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 ©

Copyright © 1998, Triola, Elementary Statistics

Addison Wesley Longman 1

Measures of Measures of Central TendencyCentral Tendency

Section 2-4 Section 2-4

M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statisitics

Addison Wesley Longman



Measure of Central Tendency



a value at the

center or middle of a data set

Measure of Central Tendency



Mean

Arithmetic Mean

AVERAGE

Definitions


Addison Wesley Longman 5FIGURE 2-7

Mean as a Balance Point



Mean

FIGURE 2-7

Mean as a Balance Point



Notation



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



Notation

x is pronounced ‘x-bar’ and denotes the mean of a set ofsample values

denotes the summation of a set of values






µ is pronounced ‘mu’ and denotes the mean of all values

Notation

x is pronounced ‘x-bar’ and denotes the mean of a set ofsample values

denotes the summation of a set of values




in a population



Definitions Mean



Definitions Mean

the value obtained by adding the scores and dividing the total by the number of scores



x =

Definitions Mean


n x

Sample



x =

Definitions Mean


n x

Sample

Nµ = xPopulation



Calculators can calculate the mean of data

Definitions Mean


nx =

xSample

Nµ = xPopulation



Examples



use class mark of classes for variable x

Mean from a Frequency Table





x = Formula 2-2f

(f • x)





x = Formula 2-2f

(f • x)x = class mark





x = Formula 2-2f


f = frequency





x = Formula 2-2f


f = frequency

f = n



Weighted Mean



Weighted Mean

x =w

(w • x)



Examples



Definitions Median

the middle value when scores are arranged in (ascending or descending) order



Definitions Median


often denoted by x (pronounced ‘x-tilde’)~



Definitions Median


often denoted by x (pronounced ‘x-tilde’)

is not affected by an extreme value

~



• 5 5 5 3 1 5 1 4 3 5 2 • 1 1 2 3 3 4 5 5 5 5 5 (in order)

exact middle

Examples

MEDIAN is 4



• 1 1 3 3 4 5 5 5 5 5 no exact middle -- shared by two numbers

4 + 5

2= 4.5

• 5 5 5 3 1 5 1 4 3 5 2 • 1 1 2 3 3 4 5 5 5 5 5 (in order)

exact middle MEDIAN is 4

Examples

MEDIAN is 4.5



Definitions Mode

the score that occurs most frequently

BimodalMultimodalNo Mode



Examples

• Mode is 5

• Bimodal (2 & 6)

• No Mode

a. 5 5 5 3 1 5 1 4 3 5

b. 2 2 2 3 4 5 6 6 6 7 9

c. 2 3 6 7 8 9 10



Examples

• Mode is 5

• Bimodal (2 & 6)

• No Mode

a. 5 5 5 3 1 5 1 4 3 5

b. 2 2 2 3 4 5 6 6 6 7 9

c. 2 3 6 7 8 9 10

d. 2 2 3 3 3 4

e. 2 2 3 3 4 4 5 5



Examples

• Mode is 5

• Bimodal (2 & 6)

• No Mode

a. 5 5 5 3 1 5 1 4 3 5

b. 2 2 2 3 4 5 6 6 6 7 9

c. 2 3 6 7 8 9 10

d. 2 2 3 3 3 4

e. 2 2 3 3 4 4 5 5

• Mode is 3

• No Mode



• 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



Midrange

the value halfway between the highest and lowest scores

Definitions



Midrange

the value halfway between the highest and lowest scores

Definitions

Midrange =highest score + lowest score

2



• 5 5 5 3 1 5 1 4 3 5 2 • 1 1 2 3 3 4 5 5 5 5 5 (in order)

Midrange is (5 + 1)/2 = 3

Examples



Carry one more decimal place than is present in the original set of data

Round-off rule for measures of central

tendency



Advantages - Disadvantages

Best Measure of Central Tendency

Table 2-6



Table 2-6



Skewness

Mode = Mean = Median

SYMMETRIC

Figure 2-8 (b)



Skewness


SKEWED LEFT(negatively)

SYMMETRIC

Mean Mode Median

Figure 2-8 (b)

Figure 2-8 (a)



Skewness


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)

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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 ©