Measure of Central Tendency

Measures of Central TendencyMeasures of Central Tendency

Unit 1Unit 1

What are the objectives of study of What are the objectives of study of Averages or Measures of Central Averages or Measures of Central

Tendency?Tendency?

There are two main objectives:There are two main objectives:

1. 1. To get one single value that describes the To get one single value that describes the characteristics of the entire data.characteristics of the entire data.

To facilitate comparisonTo facilitate comparison

CHARACTERISTICS OF A GOOD CHARACTERISTICS OF A GOOD AVERAGEAVERAGE

1.It should be easy to understand.1.It should be easy to understand. 2. It should be simple to compute.2. It should be simple to compute. 3. It should be based on all the observations.3. It should be based on all the observations. 4. It should be rigidly defined.4. It should be rigidly defined. 5. It should be capable of further algebraic 5. It should be capable of further algebraic

treatment.treatment. 6.It should not be affected by extreme values.6.It should not be affected by extreme values.

There are the five different There are the five different Measures of Central TendencyMeasures of Central Tendency

1. Arithmetic Mean1. Arithmetic Mean 2. Median2. Median 3. Mode3. Mode 4.Geometric Mean4.Geometric Mean 5. Harmonic Mean5. Harmonic Mean

Depending upon the need and nature of study, Depending upon the need and nature of study,

proper measure is chosenproper measure is chosen

1. ARITHMETIC MEAN( A.M.)1. ARITHMETIC MEAN( A.M.)

It is most popular and widely used method for It is most popular and widely used method for representing the entire data.representing the entire data.

Calculation of Arithmetic mean- Calculation of Arithmetic mean- Ungrouped dataUngrouped data

i)i) Direct methodDirect method

ii)ii) Short-cut method.Short-cut method.

Calculation of Arithmetic mean- Calculation of Arithmetic mean- Ungrouped data( Direct method)Ungrouped data( Direct method)

If XIf X11, X, X22,…………….,X ,…………….,X nn denotes n no. of observations, denotes n no. of observations, the A.M. denoted by X is defined as:the A.M. denoted by X is defined as:

X = XX = X11+ X+ X2 2 +…….+ X +…….+ X nn

nn ∑ ∑ X X = = nn

i=1

n

Example:Example:The following figures relate to the monthly output of cloth of a The following figures relate to the monthly output of cloth of a

factory in a given year: Compute average monthly output.factory in a given year: Compute average monthly output.

MonthsMonths Output ( in ‘000 metres)Output ( in ‘000 metres)

JanJan 8080

FebFeb 8888

MarchMarch 9292

AprilApril 8484

MayMay 9696

JuneJune 9292

JulyJuly 9696

AugustAugust 100100

SeptSept 9292

OctOct 9494

NovNov 9898

DecDec 8686

Short-cut MethodShort-cut Method

To simplify the manual calculations, we may To simplify the manual calculations, we may sometimes usesometimes use

Change of Origin: We add or subtract (usually Change of Origin: We add or subtract (usually subtract) a constant to the individual subtract) a constant to the individual observation. observation.

Change of Scale: This is achieved by Change of Scale: This is achieved by multiplying or dividing each individual multiplying or dividing each individual observation by a constant.observation by a constant.

Combination of the above twoCombination of the above two

Solution by using short Cut MethodSolution by using short Cut Method

XX d = X -Ad = X -A d*= x-A d*= x-A

cc

8080

8888

9292

8484

9696

9292

9696

100100

9292

9494

9898

8686

Effect of change of origin and scale Effect of change of origin and scale on A.M.on A.M.

Remember A.M. is not independent of the Remember A.M. is not independent of the change of origin. This means that if same change of origin. This means that if same constant is subtracted from each observation it constant is subtracted from each observation it must be added in the final answer.must be added in the final answer.

Remember A.M. is not independent of the Remember A.M. is not independent of the change of scale. This means that if each change of scale. This means that if each observation is divided by the same constant observation is divided by the same constant the final answer must be multiplied by the the final answer must be multiplied by the same.same.

Theoretically we can select any value as Theoretically we can select any value as

Assumed mean. However, for the purpose of Assumed mean. However, for the purpose of

Simplification of calculation work, the selectedSimplification of calculation work, the selected

value should be as nearer to the value X asvalue should be as nearer to the value X as

possible.possible.

X = A+ X = A+ ∑d* x c∑d* x c

nn

Formula Of Arithmetic Mean For Formula Of Arithmetic Mean For Ungrouped data with frequencyUngrouped data with frequency

Suppose there be n values XSuppose there be n values X11, X, X22,…………….,X ,…………….,X nn

out of which Xout of which X1 1 has occurred fhas occurred f11 times, X times, X2 2 has has

occurred foccurred f22 times, …………., X times, …………., Xnn has has

occurred foccurred fnn times times

X = ∑ f .X X = ∑ f .X

∑ ∑ff

ApplicationApplicationThe following is the frequency distribution of age of 670 studentsThe following is the frequency distribution of age of 670 students

of a school Compute the arithmetic mean of the data.of a school Compute the arithmetic mean of the data.

Age (in years) XAge (in years) X No. of students No. of students (frequency)(frequency)

55 2525

66 4545

77 9090

88 165165

99 112112

1010 9696

1111 8181

1212 2626

1313 1818

1414 1212

Direct MethodDirect Method

Age (in Age (in years) Xyears) X

No. of students No. of students (frequency)(frequency)

f.xf.x

55 2525

66 4545

77 9090

88 165165

99 112112

1010 9696

1111 8181

1212 2626

1313 1818

1414 1212

∑∑f.x =5918f.x =5918

Short -CutShort -Cut

Calculation of Arithmetic Mean of Calculation of Arithmetic Mean of the Grouped Datathe Grouped Data

Class IntervalsClass Intervals FrequencyFrequency

0-100-10 33

10-2010-20 88

20-3020-30 1212

30-4030-40 1515

40-5040-50 1818

50-6050-60 1616

60-7060-70 1111

70-8070-80 55

Direct methodDirect method

X = ∑ f .X X = ∑ f .X

∑ ∑ff

X= mid –point of various classesX= mid –point of various classes

Short-cut methodShort-cut method

Class Class IntervalsIntervals

FrequencyFrequency Mid Mid values xvalues x

fxfx

0-100-10 33

10-2010-20 88

20-3020-30 1212

30-4030-40 1515

40-5040-50 1818

50-6050-60 1616

60-7060-70 1111

70-8070-80 55

Short-cut methodShort-cut method

Class Class IntervalsIntervals

FrequencyFrequency Mid Mid values xvalues x

d=x-35d=x-35 fdfd

0-100-10 33

10-2010-20 88

20-3020-30 1212

30-4030-40 1515

40-5040-50 1818

50-6050-60 1616

60-7060-70 1111

70-8070-80 55

X =A+ ∑f.d X =A+ ∑f.d

∑ ∑ff

= 35+ 660= 35+ 660

8888

= 42.5= 42.5

Correcting Incorrect ValuesCorrecting Incorrect Values

E.G. 1 The average weekly wage for a group of E.G. 1 The average weekly wage for a group of 25 persons working in a factory was calculated 25 persons working in a factory was calculated to be Rs. 378.40. It was later discovered that to be Rs. 378.40. It was later discovered that one figure was misread as 160 instead of the one figure was misread as 160 instead of the correct value Rs. 200. Calculate the correct correct value Rs. 200. Calculate the correct average wageaverage wage . 380. 380

2.The mean of 200 observations was 50. Later on, it was 2.The mean of 200 observations was 50. Later on, it was discovered that two observations were wrongly read as 92 and discovered that two observations were wrongly read as 92 and 8 instead of 192 and 88. Find out the correct mean.8 instead of 192 and 88. Find out the correct mean.

50.950.9

Properties of Arithmetic MeanProperties of Arithmetic Mean

1.1. The sum of deviations of all the observations The sum of deviations of all the observations taken from their arithmetic mean is always taken from their arithmetic mean is always zero.zero.

i.e. ∑ (X-X ) =0 , where X =Actual meani.e. ∑ (X-X ) =0 , where X =Actual mean

ExampleExample

XX X-XX-X

1010 -20-20

2020 -10-10

3030 00

4040 +10+10

5050 +20+20

∑∑X = 150X = 150 ∑ ∑ (X- X ) = 0(X- X ) = 0

According to this property, the arithmetic mean According to this property, the arithmetic mean

serves as a point of balance or centre of gravity serves as a point of balance or centre of gravity

of the distribution; since sum of the positiveof the distribution; since sum of the positive

deviations( i.e. deviations of observations whichdeviations( i.e. deviations of observations which

are greater than X ) is equal to the sum ofare greater than X ) is equal to the sum of

negative deviations ( i.e. deviations of negative deviations ( i.e. deviations of

observations which are less than X .observations which are less than X .

2. The sum of the squares of deviations is 2. The sum of the squares of deviations is minimum when taken from their arithmetic minimum when taken from their arithmetic mean.mean.

∑ ∑ ( X- A)( X- A)2 2 is minimum if A= Xis minimum if A= X

ExampleExample

XX X-XX-X (X-X)(X-X)22

22 -2-2 44

33 -1-1 11

44 00 00

55 +1+1 11

66 +2+2 44

∑∑X = 20X = 20 ∑ ∑ (X-X )=0(X-X )=0 ∑∑((X-X)((X-X)2 2 =10=10

If the deviation is taken from any other valueIf the deviation is taken from any other value

(except actual mean), the sum of the squared(except actual mean), the sum of the squared

deviation would be greater than 10.deviation would be greater than 10.

3. 3. Arithmetic mean is capable of being treated Arithmetic mean is capable of being treated algebricallyalgebrically..

If XIf X1 1 and nand n1 1 are mean and number of are mean and number of

observations of a group1 and if Xobservations of a group1 and if X2 2 and nand n2 2 are are

mean and number of observations of a group mean and number of observations of a group 2, then the mean X of the combined series 2, then the mean X of the combined series [ n[ n11+n+n22] observations is given by ] observations is given by

X =X =[n1X1+n2X2]

[n1+n2]

Illustration 1Illustration 1

There are two of a company employing 100 and There are two of a company employing 100 and

80 employees respectively. If average monthly80 employees respectively. If average monthly

Salary paid by two branches are Rs.4570 and Salary paid by two branches are Rs.4570 and

Rs.6750 Respectively, find the average monthly Rs.6750 Respectively, find the average monthly

salary of the employees of the company as a salary of the employees of the company as a

whole.whole.

Ans:5538.89Ans:5538.89


The average marks in statistics of 100 students ofThe average marks in statistics of 100 students of

a class was 72. The average marks of boys was a class was 72. The average marks of boys was

75, while their number was 70. 75, while their number was 70.

Find the average marks of girls in the class.Find the average marks of girls in the class.

Ans: 65Ans: 65


The mean age of a combined group of men andThe mean age of a combined group of men and

women is 30 years. If the mean age of the group of women is 30 years. If the mean age of the group of men is 32 and that of the group of women is 25. men is 32 and that of the group of women is 25. Find the percentage of men and women in the Find the percentage of men and women in the group.group.

Ans: 71.43 and 28.57Ans: 71.43 and 28.57


The average rainfall for a week, excluding The average rainfall for a week, excluding

Sunday, was 10 cms. Due to heavy rainfall on Sunday, was 10 cms. Due to heavy rainfall on

Sunday, the average for the week rose to 15 cms. Sunday, the average for the week rose to 15 cms. How much rainfall was on Sunday?How much rainfall was on Sunday?

Ans:45cms.Ans:45cms.

Illustration 5Illustration 5There are 130 teachers and 100 non-teaching employees in a college.There are 130 teachers and 100 non-teaching employees in a college.

The respective distribution of their monthly salaries are given in theThe respective distribution of their monthly salaries are given in the

Following table.Following table.

TeachersTeachers Non-teaching EmployeesNon-teaching Employees

Monthly Monthly Salary (Rs)Salary (Rs)

FrequencyFrequency Monthly Monthly Salary (Rs)Salary (Rs)

FrequencyFrequency

4000-50004000-5000 1010 1000-20001000-2000 2121

5000-60005000-6000 1616 2000-30002000-3000 4545

6000-70006000-7000 2222 3000-400003000-40000 2828

7000-80007000-8000 6767 40000-500040000-5000 66

8000-90008000-9000 1515

TotalTotal 130130 TotalTotal 100100

From the above data find: From the above data find:

i)i) Average monthly salary of a teacher.Average monthly salary of a teacher.

ii)ii) Average monthly salary of a non-teaching Average monthly salary of a non-teaching

Employee.Employee.

iii) Average monthly salary of a college iii) Average monthly salary of a college employeeemployee

( teaching and non-teaching ). ( teaching and non-teaching ).

To find Missing FrequencyTo find Missing FrequencyThe following is the distribution of weights ( in lbs.) of 60 The following is the distribution of weights ( in lbs.) of 60

students of a class : students of a class :

WeightsWeights No. of No. of studentsstudents

93-9793-97 22

98-10298-102 55

103-107103-107 1212

108-112108-112 ??

113-117113-117 1414

118-122118-122 ??

123-127123-127 33

128-132128-132 11

TotalTotal 6060

If the mean weight of the students is 110.917, If the mean weight of the students is 110.917, find the missing frequencies.find the missing frequencies.

Ans: 17 and 6Ans: 17 and 6

ExampleExampleFind out the missing item (x) of the following frequencyFind out the missing item (x) of the following frequency

distribution whose arithmetic mean is 11.37.distribution whose arithmetic mean is 11.37.

XX ff

55 22

77 44

xx 2929

1111 5454

1313 1111

1616 88

2020 44

Merits and Demerits of Arithmetic Merits and Demerits of Arithmetic MeanMean

Out of all averages arithmetic mean is the mostOut of all averages arithmetic mean is the most

Popular average in statistics because of thePopular average in statistics because of the

following following meritsmerits::1.1. It is simplest average to understand andIt is simplest average to understand and

easiest to compute.easiest to compute.

2.2. Arithmetic mean isArithmetic mean is rigidly defined by a mathematical rigidly defined by a mathematical

formula. As a result of that everyone who compute the formula. As a result of that everyone who compute the average from a given set of data get the same answer.average from a given set of data get the same answer.

3.3. Calculation of arithmetic mean is based on all theCalculation of arithmetic mean is based on all the

observations and hence, it can be considered as observations and hence, it can be considered as

representative of the given data.representative of the given data.

4. Being determined by a rigid formula, it is capable 4. Being determined by a rigid formula, it is capable of being treated mathematically and hence, widely of being treated mathematically and hence, widely

used in statistical analysis.used in statistical analysis.

5.5. The mean is typical in the sense that it is the centre The mean is typical in the sense that it is the centre

of gravity, balancing either side of it.of gravity, balancing either side of it.

DemeritsDemeritsAlthough, arithmetic mean satisfies most of the Although, arithmetic mean satisfies most of the

properties of ideal average, it has certain drawbacks andproperties of ideal average, it has certain drawbacks and

should be used with care. should be used with care. Some demerits of arithmeticSome demerits of arithmetic

mean are:mean are:

Since the value of mean depends upon each and Since the value of mean depends upon each and

every item of the series, extreme values( i.e. very every item of the series, extreme values( i.e. very small and large items) affect the value of the small and large items) affect the value of the average. Hence, it can not represent data consisting average. Hence, it can not represent data consisting

of some extreme observations.of some extreme observations.

22. It can neither be determined by inspection nor by. It can neither be determined by inspection nor by

graph.graph.

3. Arithmetic mean can not be computed for a 3. Arithmetic mean can not be computed for a qualitative data; like data on intelligence, honesty, qualitative data; like data on intelligence, honesty, smoking habit etc.smoking habit etc.

4.4. Arithmetic mean can not be computed when classArithmetic mean can not be computed when class

intervals have open ends. In such cases, to compute intervals have open ends. In such cases, to compute mean some assumptions regarding the size of the mean some assumptions regarding the size of the class interval (width) of the open-end classes class interval (width) of the open-end classes should be made. should be made.

MedianMedian

The median by definition refers to the middle The median by definition refers to the middle value in a distribution when they are arranged in their value in a distribution when they are arranged in their ascending or descending order of their magnitude.ascending or descending order of their magnitude.

In other words, median of distribution is that valueIn other words, median of distribution is that valueWhich divides it into two equal parts. It is calledWhich divides it into two equal parts. It is calledPositional average because its value depends upon thePositional average because its value depends upon thePosition of an item and not on its magnitude.Position of an item and not on its magnitude.

Weighted Arithmetic MeanWeighted Arithmetic Mean

X = X = ƩWXƩWX

ƩWƩW

Determination of Median Determination of Median

a)a) When individual observations are givenWhen individual observations are given::

Example, If the income of five employees areExample, If the income of five employees are

Rs. 900, 950, 1020, 1200 and 1280. CalculateRs. 900, 950, 1020, 1200 and 1280. Calculate

the median.the median.

Example on Calculation of Median Example on Calculation of Median for even no. of observationsfor even no. of observations

If the income of six employees areIf the income of six employees are

Rs. 900, 950, 1020, 1200 , 1280, 1300. Rs. 900, 950, 1020, 1200 , 1280, 1300.

Calculate the median.Calculate the median.

Note: there is no single middle position valueNote: there is no single middle position value

and median is taken to be the A.M. of two and median is taken to be the A.M. of two

middle most items.middle most items.

Hence, in case of median may be found byHence, in case of median may be found by

averaging two middle position values.averaging two middle position values.

Observations:Observations: The following steps are involved in the calculationThe following steps are involved in the calculation of Medianof Median The given observations are arranged in eitherThe given observations are arranged in either ascending or descending order of magnitude.ascending or descending order of magnitude.ii Given that there are n observations, the median is ii Given that there are n observations, the median is

given by:given by: (n+1) th observation , when n is odd.(n+1) th observation , when n is odd. 22iii When n is even, median is n th observation ie iii When n is even, median is n th observation ie

A.M of middle values. 2A.M of middle values. 2

IllustrationsIllustrations

Obtain the value of median from the followingObtain the value of median from the following

data:data:

391 384 591 407 672 522 777 753 2488391 384 591 407 672 522 777 753 2488

14901490

b) Computation of Median – Discrete b) Computation of Median – Discrete seriesseriesExample: From the Example: From the

following data find following data find the value of medianthe value of median

Income Income (Rs.)(Rs.)

No. of No. of personspersons

10001000 2424

15001500 2626

800800 1616

20002000 2020

25002500 66

18001800 3030

Income (Rs.)Income (Rs.) No. of personsNo. of persons Cumulative Cumulative Frequency(c.f.)Frequency(c.f.)

800800 1616

10001000 2424

15001500 2626

18001800 3030

20002000 2020

25002500 66

StepsSteps

i.i. Arrange the data in ascending or decendingArrange the data in ascending or decending

order of magnitude.order of magnitude.

ii Find out the cumulative frequenciesii Find out the cumulative frequencies

iii iii Apply the formula: Median = size of n+1Apply the formula: Median = size of n+1

22

iv Now look at the cumulative frequency iv Now look at the cumulative frequency column and find the total which is either equalcolumn and find the total which is either equal

to n+1 or next higher to that and determineto n+1 or next higher to that and determine

22

the value of the variable corresponding to it. the value of the variable corresponding to it. That gives the value of median.That gives the value of median.

c) Calculation of Median- c) Calculation of Median- Continuous SeriesContinuous SeriesExample: Calculate the Example: Calculate the

median of the following median of the following frequency Distribution.frequency Distribution.

ClassesClasses FrequencyFrequency c.fc.f

0-100-10 55

10-2010-20 1212

20-3020-30 1414

30-4030-40 1818

40-5040-50 1313

50-6050-60 88

Determine the median class. In grouped data useDetermine the median class. In grouped data use

(n/2) as the median class and not n+1 (n/2) as the median class and not n+1

22

Apply the Formula:Apply the Formula:Median= L + N/2 – c.f x iMedian= L + N/2 – c.f x i

ff

L= lower boundary of the median classL= lower boundary of the median class

c.f = cumulative frequency of the class preceding to the c.f = cumulative frequency of the class preceding to the median class.median class.

f = frequency of the median class.f = frequency of the median class.

i = The class interval of the median class.i = The class interval of the median class.

Median= Median=

More Illustration: calculate medianMore Illustration: calculate median

MarksMarks No. of studentsNo. of students

5-105-10 77

10-1510-15 1515

15-2015-20 2424

20-2520-25 3131

25-3025-30 4242

30-3530-35 3030

35-4035-40 2626

40-4540-45 1515

45-5045-50 1010

Illustration :calculate medianIllustration :calculate median

Weight Weight (in (in

grams)grams)

No. of applesNo. of apples

410-419410-419 1414

420-429420-429 2020

430-439430-439 4242

440-449440-449 5454

450-459450-459 4545

460-469460-469 1818

470-479470-479 77

Example of open-end classExample of open-end classClass Class

IntervalsIntervalsfrequencyfrequency c.f.c.f.

Less than 425Less than 425 22425-475425-475 88475-525475-525 3333525-575525-575 8080575-625575-625 170170625-675625-675 213213675-725675-725 213213725-775725-775 145145775-825775-825 9191825-875825-875 4545

Determination of Missing FrequenciesDetermination of Missing Frequencies

If the frequencies of some classes are missing,If the frequencies of some classes are missing,

however, the median of the distribution ishowever, the median of the distribution is

known, then these frequencies can be determinedknown, then these frequencies can be determined

by the use of median formula.by the use of median formula.

ExampleExampleThe following table gives the distribution of daily wages of 900The following table gives the distribution of daily wages of 900

workers. However, the frequencies of the classes 40-50 and 60-70 areworkers. However, the frequencies of the classes 40-50 and 60-70 are

Missing. Missing. If the median of the distribution is Rs.59.25If the median of the distribution is Rs.59.25, find the missing, find the missing

frequencies.frequencies.

Wages (Rs.)Wages (Rs.) FrequencyFrequency

30-4030-40 120120

40-5040-50 ??

50-6050-60 200200

60-7060-70 ??

70-8070-80 185185

ff11=145, f=145, f22=250=250

Calculaton of Median when class Calculaton of Median when class Intervals are UnequalIntervals are Unequal

Calculate the median from the following data:Calculate the median from the following data:


0-100-10 55

10-3010-30 1515

30-6030-60 3030

60-8060-80 88

80-9080-90 22

Note: when the class intervals are Note: when the class intervals are unequal, unequal, there there is no need to make any adjustment to make it is no need to make any adjustment to make it equal.equal.

Ans:40Ans:40

Merits and Demerits of MedianMerits and Demerits of MedianMeritsMerits1)1) It is easy to understand and simple to calculate.It is easy to understand and simple to calculate.

2)2) Median can be determined even when class intervals have open-Median can be determined even when class intervals have open-ends (since only the position not the values of item must be ends (since only the position not the values of item must be known.)known.)

3)3) The median is recommended if the classes are not equal width, The median is recommended if the classes are not equal width, since it is easier to compute than mean.since it is easier to compute than mean.

4)4) Extreme values do not influence the median. In fact when Extreme values do not influence the median. In fact when extreme values are present in the data , the median is more extreme values are present in the data , the median is more satisfactory measure of average than mean. For example, the satisfactory measure of average than mean. For example, the median of 10,20,30, 40 and 150 would be 30 whereas mean is median of 10,20,30, 40 and 150 would be 30 whereas mean is 50. 50. Hence, very often when extreme values are present in a set of Hence, very often when extreme values are present in a set of observations, the median is a more satisfactory measure of observations, the median is a more satisfactory measure of central tendency. central tendency.

5)5) The value of median can be determined The value of median can be determined graphically whereas the value of A.M. can not graphically whereas the value of A.M. can not be determined graphically.be determined graphically.

Demerits or limitations of MedianDemerits or limitations of Median

1.1. For calculating median, it is necessary to arrange For calculating median, it is necessary to arrange the data in order of magnitude, which may be the data in order of magnitude, which may be cumbersome task, particularly when the number of cumbersome task, particularly when the number of observations are very large.observations are very large.

2.2. It is not capable of algebraic treatment. For example It is not capable of algebraic treatment. For example median can not be used for determining the median can not be used for determining the combined median of two or more groups as is combined median of two or more groups as is possible in case of mean.possible in case of mean.

3.3. It is not based on all the values.It is not based on all the values.

Related Positional AveragesRelated Positional Averages

Besides median, there are other measures which Besides median, there are other measures which dividedivide

a series into equal parts. Important among these area series into equal parts. Important among these are

quartiles, deciles and percentiles.quartiles, deciles and percentiles.

QuartilesQuartiles

Quartiles are those values of the variable which divide theQuartiles are those values of the variable which divide the

total frequency into four equal parts, deciles divide thetotal frequency into four equal parts, deciles divide the

total frequency into 10 equal parts and the percentiles divide total frequency into 10 equal parts and the percentiles divide

the total frequency into 100 equal parts .the total frequency into 100 equal parts .

Computation of QuartilesComputation of QuartilesSince there values are needed to divide a distribution into four Since there values are needed to divide a distribution into four

equal parts, there are three quartiles , equal parts, there are three quartiles , QQ11, , QQ2 2 andand QQ3, 3, known as the known as the

first, first, second and third quartiles respectively.second and third quartiles respectively.

In Individual observations and discrete seriesIn Individual observations and discrete series

QQ11=Size of N+1 th item.=Size of N+1 th item.

44

QQ22=Size of N+1 th item.=Size of N+1 th item.

22

QQ33=Size of 3(N+1) th item=Size of 3(N+1) th item

44

ExampleExample

Price of a commodity in 8 different shops are asPrice of a commodity in 8 different shops are as

follows. Calculate the quartiles.follows. Calculate the quartiles.

Price (Rs): 208, 205,212,209, 207,210,208,206.Price (Rs): 208, 205,212,209, 207,210,208,206.

Solution:Solution:

Arrange in their ascending order of magnitude.Arrange in their ascending order of magnitude.

205,206,207,208,209,210,212205,206,207,208,209,210,212

The first quartile is-The first quartile is-

QQ11=Size of N+1 =Size of N+1 th th value in the seriesvalue in the series

44 =(8+1) =(8+1) thth value value 44 = (2.25) = (2.25) th th valuevalue = 2= 2ndnd value + 0.25( 3 value + 0.25( 3rdrd value- 2 value- 2ndnd value ) value ) = 206+ 0.25(207-206)= 206+ 0.25(207-206) = 206.25/-= 206.25/-

Similarly, calculate Similarly, calculate QQ2 2 andand QQ33..

For Continuous SeriesFor Continuous Series

QQ11=Size of N th item.=Size of N th item.

44

QQ22=Size of N th item.=Size of N th item.

22

QQ33=Size of 3(N) th item=Size of 3(N) th item

44

Deciles and PercentilesDeciles and Percentiles

DD11 = (N+1) th item ( = (N+1) th item ( in individual and discrete series)in individual and discrete series)

1010

DD1 = 1 = N th item ( N th item ( in continuous series)in continuous series)

1010………………....

PP11 = (N+1) th item ( = (N+1) th item ( in individual and discrete series)in individual and discrete series)

100100

PP11 = N th item ( = N th item ( in continuous series)in continuous series)

100100

IllustrationIllustration

From the following data compute the value of upperFrom the following data compute the value of upper

quartiles (Qquartiles (Q33) and lower quartiles (Q) and lower quartiles (Q11) ,D) ,D22, P, P55 and and

PP90.90.MarksMarks No. of StudentsNo. of Students

Below 10Below 10 88

10-2010-20 1010

20-4020-40 2222

40-6040-60 2525

60-8060-80 1010

Above 80Above 80 55

MarksMarks No. of StudentsNo. of Students C.f.C.f.

Below 10Below 10 88 88

10-2010-20 1010 1818

20-4020-40 2222 4040

40-6040-60 2525 6565

60-8060-80 1010 7575

Above 80Above 80 55 80=N80=N

QQ11=Size of N th item = 80/4= 20=Size of N th item = 80/4= 20thth item. item.

44

Hence QHence Q1 1 lies in the class 20-40.lies in the class 20-40.

QQ11= L+ N/4–c.f. x i= L+ N/4–c.f. x i

ff

=21.82=21.82

QQ3 3 = L+ N/4–c.f. x I = L+ N/4–c.f. x I

ff

= 56.= 56.

DD2 = 2 = Size of 2N th item = 16Size of 2N th item = 16thth item. item.

1010

Hence DHence D22 lies in the class 10-20. lies in the class 10-20.

DD2= 2= L + 2N/10 –c.f. X iL + 2N/10 –c.f. X i

ff

= 18= 18

PP5 = 5 = Size of 5N th item = 4Size of 5N th item = 4thth

1010

Hence PHence P5 5 lies in the class 0-10.lies in the class 0-10.

MODEMODEMode is that value which occurs maximum number of times in a Mode is that value which occurs maximum number of times in a

distribution. In other words, modal value is that value in a seriesdistribution. In other words, modal value is that value in a series

of observations which occurs with the highest frequency.of observations which occurs with the highest frequency.

For example, what is the mode of the following series For example, what is the mode of the following series 3,5,8,5,4,5,9,3.3,5,8,5,4,5,9,3.

Ans. is 5 , since this value occurs Ans. is 5 , since this value occurs more frequently more frequently than any of the than any of the

others.others.

ExampleExample

Calculate modal size of Calculate modal size of shoes from the followingshoes from the following

data:data:

Size of shoesSize of shoes No. of shoesNo. of shoes

55 1010

66 2020

77 2525

88 4040

99 2222

1010 1515

1111 66

Remarks:Remarks:i) When there are two or more values having the same i) When there are two or more values having the same

maximum frequency, one cannot say which is the modal maximum frequency, one cannot say which is the modal value and hence mode is said to be ill-defined. Such a series value and hence mode is said to be ill-defined. Such a series is also known as is also known as bi-modal or multi-modalbi-modal or multi-modal..

For example, observe the following data:For example, observe the following data: Income( Rs): 110, Income( Rs): 110, 120120,,130130,,120120,110,140,,110,140,130130,,120120,,130,130,140.140. Since 120 and 130 have the same maximum frequency i.e. 3,Since 120 and 130 have the same maximum frequency i.e. 3, mode will be 120 and 130. So, in this case mode is ill-mode will be 120 and 130. So, in this case mode is ill-

defined.defined.

When mode is ill-defined, its value may be determined by theWhen mode is ill-defined, its value may be determined by the following formula based upon the relationship between following formula based upon the relationship between

mean, median and mode:mean, median and mode:

Mode = 3Median – 2 Mean.Mode = 3Median – 2 Mean.

Remarks:Remarks:

ii)ii) If the frequency of each possible value is If the frequency of each possible value is same, there is no mode.same, there is no mode.

Ex: Ex: XX 55 1010 1515 2020 2525 3030 3535 4040

ff 66 66 66 66 66 66 66 66

Calculation of Mode- Continuous SeriesCalculation of Mode- Continuous Series

Step 1Step 1

By inspection identify the modal class ( the classBy inspection identify the modal class ( the class

having the highest frequency).having the highest frequency).

Step 2Step 2

Determine the value of mode by Applying theDetermine the value of mode by Applying the

following formulafollowing formula

M= L+ M= L+ ∆∆1 x i 1 x i

∆ ∆11+ ∆+ ∆2 2

Where L = lower boundary of the modal classWhere L = lower boundary of the modal class

∆∆1= 1= the difference between the frequency of thethe difference between the frequency of the

modal class and the frequency of the pre-modal class and the frequency of the pre-

modal class( i.e. preceding class).modal class( i.e. preceding class).

∆∆2 = 2 = the difference between the frequency of thethe difference between the frequency of the

modal class and the frequency of the post-modal class and the frequency of the post-

modal class( i.e. succeeding class).modal class( i.e. succeeding class).

i = the class interval of the modal class i = the class interval of the modal class

IllustrationIllustration

Calculate mode from the Calculate mode from the following data:following data:


Above 0Above 0 8080Above 10Above 10 7777Above 20Above 20 7272Above 30Above 30 6565Above 40Above 40 5555Above 50Above 50 4343Above 60Above 60 2828Above 70Above 70 1616Above 80Above 80 1010Above 90Above 90 88Above 100Above 100 00

Solution:Solution:

Since it is more than typeSince it is more than type

cumulative frequencycumulative frequency

distribution, we willdistribution, we will

convert it into simpleconvert it into simple

frequency distribution.frequency distribution.


0- 100- 10 3310-2010-20 5520-3020-30 7730-4030-40 101040-5040-50 121250-6050-60 151560-7060-70 121270-8070-80 6680-9080-90 2290-10090-100 88

By inspection the modal class is 50-60.By inspection the modal class is 50-60.

M= L+ M= L+ ∆∆1 x i 1 x i

∆ ∆11+ ∆+ ∆2 2

==

Find the Mode of the data given Find the Mode of the data given below:below:

Weight (kg)Weight (kg) No. of studentsNo. of students

93-9793-97 22

98-10298-102 55

103-107103-107 1212

108-112108-112 1717

113-117113-117 1414

118-122118-122 66

123-127123-127 33

128-132128-132 11

By inspection mode lies in the class 108-112.By inspection mode lies in the class 108-112.

But the boundaries of this class is 107.5-112.5.But the boundaries of this class is 107.5-112.5.

Mode= L + Mode= L + ∆∆1 x i 1 x i

∆∆11+ ∆+ ∆22

L = 107.5L = 107.5

∆∆1 = 1 = (17-12)=5(17-12)=5

∆∆2 = 2 = (17-14)=3(17-14)=3

Mode = 107.5+ 3.125 = 110.625.Mode = 107.5+ 3.125 = 110.625.

Calculation of Mode by Method of Calculation of Mode by Method of Grouping & AnalysisGrouping & Analysis

In frequency distribution, sometimes mode can not beIn frequency distribution, sometimes mode can not be

determined just by inspection ( looking at the table) when thedetermined just by inspection ( looking at the table) when the

maximum frequency and the preceding it and succeeding it maximum frequency and the preceding it and succeeding it

is very small. In such cases it is desirable to prepare a is very small. In such cases it is desirable to prepare a

Grouping table and an analysis table. These tables help us in Grouping table and an analysis table. These tables help us in

determining the mode.determining the mode.

Ex-Discrete SeriesEx-Discrete Series

Calculate the value of mode for the following table:Calculate the value of mode for the following table:

Marks (X)Marks (X) FrequencyFrequency

1010 88

1515 1212

2020 3636

2525 3535

3030 2828

3535 1818

4040 99

Solution: Solution: Since it is difficult to say by inspection which is the Since it is difficult to say by inspection which is the modal value, we prepare grouping and analysis tables.modal value, we prepare grouping and analysis tables.

Steps of preparing a grouping table:Steps of preparing a grouping table:1.1. Prepare a table consisting of six columns in addition to a Prepare a table consisting of six columns in addition to a

column for various values of X.column for various values of X.2.2. In the first column write the frequencies against the various In the first column write the frequencies against the various

values of X as given in the question and mark the highest values of X as given in the question and mark the highest frequency.frequency.

3.3. In second column, starting from the top, take the sum of In second column, starting from the top, take the sum of frequencies which are grouped in twos and mark the highest frequencies which are grouped in twos and mark the highest frequency.frequency.

4. In 34. In 3rdrd column, leave the first frequency and then group the column, leave the first frequency and then group the remaining in twos and take the sum of frequencies and mark remaining in twos and take the sum of frequencies and mark the highest frequency.the highest frequency.

5. 5. In col 4, group the frequencies in threes, take the sum of In col 4, group the frequencies in threes, take the sum of frequencies of each group and mark the highest frequency.frequencies of each group and mark the highest frequency.

6. In col 5, leave the first frequency, then group them in threes. 6. In col 5, leave the first frequency, then group them in threes. Take the sum of frequencies of each group and mark the Take the sum of frequencies of each group and mark the highest frequency.highest frequency.

7. In col 6, leave the first two frequencies, then group them in 7. In col 6, leave the first two frequencies, then group them in threes. Take the sum of frequencies of each group and mark threes. Take the sum of frequencies of each group and mark the highest frequency.the highest frequency.

XX Col 1(f)Col 1(f) Col 2Col 2 Col 3Col 3 Col 4Col 4 Col 5Col 5 Col 6Col 6

1010 88

56561515 1212

83832020 3636 **

99992525 3535

81813030 2828

55553535 1818

4040 99

20

71

46

48

63

27

After this grouping, prepare an After this grouping, prepare an Analysis tableAnalysis table

Col Col nono

1010 1515 2020 2525 3030 3535 4040

11 //

22 // //

33 // //

44 // // //

55 // // //

66 // // //

TotalTotal 11 44 55 33 11

The values against which frequencies are the The values against which frequencies are the highest are marked in the grouping table and highest are marked in the grouping table and entered by means of a bar (/) in the Analysis entered by means of a bar (/) in the Analysis table.table.

Total is maximum corresponding to the value Total is maximum corresponding to the value 25.So mode is 25.25.So mode is 25.

Calculate mode of the following Calculate mode of the following distributiondistribution

Class IntervalsClass Intervals FrequencyFrequency

0-100-10 77

10-2010-20 1515

20-3020-30 1818

30-4030-40 3030

40-5040-50 2929

50-6050-60 44

60-7060-70 33

70-8070-80 11

Step 1: Identify the modal class. By inspection, it Step 1: Identify the modal class. By inspection, it is difficult to locate. Hence, modal class will be is difficult to locate. Hence, modal class will be determined by method of grouping and analysis.determined by method of grouping and analysis.

XX Col 1(f)Col 1(f) Col 2Col 2 Col 3Col 3 Col 4Col 4 Col 5Col 5 Col 6Col 6

0-100-10 77

2222

404010-2010-20 1515

3333

636320-3020-30 1818

4848

777730-4030-40 3030

5959

636340-5040-50 2929

3333

363650-6050-60 44

77

8860-7060-70 33

4470-8070-80 11

Column Column nono

0-100-10 10-2010-20 20-3020-30 30-4030-40 40-5040-50 50-6050-60 60-7060-70 70-8070-80

11 //

22 // //

33 // //

44 // // //

55 // // //

66 // // //

TotalTotal 11 33 66 33 11

From the analysis table, the modal class is From the analysis table, the modal class is 30-40.30-40.

Mode= L + Mode= L + ∆∆1 x i 1 x i

∆∆11+ ∆+ ∆22

LL= = 30 , ∆30 , ∆1 1 =12 , ∆=12 , ∆2 2 == 11

==

Merits and Demerits of ModeMerits and Demerits of Mode

1.1. It is easy to understand and easy to calculate. In many cases It is easy to understand and easy to calculate. In many cases it can be located just by inspection.it can be located just by inspection.

2.2. It is not effected by extreme values. For example, the mode It is not effected by extreme values. For example, the mode of 10, 2, 5, 10,5, 60, 5, 10, 60,10 is of 10, 2, 5, 10,5, 60, 5, 10, 60,10 is 10 as it has occurred 10 as it has occurred most often in the data.most often in the data.

3.3. It can be determined even if the distribution has open-end It can be determined even if the distribution has open-end classes.classes.

4. It is a value around which there is more concentration of 4. It is a value around which there is more concentration of

observation and hence the best representative of dataobservation and hence the best representative of data..

DemeritsDemerits

1.1. It is not based on all the observations.It is not based on all the observations.

2.2. It is not capable of further mathematical It is not capable of further mathematical treatment.treatment.

3.3. The value of mode cannot always be The value of mode cannot always be determined. In some cases we may have a determined. In some cases we may have a bimodal or multimodal series.bimodal or multimodal series.

1.a) For a distribution, mode and mean are 32.1 1.a) For a distribution, mode and mean are 32.1 and 35.4 respectively. Find the value of and 35.4 respectively. Find the value of median.median.

Ans 34.3Ans 34.3

b) Given median = 20.6 mode = 26, find meanb) Given median = 20.6 mode = 26, find mean

Ans. 17.9Ans. 17.9

Geometric Mean ( G.M.)Geometric Mean ( G.M.)

The Geometric Mean of a series of n positive The Geometric Mean of a series of n positive numbers is defined as the nth root of their numbers is defined as the nth root of their product.product.

G.M.= n (XG.M.= n (X1 1 )x (X)x (X2.2.) x………(X ) x………(X nn))

G.M.= (XG.M.= (X1. 1. XX2……… 2……… X X n) n) 1/n1/n

For example G.M. of 3 values 2,3,4 would be:For example G.M. of 3 values 2,3,4 would be:

G.M.= 2x3x4G.M.= 2x3x4

= 2.885= 2.885

When the When the number of items are largenumber of items are large, the task of , the task of multiplying the numbers and of extracting the root multiplying the numbers and of extracting the root becomes excessively difficult. To simplify calculations becomes excessively difficult. To simplify calculations logarithms are used. Geometric mean then calculated as logarithms are used. Geometric mean then calculated as followsfollows::

log G.M.= log Xlog G.M.= log X11+ log X+ log X2 2 +……. + log X +……. + log X nn

nn log G.M.= log G.M.= ∑ ∑ log X log X nn G.M.= Antilog G.M.= Antilog ∑ ∑ log X log X nn

Weighted Geometric MeanWeighted Geometric Mean

If the various observations are not equal importanceIf the various observations are not equal importancein the data, we calculate weighted arithmetic mean.in the data, we calculate weighted arithmetic mean.Weighted Geometric Mean of the observations Weighted Geometric Mean of the observations

XX1. 1. XX2……… 2……… X X n n with respective weights as wwith respective weights as w11, w, w22, , ….w….wnn is is

G.M.= Antilog G.M.= Antilog ∑w. ∑w. log Xlog X ∑∑ww

Harmonic MeanHarmonic Mean

Harmonic mean is defined as the reciprocal of theHarmonic mean is defined as the reciprocal of theArithmetic mean of the reciprocals of the data, Arithmetic mean of the reciprocals of the data,

( none of which is zero).( none of which is zero).

If there are n observations XIf there are n observations X1. 1. XX2……… 2……… X X n , n , theirtheir

Harmonic Mean is defined as:Harmonic Mean is defined as:

H.M.H.M.== n n

1/ 1/ XX1 1 + + 1/ 1/ XX22+…….. ++…….. +1/ 1/ XXnn

ExampleExample

1.Calculate the harmonic mean of 8 and 10.1.Calculate the harmonic mean of 8 and 10.

2. Ram goes from his house to office on a car at 2. Ram goes from his house to office on a car at a speed of 60km/hour and returns at a speed of a speed of 60km/hour and returns at a speed of 40km/hour. Find his average speed 40km/hour. Find his average speed

Relation between A.M., G.M. and H.M.Relation between A.M., G.M. and H.M.

G.M. = A.M. x H.M.G.M. = A.M. x H.M.

ApplicationApplication

If the arithmetic mean of two positive numbers isIf the arithmetic mean of two positive numbers is

15 and their geometric mean is 9, find their 15 and their geometric mean is 9, find their

harmonic mean.harmonic mean.

Measure of Central Tendency

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