete Kwiatowski Johnathan Smith Chris Strausser Bob Gregory Jimmy Lake Jeff Choate Brent Pease Keith Bhonapa Joran Paopao

4 Motivating Example Histogram of Husky Wins Histogram of Coach Salaries Frequency Frequency Wins Salary $ 4

5 Two options for describing the center and spread of a distribution Median and Quartiles The median and quartiles split a distribution into four equal parts 25% of the observations are smaller than the first quartile 50% of the observations are smaller than the median 75% of the observations are smaller than the third quartile Advantage: The median and quartiles are not affected by outliers Disadvantage: It is more difficult to use the median and quartiles to calculate properties of the distribution Mean and standard deviation The mean is the average of all the observations, while the standard deviation is a measure of the spread of the values around the mean Advantage: The mean and standard deviation are very important for calculating the properties of distributions (like normal distributions: next week) Disadvantage: The mean and standard deviation are affected by outliers 5

6 Formal definitions The median M is the midpoint of a distribution, the number such that half the observations are smaller and the other half are larger The first quartile Q1 is the median of the observations whose positions in the ordered list is to the left of the location of the overall median The third quartile Q3 is the median of the observations whose positions in the ordered list is to the right of the location of the overall median The five-number summary of a distribution consists of the minimum, first quartile, median, third quartile, and maximum, written in order from smallest to largest 6

7 Example #1: Football team wins Don t forget: Write the data in increasing order: If there are two numbers in the middle, the median is the average of the two numbers The median win total is 7 (M = 7) Key idea #1: To find the median, take the n+1 2 number in the ordered observation list where n is the number of observations 7

8 Example #1: Football team wins Don t forget: Write the data in increasing order: Q M Q The first quartile is the median of the numbers to the left of the overall median. Q1=5.5 The third quartile is the median of the numbers to the right of the overall median. Q3=8.5 The five number summary is {0, 5.5, 7, 8.5, 12} 8

9 Example #2: Coach Salaries Let s write the salaries in order (and divide by 1000) First we find the median. It will be the = 5.5 th number. M= M Then we find the quartiles. Q1=345 and Q3= Q1 = M Q3 = The five number summary (in $) is: {110000, , , , } 9

10 Boxplots UW Husky Footbal Team Wins A boxplot is a graph of the five-number summary A central box spans the quartiles A line in the box marks the median 2 Lines extend from the box out to the maximum and minimum observations Wins

11 What is the median? Championships Angels 1 A s 4 Blue Jays 2 Indians 2 Mariners 0 Orioles 3 Rangers 0 Rays 0 Red Sox 8 Royals 1 Tigers 5 Twins 2 White Sox 3 Yankees 27 11

12 What is the 1st Quartile? Championships Angels 1 A s 4 Blue Jays 2 Indians 2 Mariners 0 Orioles 3 Rangers 0 Rays 0 Red Sox 8 Royals 1 Tigers 5 Twins 2 White Sox 3 Yankees 27 12

13 What is the 3rd Quartile? Championships Angels 1 A s 4 Blue Jays 2 Indians 2 Mariners 0 Orioles 3 Rangers 0 Rays 0 Red Sox 8 Royals 1 Tigers 5 Twins 2 White Sox 3 Yankees 27 13

14 Draw a Boxplot Championships Angels 1 A s 4 Blue Jays 2 Indians 2 Mariners 0 Orioles 3 Rangers 0 Rays 0 Red Sox 8 Royals 1 Tigers 5 Twins 2 White Sox 3 Yankees 27 14

15 Draw a Boxplot AL Championship Wins AL Championship Wins Frequency Wins 15

16 More formal definitions The sample mean, X, of a set of observations is the average of the observations To find the mean of n observations, add the values and divide by n The sample standard deviation, s, is a measure of the spread of the values around the sample mean Use s to describe the spread only when you use the sample mean to describe the center s = 0 only when there is no spread. This happens only when all observations have the same value As the observations become more spread around the sample mean, s gets larger 16

17 More formal definitions The sample standard deviation, s, is a measure of the spread of the values around the sample mean It is calculated by finding an average of the squared distances and then taking the square root. To find the standard deviation of n observations: 1 Find the distance of each observation from the mean and square each of these distances. 2 Average the distances by dividing their sum by n 1. This average squared distance is called the variance. 3 The standard deviation s is the square root of this average squared distance. 17

18 Example #1: Football team wins To compute the sample mean, add up all 24 observations and divide by 24 In this case, X = = 6.71 The standard deviation s = 2.88 There is some spread across observations, but not too much We ll learn more about interpreting standard deviations next class Notice that the sample mean (6.71) is relatively close to the median (7) Rule of Thumb The mean and median are similar for symmetric distributions that don t have outliers 18

19 Step 1 Example #1: SD calculation Wins Win Wins (Wins Wins) Sum

20 Example #1: SD calculation Step 2: Find the Variance Var = = 8.30 Step 3: Find the Standard Deviation SD = Var = 8.30 =

21 Example #2: Head coach salaries To compute the sample mean, add up all 10 observations and divide by 10 In this case, X = = $668, 500 The standard deviation is s = $1, 038, 666 There is a lot of spread around the sample mean Notice that the sample mean ($668,500) is much larger than the median ($400,000) Rule of Thumb the mean will be larger than the median for right-skewed distributions (or if there is an outlier on the right tail). The mean is smaller for left-skewed distributions. 21

22 Examples #1 and # 2 Histogram of Husky Wins Histogram of Coach Salaries Frequency Median Mean Frequency Median Mean Wins Salary $ 22

23 Recommendations The mean and standard deviation are strongly affected by skewed distributions and outliers The median and quartiles are less affected So, the five-number summary is usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers Use the sample mean and standard deviation for reasonably symmetric distributions that are free of outliers 23

24 What is the mean? Championships Angels 1 A s 4 Blue Jays 2 Indians 2 Mariners 0 Orioles 3 Rangers 0 Rays 0 Red Sox 8 Royals 1 Tigers 5 Twins 2 White Sox 3 Yankees 27 24

25 Why is the mean larger than the median? Championships Angels 1 A s 4 Blue Jays 2 Indians 2 Mariners 0 Orioles 3 Rangers 0 Rays 0 Red Sox 8 Royals 1 Tigers 5 Twins 2 White Sox 3 Yankees 27 25

26 Homework Read Chapter 12 Do problems 12.1 (also give the five-number summary), 12.3, 12.11, 12.13, 12.19,