Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Measures of Central Tendency Section 2-4 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statisitics Addison Wesley Longman
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 Measure of Central Tendency
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 a value at the center or middle of a data set Measure of Central Tendency
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Mean Arithmetic Mean AVERAGE Definitions
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 FIGURE 2-7 Mean as a Balance Point
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Mean FIGURE 2-7 Mean as a Balance Point
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Notation
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Notation denotes the summation of a set of values
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 Notation denotes the summation of a set of values x is the variable usually used to represent the individual data values
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Notation denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Notation denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 Notation x is pronounced ‘x-bar’ and denotes the mean of a set of sample values denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 µ is pronounced ‘mu’ and denotes the mean of all values Notation x is pronounced ‘x-bar’ and denotes the mean of a set of sample values denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population in a population
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Definitions Mean
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 Definitions Mean the value obtained by adding the scores and dividing the total by the number of scores
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 x = Definitions Mean the value obtained by adding the scores and dividing the total by the number of scores n x Sample
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 x = Definitions Mean the value obtained by adding the scores and dividing the total by the number of scores n x x Sample N µ = x x Population
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Calculators can calculate the mean of data Definitions Mean the value obtained by adding the scores and dividing the total by the number of scores n x = x x Sample N µ = x x Population
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19 Examples
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20 use class mark of classes for variable x Mean from a Frequency Table
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f (f x)
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f (f x) x = class mark
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f (f x) x = class mark f = frequency
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24 use class mark of classes for variable x Mean from a Frequency Table x = Formula 2-2 f (f x) x = class mark f = frequency f = n f = n
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25 Weighted Mean
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26 Weighted Mean x = w (w x)
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27 Examples
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 28 Definitions Median the middle value when scores are arranged in (ascending or descending) order
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 29 Definitions Median the middle value when scores are arranged in (ascending or descending) order often denoted by x (pronounced ‘x-tilde’) ~
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 30 Definitions Median the middle value when scores are arranged in (ascending or descending) order often denoted by x (pronounced ‘x-tilde’) is not affected by an extreme value ~
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman (in order) exact middle Examples MEDIAN is 4
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman no exact middle -- shared by two numbers = (in order) exact middle MEDIAN is 4 Examples MEDIAN is 4.5
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 33 Definitions Mode the score that occurs most frequently Bimodal Multimodal No Mode
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 34 Examples Mode is 5 Bimodal (2 & 6) No Mode a b c
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 35 Examples Mode is 5 Bimodal (2 & 6) No Mode a b c d e
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 36 Examples Mode is 5 Bimodal (2 & 6) No Mode a b c d e Mode is 3 No Mode
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 37 Mode is “O” blood type Blood types: O 35 A 14 B 16 AB 10 Remark: Mode is the only measure of central tendency that can be used with nominal data
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 38 Midrange the value halfway between the highest and lowest scores Definitions
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 39 Midrange the value halfway between the highest and lowest scores Definitions Midrange = highest score + lowest score 2
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman (in order) Midrange is (5 + 1)/2 = 3 Examples
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 41 Carry one more decimal place than is present in the original set of data Round-off rule for measures of central tendency
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 42 Advantages - Disadvantages Best Measure of Central Tendency Table 2-6
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 43 Table 2-6
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 44 Skewness Mode = Mean = Median SYMMETRIC Figure 2-8 (b)
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 45 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-8 (b) Figure 2-8 (a)
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 46 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-8 (b) Figure 2-8 (a) Figure 2-8 (c)