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Measures Of Central Tendency

Quantitative Aptitude & Business Statistics

Quantitative aptitude & Business Statistics: Measures Of Central

2

Statistics in Plural Sense as Statistical data.

Statistics in Plural Sense refers to numerical data of any phenomena placed in relation to each other.

For example ,numerical data relating to population ,production, price level, national income, crimes, literacy ,unemployment ,houses etc.,

Statistical in Singular Scene as Statistical method.


3

According to Prof.Horace Secrist:

“By Statistics we mean aggregate of facts affected to marked extend by multiplicity of causes numerically expressed, enumerated or estimated according to reasonable standard of accuracy ,collected in a systematic manner for a pre determined purpose and placed in relation to each other .”


4

Measures of Central Tendency


5

Def:Measures of Central Tendency A single expression

representing the whole group,is selected which may convey a fairly adequate idea about the whole group.

This single expression is

known as average.


6

Averages are central part of distribution and, therefore ,they are also called measures of central tendency.


7

Types of Measures central tendency:

There are five types ,namely 1.Arithmetic Mean (A.M) 2.Median 3.Mode 4.Geometric Mean (G.M) 5.Harmonic Mean (H.M)


8

Features of a good average 1.It should be rigidly defined 2.It should be easy to

understand and easy to calculate

3.It should be based on all the observations of the data


9

4.It should be easily subjected to further mathematical calculations

5.It should be least affected by fluctuations of sampling


10

Arithmetic Mean (A.M) The most commonly used measure of central tendency. When people ask about the “average" of a group of scores, they usually are referring to the mean.


11

The arithmetic mean is simply dividing the sum of variables by the total number of observations.


12

Arithmetic Mean for raw data is given by

n

x

nX

n

ii

xxxx n

∑=++++ == 1......321


13

Find mean for the data 17,16,21,18,13,16,12 and 11


14

Arithmetic Mean for Discrete Series

∑

∑

=

=++++ =++++

= n

ii

n

iii

n

xfxfxfxf

f

xf

ffffX nn

1

1

321

......

....332211


15

Arithmetic Mean for Continuous Series

CN

fdAX ×+= ∑


16

Calculation of Arithmetic mean in case of Continuous Series

Marks 0-10

10-20

20-30

30-40

40-50

50-60

No. of Students

10 20 30 50 40 30

From the following data calculate Arithmetic mean


17

Marks Mid values

(X)

No.of Students

(f)

d= X-45 10

f.d

0-10 5 10 -4 -40 10-20 15 20 -3 -60 20-30 25 30 -2 -60 30-40 35 50 -1 -50


18

Marks Mid values

(X)

No.of Students

(f)

d= X-45 10

f.d

40-50 45 40 0 0 50-60 55 30 1 30

N=180 ∑fd=-180


19

Solution

Let us take assumed mean =45

Calculation from assumed mean

Mean =

35180

10*18045x

=

−+=×+=

− ∑ CN

fdA


20

Calculation Of Arithmetic Mean in case of Less than series

Marks less than /up to

10 20 30 40 50 60

No. of students

10 30 60 110 150 180


21

Solution: Let us first convert Less than series into continuous series as follows

Marks 0-10 10-20

20-30

30-40

40-50

50-60

No. of students

10 20 30 50 40 30 180-150=30


22

Calculation Of Arithmetic Mean in case of more than series

Marks more than

0 10 20 30 40 50 60

No. of students

180 170 150

120 70 30 0


23

Solution: Let us first convert More than series into continuous series as follows

Marks 0-10 10-20

20-30

30-40

40-50 50-60

No. of students

10 20 30 50 40 30

180-170=10 170-150=20

70-30=40

30-0=30


24

Calculation of Arithmetic Mean in case of Inclusive series

From the following data ,calculate Arithmetic Mean

Marks 1-10 11-20 21-

30 31-40

41-50

51-60

No. of Students

10

20 30 50 40 30


25

Solution

Let us take assumed mean =45.5

Calculation from assumed mean

Mean = 35

18010*18045x

=

−+=×+=

− ∑ CN

fdA


26

Marks Mid values

No.of Students

d=X-45.5 10

f.d

0.5-10.5 5.5 10 -4 -40 10.5-20.5 15.5 20 -3 -60 20.5-30.5 25.5 30 -2 -60 30.5-40.5 35.5 50 -1 -50 40.5-50.5 45.5 40 0 0 50.5-60.5 55.5 30 1 30

N=180 ∑fd= -180


27

Calculation of Arithmetic Mean in case of continuous exclusive series when class intervals are unequal

From the following data ,calculate Arithmetic Mean

Marks 0-10 10-30 30-40 40-50 50-60

No. of Students

10

60 50 40 20


28

Since class intervals are unequal, frequencies have been adjusted to make the class intervals equal on the assumption that they are equally distributed throughout the class

Let us take assumed mean =45


29

Calculation of Deviations from assumed mean

Mean=

778.32180

1022045x

=

−+=×+=

− ∑ XCN

fdA


30

Marks Mid values

No. of Students

d= X-45.5 10

f.d

0-10 5 10 -4 -40 10-20 15 30 -3 -90 20-30 25 30 -2 -60 30-40 35 50 -1 -50 40-50 45 40 0 0 50-60 55 20 1 30

N=180 ∑fd=-220


31

Combined Arithmetic Mean (A.M)

An average daily wages of 10 workers in a factory ‘A’ is Rs.30 and an average daily wages of 20 workers in a factory B’ is Rs.15.Find the average daily wages of all the workers of both the factories.


32

Solution

Step 1;N1=10 N2=20

Step2:

=20

15;30 21 == XX

21

221112 NN

XNXNX++

=


33

Weighted Arithmetic Mean

The term ‘ weight’ stands for the relative importance of the different items of the series. Weighted Arithmetic Mean refers to the Arithmetic Mean calculated after assigning weights to different values of variable. It is suitable where the relative importance of different items of variable is not same


34

Weighted Arithmetic Mean is specially useful in problems relating to

1)Construction of Index numbers. 2)Standardised birth and death rates


35

Weighted Arithmetic Mean is given by

∑∑∑ =

WXW

X w

.


36

Mathematical Properties of Arithmetic Mean

1.The Sum of the deviations of the items from arithmetic mean is always Zero. i.e.

2.The sum of squared deviations of the items from arithmetic mean is minimum or the least

( ) 0=−∑ XX

( ) 02≤−∑ XX


37

3.The formula of Arithmetic

mean can be extended to

compute the combined

average of two or more

related series


38

4.If each of the values of a

variable ‘X’ is increased or decreased by some constant C, the arithmetic mean also increased or decreased by C .


39

Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k .


40

When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity


41

Merits Of Arithmetic Mean

1.Its easy to understand and easy to calculate.

2.It is based on all the items of the samples.

3.It is rigidly defined by a mathematical formula so that the same answer is derived by every one who computes it.


42

4.It is capable for further algebraic treatment so that its utility is enhanced


43

6.The formula of arithmetic mean can be extended to compute the combined average of two or more related series.


44

7.It has sampling stability .It is least affected by sampling fluctuations


45

Limitations of Arithmetic Mean

1.Affected by extreme values i.e . Very small or very big values in the data unduly affect the value of mean because it is based on all the items of the series.


46

2.Mean is not useful for studying the qualitative phenomenon.


47

Median

The middle score of the distribution when all the scores have been ranked.

If there are an even number of scores, the median is the average of the two middle scores.


48

In an ordered array, the median is the “middle” number If n or N is odd, the median is the

middle number If n or N is even, the median is the

average of the two middle numbers


49

Potential Problem with Means

Mean

Mean

Median

Median


50

Median

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5


51

Median for raw data

When given observation are even First arrange the items in ascending

order then

Median (M)=Average of Item

21

2+

+=NN


52

Find the Median for the raw data

25,55,5,45,15 and 35 Solution ;Arrange the items 5,15,25,35,45,55,here N=6 Median =Average of 3rd and 4th

item=30


53

Median for raw data

When given observation are odd First arrange the items in ascending

order then

Median (M)=Size of Item

21+

=N


54

Median for continuous series

cf

mN

LM ×

−+= 2

Where M= Median; L=Lower limit of the Median Class,m=Cumulative frequency above median class f=Frequency of the median class N=Sum of frequencies


55

Quartiles

The values of variate that divides the series or the series or the distribution into four equal parts are known as Quartiles .


56

The first Quartile (Q1),known as a lower Quartile is the value of variate below which 25% of the observations.


57

The Second Quartile known as middle Quartile(Q2)known as middle Quartile or median ,the value of variates below which 50% of the observations


58

The Third Quartile known as Upper Quartile(Q3)known as middle Quartile or median ,the value of variates below which 75 % of the observations.


59

thNSizeQ4

11

+= Item

thNSizeQ4

)1(33

+= Item


60

Octiles

The values of variate that divides the series or the distribution into eight equal parts are known as Octiles .

Each octile contains 12.5% of the total number of observations .


61

Since seven points are required to divide the data into 8 equal parts ,we have 7 octiles.


62

thNjSizeOj 8)1( +

= Item

thNSizeO8

)1(44

+= Item


63

Deciles

The values of variate that divides the series or the distribution into Ten equal parts are known as Deciles .

Each Decile contains 10% of the total number of observations .


64

Since 9 points are required to divide the data into 10 equal parts ,we have 9 deciles(D1 to D9)


65

thNjSizeDj 10)1( +

= Item

thNSizeD10

)1(55

+= Item


66

Percentiles The values of variate that divides

the series or the distribution into hundred equal parts are known as Percentiles .

Each percentile contains 10% of the total number of observations .

Since 99 points are required to divide the data into 10 equal parts ,we have 99 deciles(p1 to p99)


67

thNjSizePj 100)1( +

= Item

thNSizep100

)1(5050

+= Item


68

Relation Ship Between Partition Values 1.Q1=O2=P25 value of variate which

exactly 25% of the total number of observations

2.Q2=D5=P50,value of variate which exactly 50% of the total number of observations.

3. Q3=O6=P75,value of variate which exactly 75% of the total number of observations


69

Calculation of Median in case of Continuous Series

Marks 0-10 10-20 20-30 30-40 40-50 50-60

No. of Students

10 20 30 50 40 30

From the following data calculate Median


70

Marks No. of Students

(f)

Cumulative Frequencies

(c.f.) 0-10 10 10 10-20 20 30 20-30 30 60 30-40 50 110 40-50 40 150 50-60 30 180

N=180


71

Calculate size of N/2

902

1802

==N


72

1050

602

180

30 ×

−+=M

36630 =+=M


73

Merits of Median 1.Median is not affected by

extreme values . 2.It is more suitable average

for dealing with qualitative data ie.where ranks are given.

3.It can be determined by graphically.


74

Limitations of Median 1.It is not based all the items of

the series . 2.It is not capable of algebraic

treatment .Its formula can not be extended to calculate combined median of two or more related groups.


75

0 X

Y

M

Less than Cumulative curve

More than Cumulative Curve

Median By Graph

Q3 Q1 CI

Frequency N/2

3N/4

N/4


76

Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or

categorical data There may be no mode or several

modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode


77

Mode The most frequent score in the

distribution. A distribution where a single

score is most frequent has one mode and is called unimodal.


78

A distribution that consists of only one of each score has n modes.

When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie.


79

Calculate the mode from the following data of marks obtained by 10 students.

20,30,31,32,25,25,30,31,30,32

Mode (Z)=30


80

Mode for Continuous Series

cfff

ffLZ ×

−−

−+=

201

01

2Where Z= Mode ;L=Lower limit of the Mode Class f0 =frequency of the pre modal class f1=frequency of the modal class f2=frequency of the post modal class C=Class interval of Modal Class


81

Calculation of Mode :Continuous Series

Marks 0-10

10-20

20-30

30-40

40-50

50-60

No. of Students

10 20 30 50 40 30

From the following data calculate Mode


82

Marks No. of Students

(f) 0-10 10 10-20 20 20-30 30 30-40 50 f1 40-50 40 50-60 30

N=180

f0

f2


83

667.36667.630

104030502

605030

2 201

01

=+=

×

−−×−

+=

×

−−

−+=

Z

cfff

ffLZ


84

x 0

Y

Z 10 20 30 40 50 60

10

20

30

40

50

Calculation Mode Graphically


85

Relationship between Mean, Median and Mode

The distance between Mean and Median is about one third of distance between the mean and the mode.


86

Karl Pearson has expressed the relationship as follows. Mean –Mode=(Mean-Median)/3 Mean-Median=3(Mean-Mode)

Mode =3Median-2Mean Mean=(3Median-Mode)/2


87

Example

For a moderately skewed distribution of marks in statistics for a group of 200 students ,the mean mark and median mark were found to be 55.60 and 52.40.what is the modal mark?


88

Solution

Since in this case mean=55.60and median =52.40 applying ,we get

Mode=3median -2Mean =3(52.40)-2(55.60) Mode =46


89

Example

If Y=2+1.50X and mode of X is 15 ,What is mode of Y

Solution Y m=2+1.50*15=24.50


90

Merits of Mode

1.Mode is the only suitable average e.g. ,modal size of garments, shoes.,etc

2.It is not affected by extreme values.

3.Its value can be determined graphically.


91

Limitations of Mode

1.In case of bimodal /multi modal series ,mode cannot be determined.

2.It is not capable for further algebraic treatment, combined mode of two or more series cannot be determined.


92

3.It is not based on all the items of the series

4.Its value is significantly affected by the size of the class intervals


93

Geometric mean

nn

ii

nniG

x

xxxxx/1

1

21

=

=

∏=


94

Take the logarithms of each item of variable and obtain their total i.e ∑ log X

Calculate G M as follows

= ∑

nX

AntiMGlog

log.


95

Computation of G.M -Discrete Series Take the logarithms of each item of

variable and multiply with the respective frequencies obtain their total

i.e ∑ f .log X Calculate G M as follows

= ∑

NXf

AntiMGlog.

log.


96

Merits of Geometric Mean

1.It is based on all items of the series .

2 It is rigidly defined 3.It is capable for algebraic

treatment.


97

4.It is useful for averaging ratios and percentages rates are increase or decrease


98

Limitations of Geometric Mean

1.Its difficult to understand and calculate.

2.It cannot be computed when there are both negative and positive values in a series


99

3.It is biased for small values as it gives more weight to small values .


100

Calculation of G.M :Individual Series

From the following data calculate Geometric Mean Roll No 1 2 3 4 5 6

Marks 5 15 25 35 45 55


101

Computation of G.M :Individual Series

X log X 5 0.6990 15 1.1761 25 1.3979 35 1.5441 45 1.6532 55 1.7404

∑log X=8.2107


102

36.23)3685.1log(

62107.8

loglog.

==

=

= ∑

Anti

Al

nX

AntiMG


103

Find the average rate of increase population which in the first decade has increased by 10% ,in the second decade by 20% and third by 30%


104

Decade % rise Population at the end of the decade

logx

1 2 3

10 20 30

110 120 130

2.0414 2.0792 2.1139

∑log X=6.2345


105

8.119)0782.2log(

2345.6

loglog.

==

=

= ∑

Anti

Al

nX

AntiMG

Average Rate of increase in Population is 19.8%


106

Weighted Geometric Mean

=

∑∑

wXw

AntiMGlog.

log.


107

Harmonic Mean (H.M)

Harmonic Mean of various items of a series is the reciprocal of the arithmetic mean of their reciprocal .Symbolically,

nXXXX

NMH1.......111

.

321

++++=


108

Where X1,X2,X3…….X n refer to the value of various series.

N= total no. of series


109

Merits of Harmonic Mean

1.It is based on all items of the series .

2 It is rigidly defined 3.It is capable for algebraic

treatment.


110

4.It is useful for averaging measuring the time ,Speed etc


111

Limitations of Harmonic Mean

1.Its difficult to understand and calculate.

2.It cannot be computed when one or more items are zero


112

3.It gives more weight to smallest values . Hence it is not suitable for analyzing economic data .


113

Calculation of H.M :Individual Series

From the following data calculate Harmonic Mean Roll No

1 2 3 4 5 6

Marks

5 15 25 35 45 55


114

Computation of H.M :Individual Series

X l/x 5 0.2000

15 0.0666 25 0.0400 35 0.0286 45 0.0222 55 0.0182

∑(1/x)=0.3756


115

9744.153576.06

11

=

=

=∑ =

n

ii

H

x

nx


116

Compute AM ,GM and HM for the numbers 6,8,12,36

AM=(6+81+12++36)/4=15.50 GM=(6.8.12.36)1/4=12

H.M=9.93

361

121

81

61

4.+++

=MH


117

Weighted Harmonic Mean

∑∑=

)(i

i

i

Xww

HM


118

Find the weighted AM and HM of first n natural numbers ,the weights being equal to the squares of the Corresponding numbers.

X 1 2 3 …n

W 12 22 32 ..n2


119

Weighted ∑∑=

WiXiWi

AM.

)12(2)1(3

++

=nnn

++

+

=

++++

++++

6)12)(1(

4)1(.....321.....321

22

2222

3333

nnn

nnnn


120

∑∑=

)(i

i

i

Xww

HM

312

2)1(

6)12)(1(

.....321.....321 23222

+=

+

++

=

++++

++++

n

nn

nnnnn


121

The AM and GM of two observations are 5 and 4 respectively ,Find the two observations.

Solution : Let the Two numbers are a and b given

( a+b)/2=10 ;a + b=10 GM=4 ab=16 (a-b)2=(a+b)2-4ab=100-64=36

a-b=6 a=8 and b=2


122

The relationship between AM ,GM and HM

G2=A.H


123

1.The empirical relationship among mean, median and mode is ______

(a) mode=2median–3mean (b) mode=3median-2mean (c) mode=3mean-2median (d) mode=2mean-3median


124

1. The empirical relationship among mean, median and mode is ______

(a) mode=2median–3mean (b) mode=3median-2mean (c) mode=3mean-2median (d) mode=2mean-3median


125

2. In a asymmetrical distribution ____ (a) AM = GM = HM (b) AM<GM<AM (c) AM<GM>HM (d)

HMGMAM ≠≠


126

2. In a asymmetrical distribution ____

(a) AM = GM = HM (b) AM<GM<AM (c) AM<GM>HM (d)

HMGMAM ≠≠


127

3. The points of intersection of the “less than and more than” ogive corresponds to ___

(a) mean (b) mode (c) median (d) all of above


128

.3. The points of intersection of the “less than and more than” ogive corresponds to ___

(a) mean (b) mode (c) median (d) all of above


129

•4. Pooled mean is also called

(a) mean (b) geometric mean (c) grouped mean (d) none of these


130

4. Pooled mean is also called

(a) mean (b) geometric mean (c) grouped mean (d) none of these


131

5. Relation between mean, median and mode is

(a) mean–mode=2(mean-median) (b) mean–median=3(mean–mode)

(c) mean–median=2(mean–mode

(d) mean–mode=3(mean–median)


132

5. Relation between mean, median and mode is

(a) mean–mode=2(mean-median) (b) mean–median=3(mean–mode)

(c) mean–median=2(mean–mode

(d) mean–mode=3(mean–median)


133

6. The geometric mean of 9, 81, 729 is _____

(a) 9 (b) 27 (c) 81 (d) none of these


134

6. The geometric mean of 9, 81, 729 is _____

(a) 9 (b) 27 (c) 81 (d) none of these


135

7. The mean of the data set of 1000 items is 5. From each item 3 is subtracted and then each number is multiplied by 2. The new mean will be _____

(a) 4 (b) 5 (c) 6 (d) 7


136

7. The mean of the data set of 1000 items is 5. From each item 3 is subtracted and then each number is multiplied by 2. The new mean will be

(a) 4 (b) 5 (c) 6 (d) 7


137

8. If each item is reduced by 15, AM is ____

(a) reduced by 15 (b) increased by 15 (c) reduced by 10 (d) none of these


138

8. If each item is reduced by 15, AM is ____

(a) reduced by 15 (b) increased by 15 (c) reduced by 10 (d) none of these


139

9. In a series of values if one value is 0 ____

(a) both GM and HM are zero (b) both GM and HM are intermediate (c) GM is intermediate and HM is zero (d) GM is zero and HM is intermediate


140

9. In a series of values if one value is 0 ____

(a) both GM and HM are zero (b) both GM and HM are intermediate (c) GM is intermediate and HM is zero (d) GM is zero and HM is intermediate


141

10.Histogram is useful to determine graphically the value of

(a) Mean (b) Mode (c) Median (d) all of above


142

10.Histogram is useful to determine graphically the value of

(a) Mean (b) Mode (c) Median (d) all of above


143

11.The positional measure of central Tendency

(a) Arithmetic Mean (b) Geometric Mean (c) Harmonic Mean (d) Median


144

11.The positional measure of central Tendency

(a) Arithmetic Mean (b) Geometric Mean (c) Harmonic Mean (d) Median


145

12.The average has relevance for (a) Homogeneous population (b) Heterogeneous population (c) Both (d) none


146

12.The average has relevance for (a) Homogeneous population (b) Heterogeneous population (c) Both (d) none


147

13.The sum of individual observations is Zero When taken from

(a) Mean (b) Mode (C) Median (d) All the above


148

13.The sum of individual observations is Zero When taken from

(a) Mean (b) Mode (C) Median (d) All the above


149

14.The sum of absolute deviations from median is

(a) Minimum (b) Zero (C) Maximum (d) A negative figure


150

14.The sum of absolute deviations from median is

(a) Minimum (b) Zero (C) Maximum (d) A negative figure


151

15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none


152

15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none


153

16.The calculation of Speed and velocity

(a) G.M (b) A.M (c) H.M (d) none is used


154

16.The calculation of Speed and velocity

(a)G.M (b)A.M (c)H.M (d)none is used


155

17. The class having maximum frequency is called

A) Modal class B) Median class C) Mean Class D) None of these


156

17. The class having maximum frequency is called

A) Modal class B) Median class C) Mean Class D) None of these


157

18. The mode of the numbers 7, 7, 9, 7, 10, 15, 15, 15, 10 is

A) 7 B) 10 C) 15 D) 7 and 15


158

18. The mode of the numbers 7, 7, 9, 7, 10, 15, 15, 15, 10 is

A) 7 B) 10 C) 15 D) 7 and 15


159

19. Which of the following measures of central tendency is based on only 50% of the central values?

A) Mean B) Mode C) Median D) Both (a) and (b)


160

19. Which of the following measures of central tendency is based on only 50% of the central values?

A) Mean B) Mode C) Median D) Both (a) and (b)


161

20. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?

A) 17 B) 16 C) 15.75 D) 12


162

20. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?

A) 17 B) 16 C) 15.75 D) 12


163

21. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is

A) 13 B) 10.70 C) 11 D) 11.50


164

21. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is

A) 13 B) 10.70 C) 11 D) 11.50


165

22. In case of an even number of observations which of the following is median?

A) Any of the two middle-most value.. B) The simple average of these two

middle values C) The weighted average of these two

middle values. D) Any of these


166

22. In case of an even number of observations which of the following is median?

A) Any of the two middle-most value.. B) The simple average of these two middle

values C) The weighted average of these two middle

values. D) Any of these


167

23. A variable is known to be _______ if it can assume any value from a given interval.

A) Discrete B) Continuous C) Attribute D) Characteristic


168

23. A variable is known to be _______ if it can assume any value from a given interval.

A) Discrete B) Continuous C) Attribute D) Characteristic


169

24. Ogive is used to obtain. A) Mean B) Mode C) Quartiles D) All of these


170

24. Ogive is used to obtain. A) Mean B) Mode C) Quartiles D) All of these


171

25. The presence of extreme observations does not affect

A) A.M. B) Median C) Mode D) Any of these


172

25. The presence of extreme observations does not affect

A) A.M. B) Median C) Mode D) Any of these

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Measures Of Central Tendency

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