Chapter 3 Chapter 3 Data Description McGraw-Hill, Bluman, 7 th ed, - PowerPoint PPT Presentation

Chapter 3 Chapter 3 Data Description McGraw-Hill, Bluman, 7 th ed, Chapter 3 1

Ch Chapter 3 Overview t 3 O i Introduction Introduction � 3-1 Measures of Central Tendency � 3-2 Measures of Variation � 3-3 Measures of Position � 3 4 Exploratory Data Analysis � 3-4 Exploratory Data Analysis Bluman, Chapter 3 2

Chapter 3 Objectives Chapter 3 Objectives 1. Summarize data using measures of central tendency. 2. Describe data using measures of g variation. 3. Identify the position of a data value in a 3 Identify the position of a data value in a data set. 4 Use boxplots and five number 4. Use boxplots and five-number summaries to discover various aspects of data of data. Bluman, Chapter 3 3

Introduction Introduction Traditional Statistics Traditional Statistics � Average � Variation � Position Bluman, Chapter 3 4

3 1 Measures of Central Tendency 3.1 Measures of Central Tendency � A statistic � A statistic statistic is a characteristic or measure statistic is a characteristic or measure obtained by using the data values from a sample sample. � A parameter parameter is a characteristic or measure obtained by using all the data values for a specific population. Bluman, Chapter 3 5

Measures of Central Tendency Measures of Central Tendency General Rounding Rule General Rounding Rule The basic rounding rule is that rounding should not be done until the final answer is calculated. Use of parentheses on calculators or use of spreadsheets help to avoid early rounding error. Bluman, Chapter 3 6

Measures of Central Tendency Measures of Central Tendency What Do We Mean By Average Average ? � Mean � Median � Mode � Mode � Midrange � Weighted Mean Bluman, Chapter 3 7

Measures of Central Tendency: y Mean � The mean � The mean mean is the quotient of the sum of mean is the quotient of the sum of the values and the total number of values. � The symbol is used for sample mean. X = ∑ + + + + � X X X X X = = = 1 2 3 n X X n n � For a population, the Greek letter μ (mu) is used for the mean. = ∑ + + + + � X X X X X μ = μ = = 1 2 3 N N N Bluman, Chapter 3 8

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-1 Page #106 Bluman, Chapter 3 9

Example 3 1: Days Off per Year Example 3-1: Days Off per Year The data represent the number of days off per year for a sample of individuals selected from f l f i di id l l t d f nine different countries. Find the mean. 20 26 40 36 23 42 35 24 30 20, 26, 40, 36, 23, 42, 35, 24, 30 = ∑ + + + + � X X X X X = 1 2 3 n X X n n + + + + + + + + 20 26 40 36 23 42 35 24 30 276 = = = 30.7 X 9 9 The mean number of days off is 30.7 years. y y Bluman, Chapter 3 10

R Rounding Rule: Mean di R l M The mean should be rounded to one more decimal place than occurs in the raw data. The mean, in most cases, is not an actual Th i t i t t l data value. Bluman, Chapter 3 11

Measures of Central Tendency: y Mean for Grouped Data � The mean for grouped data is calculated by multiplying the frequencies and y p y g q midpoints of the classes. = ∑ ⋅ f X m X n Bluman, Chapter 3 12

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-3 Page #107 Bluman, Chapter 3 13

Example 3-3: Miles Run Example 3 3: Miles Run Below is a frequency distribution of miles run per week run per week. Find the mean. Find the mean Class Boundaries Frequency q y 5.5 - 10.5 1 10.5 - 15.5 2 15.5 - 20.5 3 20.5 - 25.5 5 25 5 25.5 - 30.5 30 5 4 4 30.5 - 35.5 3 35.5 - 40.5 2 Σ f = 20 Bluman, Chapter 3 14

Example 3-3: Miles Run Example 3 3: Miles Run f · X m Class Frequency, f Midpoint, X m 5.5 - 10.5 1 8 8 10.5 - 15.5 2 13 26 15 5 15.5 - 20.5 20 5 3 3 18 18 54 54 20.5 - 25.5 23 5 115 25.5 - 30.5 4 28 112 30.5 - 35.5 3 33 99 35.5 - 40.5 2 38 76 Σ f = 20 Σ f Σ f X m Σ f · X m = 490 20 490 ∑ ⋅ 490 f X = = = m 24.5 miles 24.5 miles X X 20 n Bluman, Chapter 3 15

Measures of Central Tendency: Measures of Central Tendency: Median � The median median is the midpoint of the data array. The symbol for the median is MD. y y � The median will be one of the data values if there is an odd n mber of al es if there is an odd number of values. � The median will be the average of two g data values if there is an even number of values. Bluman, Chapter 3 16

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-4 Page #110 Bluman, Chapter 3 17

Example 3-4: Hotel Rooms Example 3 4: Hotel Rooms The number of rooms in the seven hotels in d downtown Pittsburgh is 713, 300, 618, 595, t Pitt b h i 713 300 618 595 311, 401, and 292. Find the median. Sort in ascending order. 292, 300, 311, 401, 596, 618, 713 292, 300, 311, 401, 596, 618, 713 Select the middle value. MD = 401 The median is 401 rooms The median is 401 rooms. Bluman, Chapter 3 18

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-6 Page #111 Bluman, Chapter 3 19

Example 3-6: Tornadoes in the U S Example 3 6: Tornadoes in the U.S. The number of tornadoes that have occurred in the United States over an 8 occurred in the United States over an 8- year period follows. Find the median. 684 764 656 702 856 1133 1132 1303 684, 764, 656, 702, 856, 1133, 1132, 1303 Find the average of the two middle values. 656, 684, 702, 764, 856, 1132, 1133, 1303 + + 764 764 856 856 1620 1620 = = = MD 810 2 2 The median number of tornadoes is 810. Th di b f t d i 810 Bluman, Chapter 3 20

Measures of Central Tendency: Measures of Central Tendency: Mode � The mode mode is the value that occurs most often in a data set. � It is sometimes said to be the most typical case case. � There may be no mode, one mode (unimodal), two modes (bimodal), or many modes (multimodal). Bluman, Chapter 3 21

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-9 Page #111 Bluman, Chapter 3 22

Example 3-9: NFL Signing Bonuses Example 3 9: NFL Signing Bonuses Find the mode of the signing bonuses of eight NFL players for a specific year eight NFL players for a specific year. The The bonuses in millions of dollars are 18.0, 14.0, 34.5, 10, 11.3, 10, 12.4, 10 18 0 14 0 34 5 10 11 3 10 12 4 10 You may find it easier to sort first. 10, 10, 10, 11.3, 12.4, 14.0, 18.0, 34.5 Select the value that occurs the most Select the value that occurs the most. The mode is 10 million dollars. The mode is 10 million dollars. Bluman, Chapter 3 23

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-10 Page #111 Bluman, Chapter 3 24

Example 3-10: Coal Employees in PA Example 3 10: Coal Employees in PA Find the mode for the number of coal employees per county for 10 selected counties in per county for 10 selected counties in southwestern Pennsylvania. 110 731 1031 84 20 118 1162 1977 103 752 110, 731, 1031, 84, 20, 118, 1162, 1977, 103, 752 N No value occurs more than once. l th There is no mode. Bluman, Chapter 3 25

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-11 Page #111 Bluman, Chapter 3 26

Example 3-11: Licensed Nuclear p Reactors The data show the number of licensed nuclear The data show the number of licensed nuclear reactors in the United States for a recent 15-year period. Find the mode. p 104 104 104 104 104 107 109 109 109 110 104 104 104 104 104 107 109 109 109 110 109 111 112 111 109 109 111 112 111 109 104 and 109 both occur the most. The data set i is said to be bimodal. id t b bi d l The modes are 104 and 109 The modes are 104 and 109. Bluman, Chapter 3 27

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-12 Page #111 Bluman, Chapter 3 28

Example 3-12: Miles Run per Week p p Find the modal class for the frequency distribution of miles that 20 runners ran in one week. of miles that 20 runners ran in one week. Class Frequency 5.5 – 10.5 1 The modal class is 10.5 – 15.5 2 20.5 – 25.5. 15.5 – 20.5 3 20.5 – 25.5 5 The mode, the midpoint 25.5 – 30.5 4 of the modal class, is 30.5 – 35.5 3 23 miles per week 23 miles per week. 35 5 35.5 – 40.5 40 5 2 2 Bluman, Chapter 3 29

Measures of Central Tendency: Measures of Central Tendency: Midrange � The midrange midrange is the average of the lowest and highest values in a data set lowest and highest values in a data set. + + Lowest Lowest Highest Highest = MR 2 Bluman, Chapter 3 30

Ch Chapter 3 t 3 Data Description a a esc p o Section 3-1 Section 3 1 Example 3-15 Page #114 Bluman, Chapter 3 31

Example 3-15: Water-Line Breaks Example 3 15: Water Line Breaks In the last two winter seasons, the city of Brownsville Minnesota reported these Brownsville, Minnesota, reported these numbers of water-line breaks per month. Find the midrange. Find the midrange. 2, 3, 6, 8, 4, 1 + 1 8 9 = = = MR 4.5 2 2 The midrange is 4.5. The midrange is 4.5. Bluman, Chapter 3 32