Measure of Centrel Tendency

8/8/2019 Measure of Centrel Tendency

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STATICTICSThe word statistics refers to quantitative information or to a

method of dealing with quantitative information.

The methods by which statistical data are analyzed are

called STATISTICAL METHODS.

STATISTICAL METHODS are applicable to a very large

number of fields-

economics,sociology,antropology,business,agriculture,psych

ology,medicines,education.


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y One of the powerful tools of analysis is to calculate a single

average value that represents the entire mass of the data. The

word average is very commonly used in day to day conversation.

An Average is a single value which is considered as the most

representative or typical value for a given set of data. Such

value lies somewhere in the middle of the group. For this

reason an average is frequently referred to as a measure of

central tendency or central value.


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y To get single value that describes the characteristics of the entire

data.y Useful to extract and summarize the characteristics of the entire data set.

y To facilitate comparison.

y Since average represents the entire data set, it is possible to make

comparison between two or more data sets. E.g. performance of asales person based on average sales over two month or two years.

y It becomes the base for computing other measures such as

dispersion, skewness, kurtosis etc.

Objectives of Averaging


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Characteristics of a good average

y It should be easy to understand.

y It should be simple to compute.

y It should be based on all the observations.

y It should be rigidly defined.

y It should have sampling stability.

y It should be capable of further algebraic treatment.

y It should not be unduly affected by the presence of

extreme values.


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Measures of Central Tendency

y Mathematical Averagesy Arithmetic mean (simple or weighted)

y

Geometric meany Harmonic mean

yAverages of Position

y Median

y Quartilesy Deciles

y Percentiles

y Mode


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ARITHMETIC MEAN The arithmetic mean (AM) of a set of observations is their sum,

divided by the number of observations.

It is generally denoted by x or AM. Population mean is denoted by .

Arithmetic mean is of two types:

Simple arithmetic mean Weighted arithmetic mean


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Bilal Textiles (Pvt.) Ltd

BTL was registered as a Pvt ltd co. with authorized capital of RS.100000 million in 1975.operations officially started in sept

1976.its products & services are categorized as Yarn Manufacturing & Weaving Garments etc.It now exports to countries in

europe,america,africa,asia,mid-east,asia.it has won Best Textile Export award in 2003 for sustained growth for 2000-03. Lets

Take the Sales data Of this for Statistcal Analysis.

Mean= 8631.138889


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Computing the Mean from a

Frequency Distributiony Consider the following distribution:

X f

30 2

29 3

28 5

27 3

26 2


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Computing the Mean from a

Frequency Distributiony How would you compute the mean?

1

1

K

i i

i

N

i

i

X f

X

f

!

y

!

!


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Merits and Demerits of

Arithmetic MeanyMerits

y Calculation of AM is simple

y Calculation is based on all observations and hence it can

be regarded as representative of the given datay It is capable of being treated mathematically and hence,

is widely used in statistical analysis

y It represents center of gravity of the distribution becauseit balances the magnitudes of observations which aregreater and less than it

y It gives good basis of comparison of two or moredistributions


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Arithmetic MeanyDemerits

y It can neither be determined by inspection nor bygraphical location

y

Arithmetic mean cannot be computed for a qualitativedata

y It is affected too much by extreme observations andhence does not adequately represent data consisting ofsome extreme observations

y AM cannot be computed when class intervals have openends

y Simple arithmetic mean gives greater importance tolarger values and lesser importance to smaller values


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Weighted Arithmetic Mean

y Theweighted mean enables us to calculate an average that takes intoaccount the importance of each value to the overall total.


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Weighted Arithmetic Mean

Example 1: An examination was held to decide the award of a scholarship

The weights of various subjects are different. The marks obtained by 3 students are

given below:

Subject Weight Students

A B C

Mathematics 4 60 57 62

Physics 3 62 61 67

Chemistry 2 55 53 60

English 1 67 77 49

Calculate the weighted AM to award the scholarship


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Solution to the exerciseSubject Weight Students

Student A Student B Student C

Marks

(xi)

xiwi Marks

(xi)

xiwi Marks

(xi)

xiwi

Mathematics 4 60 240 57 228 62 248

Physics 3 62 186 61 183 67 201

Chemistry 2 55 110 53 106 60 120

English 1 67 67 77 77 49 49

244 603 248 594 238 618


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GEOMETRICMEAN Geometric mean (GM) is the nth root of the product ofn items of a series.

Commonly used in the calculation of average rate of growth.


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Geometric Mean

y In many business and economic problems we deal withquantities that change over a period of time.In suchcases if we aim to know the average rate of change, weconsider geometric mean rather than arithmetic mean

y Example 01: If the population of the country has beengrowing at a rate of3%, 2.5%, 2.8%, 2% and 1.9%respectively over the last five years, what has been theaverage growth rate for the period.

y In this case, we need to calculate the geometric meanrather than the arithmetic mean


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Geometric Mean

y Example 2: The following table gives the annual rate ofgrowth of sales of a company in the last five years.Calculate the average growth rate over these five years.

Year Growth rate Sales at the endof the year

2003 5.0 105

2004 7.5 112.872005 2.5 115.69

2006 5.0 121.47

2007 10.0 133.61


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Solution to the example

y The average annual growth rate =

y GM =

y = (X1 x X2 x X3 x X4 x X5)1/5

y=

y = 5.9 percent

y Simplified solution:

y Log (G.M.) =

y GM = antilog{ }


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Geometric Mean

y Exercise 1: The rate of increase in population of acountry during the last three decades is 5 percent, 8

percent and 12 percent. Find the average rate of growthduring the last three decades.


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Uses, Merits and Demerits of

GMyUsesy GM is highly useful in averaging, ratios, percentages,

and rate of increase between two periods

y GM is important for construction of index numbers

y Meritsy The value of GM is not much affected by extreme

observations and is computed by taking all observations

y Useful in studying economic and social data

y Demeritsy GM cannot be computed if any item in the series is

negative or zero

y Difficult to calculate


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HARMONIC MEAN

y Based on the reciprocal of the numbers averaged.y Defined as the reciprocal of the arithmetic mean of the reciprocal of the

individual observation.

y It can be written as


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Applications of Harmonic Meany Useful for computing average rates

e.g. Average rate of increase of profits or average

speed at which any journey has been performed.


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Bilal Textiles (Pvt.) Ltd

Sales turnover for year 1989-2006

YEAR SALES(In millionRs.) (x)

1/X

1989 971.2 0.0010297

1990 928.9 0.0010765

1991 1236.4 0.0008088

1992 1514 0.0006605

1993 1990.3 0.0005024

1994 2454.7 0.0004074

1995 2987.1 0.0003348

1996 3623.6 0.000276

1997 4525.8 0.000221

1998 5170.8 0.0001934

1999 6255.4 0.00015992000 7721.4 0.000129

2001 10643.1 0.000093

2002 14008.1 0.000071

2003 15730.2 0.000063

2004 20554.2 0.000048

2005 24008.9 0.000041

2006 31036.2 0.000032

1/x = 0.0061473


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y n=18

1/x=0.0061473

Harmonic Mean = n

1/x

= 18/0.0061473= 2928.1148


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Harmonic Mean

y Merits

y

It is based on all observations of the seriesy It is suitable in case of series having wide dispersion

y Demerits

y Difficult to calculate

y It is not often used for analyzing business problems


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Relationship between AM, GMand HM

y

If all values are equal then AM = GM = HMy If values are different then AM > GM > HM

y If the values of an observation takes the values a, ar, ar2,ar3, ., arn, then (GM)2 = AM x HM


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Median

The median may be defined as the middle or central value ofthe variable when values are arranged in the order of

magnitude.

In other words, median is defined as that value of the variablethat divides the group into two equal parts, one part comprisingall values greater and the other all values lesser than the

median.

To measure the qualitative characteristics of data, other

measures of central tendency, namely median and mode are

used.

Positional averages, as the name indicates, mainly focus on the

position of the value of an observation in the data set.


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Averages of Position - Median

y Median Median may be defined as the middle valuein the data set when the elements are arranged insequential order (either ascending or descending)

y

Median forungrouped data:y If number of observations (n) is odd, then

y Median = Size or value of { }th observation

y If the number of observations are odd, then

y Median = observation in the data set


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Bilal Textiles (Pvt.) LtdBTL was registered as a Pvt ltd co. with authorized capital of RS.100000 million in 1995.operations officially

started in sept 1999.its products & services are categorized as Yarn Manufacturing & Weaving Garments etc.It

now exports to countries in europe,america,africa,asia,mid-east,asia.it has won Best Textile Export award in

2003 for sustained growth for 2000-03. Lets Take the Sales data Of this for Statistcal Analysis.SALESTURNOVERFROM1989-2006

MEDIAN :

total observations(n)=18

n/2th term=9th term=4525.8

(n/2+1)th term=10th term=5170.8

median=(4525.8+5170.8)/2

=4848.3


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Averages of Position - Mediany

Median for grouped data

y l= lower class limit of the median class interval

y cf= cumulative frequency of the class prior to themedian class interval

y f= frequency of the median class

y h= width of the median class interval

y n = total number of observations in the distribution


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Averages of Position - Median

y Exercise 2A survey was conducted to determine theage in years of120 automobiles.The result of such asurvey is given in the table below. What is the median

age of the autos?

y Solution -> next slide

Age of auto 0 4 4 8 8 12 12 16 16 20

No. of autos 13 29 48 22 8


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Partition Values Quartiles, Deciles,

Percentilesy Quartiles: The values of observations in a data set,

when arranged in an ordered sequence, can be dividedinto four equal parts, or quarters, using three quartiles

viz. Q1, Q2 and Q3.The first quartile Q1 divides thedistribution in such a way that 25 percent of the

observations have a value less than Q1and 75 percentof the values are more than Q1.

Q1 Q2 Q3


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Percentiles

y Deciles: The values of observations in a data set whenarranged in an ordered sequence can be divided intothen equal parts, using nine deciles (D1, D2, .., D9)

y Percentiles: The values of observations in a data setwhen arranged in an ordered sequence can be divided

into100

equal parts using99

percentiles (P1, P2, ..,P99)


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Percentiles

y Exercise 1: The following is the distribution of weeklywages of 600 workers in a factory

y

Find the 1st

quartile and 3rd

quartiley Find the 5th decile and 7th decile

y Find the 29th percentile and 95th percentile

y Find the median

Weeklywages (Rs.)

No. ofworkers

Weeklywages (Rs.)

No. ofworkers

Below 375 69 600 625 58

375 450 167 625 750 24

450 525 207 750 825 10

525 600 65


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Relationship Between Median ,

Quartiles, Deciles, Percentilesy Q1 = P25

y Q2 = Median

y P50 = D5

y Q3 = p75y D2 = P20

y Q4 = P100

y P90 = D9

y

Q2 = p50y P50 = Median

y D5 = Median


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Advantages & Disadvantages of

Mediany Meritsy Extreme values (outliers) do not affect the median as

strongly as they do the mean.

y

Useful when comparing sets of data.y It is unique - there is only one answer.

y Demerits

y Not as popular as mean.

y It tell us only One value which is one is middle.y It provides not good result in Algebric type of data


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MODE


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Averages of Position - Modey

Mode: Mode is that value of an observation whichoccurs most frequently in the data set, i.e. the point orclass mark with the highest frequency.

y Exercise 1: Find the mode of the distribution in theearlier example

= frequency of the modal class

= frequency of the class preceding the modal class

= frequency of the class following the modal class

= width of the modal class interval


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y BTL was registered as a Pvt ltd co. with authorized capital of RS.100000 million in 1995.operationsofficially started in sept 1999.its products & services are categorized as Yarn Manufacturing &Weaving Garments etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it haswon Best Textile Export award in 2003 for sustained growth for 2000-03. Lets Take the Sales data

Of this for Statistcal Analysis

y SALESTURNOVERFROM1989-2006 mode=0


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Advantages & Disadvantages of

Modey Meritsy Extreme values (outliers) do not affect the mode as

strongly as they do the mean & Median.

y

Vary simple measure of centrel tendency.y It can be located graphically , with help of histogram.

y Demerits

y It isVague and uncertain centrel tendency of values.

y It is not papuler as mean and mediany It provides not good result in Algebric type of data

y In case of more yhen one answer difficult to trace actualCentrel tendency of values


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RELATIONSHIPBETWEEN MEAN,MEDIAN & MODE

yDistribution in which values of mean,median,modecoincide are symmetrical distribution.

y Distribution in which values of mean,median,mode are not

equal are asymmetrical or skewed.

y The distance between mean & median is approximatelyone-third of the distance between the mean and mode.

Acc. To Karl Pearson :

Mean-median=1/3(mean-mode)

Mode=3median-2mean

Median=(2mean+mode)/3


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Relationship between Mean, Median and

Mode as Graphical Analysis

Mean=median=modeMeanMedianMode ModeMedianMean

For positively skewed distribution,Mean>Median>Mode

For negatively skewed distribution,M

ean

Measure of Centrel Tendency

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