Central Tendency (Stats)

7/28/2019 Central Tendency (Stats)

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MEASURES OF CENTRALTENDENCY ( Location ) Average age of top 50 powerful persons of 2010in India decreased from 58 years to 54


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Measures of Location orCentral Tendency.

In a distribution , the observationscluster around a central value. Thisproperty of concentration of the

observations around a central value iscalled Central Tendency.

The central Value around which there isconcentration is called measure ofcentral tendency( measure of location,Average).

Ex: Mean marks scored by 1st PGDM is 65 %


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Objectives of Averaging:To get a Single value that describes the

characteristics of the entire data.

To Facilitate Comparison. For computing various other

statistical measures such as

dispersion, skewness, kurtosisand various other basiccharacteristics of a mass data.


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Requisites of a Good Average.

1. It should be simple to Understand andeasy to calculate.

2. It should be based on all the items ofthe given data.

3. It should be rigidly defined.4. It should be capable of further

mathematical treatment.

5. It should be affected as little as possible

by fluctuations of sampling.6. It should not be affected by extreme

observations ( Values)


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1. Arithmetic Mean (A.M)

2. Median (M)

3. Mode (Z)

4. Geometric Mean (G.M)

5. Harmonic Mean (H.M)

Various measures of Central Tendency


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1. Arithmetic Mean ( A.M )

A.M=Sum of observationsNumber of observations


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Calculation of A.M

1. Ungrouped Data ( Raw Data)

2. Discrete Data3. Continuous Data


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Ungrouped Data (Raw Data):

A sample of 30 persons weight of a

particular class students are as follows.

62 58 58 52 48 53 54 63 69 63

57 56 46 48 53 56 57 59 58 53

52 56 57 52 52 53 54 58 61 63


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Discrete DataNumber of post graduates

(x)

Frequency (f)

0 2

1 2

2 4

3 1

4 1


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Continuous Data

Marks No. of students20 30 5

30 40 15

40 50 25


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Exclusive method (overlapping)In this method, the upper limits of one class-interval are the lower limit of next class. Thismethod makes continuity of data.

A student whose mark is between 20 to 29.9will be included in the 20 30 class.

Marks No. of students

20

30 5

3040 15

4050 25


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Inclusive method (non-overlaping)

A student whose mark is 29 is included in

20 29 class interval and a student whose

mark in 39 is included in 30 39 classinterval.

Marks No. of students2029 5

3039 15

4049 25


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Ungrouped Data (Raw Data)

x = observationsn = number of observations.

X =X1+ X2+ X3+ + X n

n

= Xn


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The following data gives value of equity holdings of 20 of

the Indias billionaires.

Name Equity Holdings ( M illions of Rs.) Kiran Mazumdar-shaw 2717The Nilekani family 2796The Punj family 3098Karsanbhai K.Patel& family 3144Shashi Ruia 3527K.K . Birla 3534

B. Rama Linga Raju 3862Habil F. khorakiwala 4187The Murthy family 4310Keshub Mahindra 4506The Kirloskar family 4745M.v. Subbiah family 4784Ajay G. Piramal 4923

Uday Kotak 5034S.P.Hinduja 5071Subhash Chandra 5424Adi Godrej 5561Vijay Mallya 6505V.N. Dhoot 6707

Naresh Goyal 6874


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X 2717+2796++6874

n 20

= Rs.4565.4 Millions

=X =


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X = ObservationsF = Frequency

Discrete Data

X =f x

f


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Problem on Discrete Data

The following is the frequency distribution of the number of

telephone calls received in 245 successive one-minuteintervals at an exchange:

Obtain the mean number of calls per minute.

No. of Calls 0 1 2 3 4 5 6 7Frequency 14 21 25 43 51 40 39 12


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No. of calls (x) Frequency (f) f x

0 14 0

1 21 21

2 25 50

3 43 129

4 51 204

5 40 200

6 39234

7 12 84

f=245 f x: 922


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f x 922

f 245=X = = 3.763


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Continuous Series

The calculation is illustrated with the data relating toequity holdings of the group of 20 billionaires given

Class Interval Frequency

2000-3000 2

3000-4000 5

4000-5000 6

5000-6000 4

6000-7000 3


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Class Interval Frequency (F) Mid value(X) fx

2000-3000 2 2500 50003000-4000 5 3500 17500

4000-5000 6 4500 27000

5000-6000 4 5500 22000

6000-7000 3 6500 19500f=20 fx=91000


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f x 91000

f 20==X = 4550


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Properties of Arithmetic Mean

1. The sum of the deviations, of all thevalues x, from their arithmetic mean, isalways zero

2. The product of the arithmetic mean and

the number of items gives the total ofall items.

3. If there are the arithmetic mean of two samplesof sizes n1and n2 respectively then, the

arithmetic mean of the distribution combiningthe two can be calculated as

X12 = N1 X 1 + N2 X 2

N1 + N2


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Properties of Arithmetic Mean4. The sum of squared deviations of the

items from mean is minimum, whencompared to the sum of squareddeviation of the items from any othervalue.


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Weighted Mean

The weighted meanof a set of numbersX1, X2, ..., Xn, with corresponding weightsw1, w2, ...,wn, is computed from the

following formula:


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EXAMPLE Weighted Mean

The Carter Construction Company pays its hourlyemployees $16.50, $19.00, or $25.00 per hour.There are 26 hourly employees, 14 of which arepaid at the $16.50 rate, 10 at the $19.00 rate, and2 at the $25.00 rate. What is the mean hourly rate

paid the 26 employees?


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Merits:1. Mean is based on all the items of the

given data.2. Mean is rigidly defined by a

mathematical formula.

3. Mean is capable of further algebraic

treatment.

4. Mean has good sampling stability.


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Demerits:1. Mean can be unduly affected by

extreme values.

2. Mean cannot be calculated for

open-end classes, since midpoints cannot be found for suchclasses.

3. Mean cannot be foundgraphically like median and mode


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Median (M) The median is that value of thevariable which divides the group in

two equal parts, one partcomprising all the values greaterand the other, all the values less

than median.


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Calculation of Median

1. Ungrouped Data ( Raw Data)

2. Discrete Data

3. Continuous Data


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Raw Data

Steps:1. Arrange the data in ascending

order.

2. Find n+1 value2

3. Apply the formula

M= size of n+1 item.

2


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Sales Sorted Sales9 66 9

12 1010 1213 1315 1416 1414 1514 1616 1617 1616 1724 1721 1822 1818 1919 2018 2120 2217 24

The median is the middle value ofdata sorted in order of magnitude.

(20+1)/2=10.516

Median


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Discrete Data

Steps:1. Find Cumulative Frequencies (C.F)2. Find N/2 value.

N= total Frequency3. Apply the formula M= Size of (N/2)thitem. In other words locate a valuewhich is just more than N/2 value.(Note: This is not Median)

4. Read the corresponding X value. Thisgives the value of Median.


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Continuous data

Steps:1. Find C.F

2. Find N/2 Value.

3. Locate the value which is morethan N/2 value from thecumulative frequency column.

4. Read the corresponding class.This is the median class i. e theclass where median lies.


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5. Apply the formula,

M= l+ 2

M = Median

l = Lower limit of the median class.

N = Total Frequency

c. f= cumulative frequency of the pre medianclass

f = frequency of the median class

c = width of the median class

N C .F

f

X c


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Merits:1. It is easy to understand and easy to

calculate for a non-mathematicalperson.2. It is not affected by extreme

observations.

3. Median can be calculated dealingwith a distribution with open endclasses.

4. Median can be representedgraphically.

5. Median is the only average to beused with qualitative data.


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Demerits:

1. In case of even number of

observation for an ungrouped data ,median can not be determinedgraphically.

2. Median, being a positional average ,

is not based on each and every itemof the distribution.

3. Median is not suitable for furthermathematical treatment.

4. Median doest not have samplingstability.


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Mode (Z)

. . . . . : . : : : . . . . .

---------------------------------------------------------------6 9 10 12 13 14 15 16 17 18 19 20 21 22 24.

Mode

Mode is defined as the value which is repeatedmaximum number of times in a data.

.


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Calculation of Mode

1. Ungrouped Data ( Raw Data)2. Discrete Data

3. Continuous Data


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Ungrouped data

Here, Mode is calculated by mere

inspection.


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Discrete Data

Here, Mode is calculated by mere

inspection.


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Continuous data

Steps:1. Locate the maximum frequency.

2. Read the corresponding class.

This is the modal class i.e., theclass where mode lies.

3. Apply the formula,

z= l + 1+

1

1 2

X c


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z = mode

l = lower limit of modal classf - f

f - f

f Frequency of modal classf Frequency of pre modal class

f Frequency of post modal

classc Width of the class interval

1= 1 0

2= 1 2

1 =

0 =

2 =

=


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Merits:

1. Its value can be easily ascertained

without much calculation.2. It is an average which is commonly

used in day to day life.

3. It is not affected by extreme values.4. The data need not be arranged.

5. Mode can be graphicallydetermined.

6. Mode can be calculated for datawith open-end classes.


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Demerits:

1. Mode is not based on each and

every item of the data.2. Mode is not capable of further

of algebraic treatment.

3. Mode is not rigidly defined.

4. Model value can be misleading.

5. Mode is ill defined for bimodalor multimodal distribution.

6. Mode doesnt have samplingstability.

R l ti b/ di


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Relation b/w mean, medianand mode.

Mode = mean - 3 [mean - median]

Mode = 3 median - 2 mean

Median = mode +


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Symmetrical Distribution


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NEGATIVELY OR LEFT SKEWED

Mean < Median < Mode


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POSITIVE OR RIGHT SKEWED

Mean > Median > Mode


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Geometric Mean It is defined as the nth root of

product of n positive values oritems.


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Calculation of G.MUngrouped data ( Raw Data )G.M= antilog log X

n


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Calculation of G.MGrouped data (Discrete &Continues Data )G.M= antilog f log X

N= total Frequency.

N

Example:


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Example:

Suppose you receive a 5 percentincrease in salary this year and a 15percent increase next year. The

average annual percent increase is8.886, not 10.0. Why is this so? Webegin by calculating the geometric

mean.


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The Geometric Mean Useful in finding the average change ofpercentages, ratios, indexes, or growthrates over time. It has a wide application in business and

economics because we are often interestedin finding the percentage changes in sales,salaries, or economic figures, such as theGDP, which compound or build on eachother.

The geometric mean will always be lessthan or equal to the arithmetic mean.


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Combined Geometric Mean

G = Antilog [(n1 log G1 + n2 log G2)/ (n1 + n2)]


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Geometric MeanMerits:1. Makes use of full data.2. Extreme large values havelesser impact.3. Useful for data relating to ratiosand percentages.4. Useful for rate ofchange/growth.

Demerits: (G.M)


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1. Cannot be calculated if anyobservation has the value zeroor is negative.2. Difficult to calculate and

interpret.


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AM, GM, and HM satisfy theseinequalities:

AMGMHM

Equality holds only when all theelements of the given sample areequal.


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Harmonic Mean

It is defined as the reciprocal ofmean of reciprocal of values.Calculation of H.M:

UngroupedData Grouped Datan


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H.M- Merits:

1. It is based on all the items of thegiven data.

2. It gives the best results wheretime and rates are under study.

3. It is rigidly defined.

4. It is calculated even if the seriescontains negative values.


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H.M Demerits:

1. It is difficult for layman tounderstand and interpret.

2. It has limited practicalapplication.

3. It cannot be calculated if any of

the value is zero.


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Sales Sales Executive A Sales Executive B Sales Executive C

March 14 10 6

April 12 10 16

May 6 10 7

June 8 10 15

July 13 10 10

Aug 7 10 6

Total 60 60 60

Average 10 10 10


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MEASURES OF DISPERSION

Why Study Dispersion?


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A measure of location, such as the mean or themedian, only describes the center of the data. It isvaluable from that standpoint, but it does not tell usanything about the spread of the data.For example, if your nature guide told you that theriver ahead averaged 3 feet in depth, would youwant to wade across on foot without additionalinformation? Probably not. You would want to knowsomething about the variation in the depth.A second reason for studying the dispersion in a setof data is to compare the spread in two or moredistributions.


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The scatterdness of

values from any measure of centraltendency is called Variation orDispersion

Characteristics for Ideal


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measure of dispersion1. It should be rigidly defined.2. It should be based on all the

observations.

3. It should be amenable to furthermathematical treatment.

4. It should be not be affected by

extreme observations.

f


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Measures of Dispersion:

1. Range2. Quartile Deviation

3. Mean Deviation

4. Standard Deviation


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Range:

Range is simply the difference between thehighest and lowest value in the distributionof values.

Example:

Weekly income of 10 people:

Range is maximum income minusminimum income: 330-180 = 150.180 220 280 320 280 180 350 280 330 220


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Group A: 30, 40, 40, 40, 40, 50, 50

Group B: 30, 30, 30, 40, 50, 50,50

Group C: 30, 35, 40, 40, 40, 45, 50

Range:20

Let us take two sets of observations.Set A contains marks of five students


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Set A contains marks of five studentsin Mathematics out of 25 marks and group Bcontains marks of the same student in

English out of 100 marks.

Set A: 10, 15, 18, 20, 20Set B: 30, 35, 40, 45, 50

The values of range and coefficient of

range are calculated as:


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Range Co efficient ofRangeSet : A 20 -10 = 10

Set : B 50 -30=20

Coefficient of Range:


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Coefficient of Range:

It is relative measure ofdispersion and is based on thevalue of range. It is also called

range coefficient of dispersion. Itis defined as

Coefficient of Range = Max-min

Max+min

Merits: Demerits:1 It i t b d


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1. It is the simplestmethod ofmeasuringvariation.

2. It can becalculated quicklysince only twovalues are taken

into consideration.

1. It is not based oneach and every item

of the given data.2. It can get affectedunduly by extremevalues, since only

those values areconsidered.

3. It can not becalculated for data

with open endclasses.

4. Range does nothave sampling

stabilit . Semi Interquartile Range


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( Quartile Deviation )Inter quartile range (IQR) is another range measure but thistime looks at the data in terms of quarters or percentiles.The range of data is divided into four equalpercentiles or quarters (25%).

Min Max

Q2

Median

50th Percentile

Q1

25th percentile

Q3

75th percentile

IQR

Range Calculation Of Q.D


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1. Ungrouped data ( Raw data).2. Discrete Data.

3. Continuous Data.

Raw Data:


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Raw Data:

thQ1 = Size of n+1 item.

4

thQ3 = Size of 3 n+1 item.

4

Di t D t


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Discrete Data:

Merits:


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1.It is simple to compute and easyto understand.

2. It can be computed for data with

open-end classes.3. It is not affected by extreme

values.

Demerits:


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1. It doesnt take all the values intoconsideration. It omits 50% of theitems- i.e. 25% items below Q1

and 25% items above Q3.2. It is not much capable of further

algebraic treatment.

3. It doesnt have sampling stability.

3. Mean Deviation


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It is defined as the mean

of absolute deviations of variousitems from either mean or medianor mode.

Calc lation of M D


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Calculation of M.D

1. Raw Data2. Discrete Data

3. Continuous Data.

Merits of M.D:


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1. It is based on every item of theseries.

2. It is rigidly defined.

3. It is not much affected byextreme values.

Demerits of M.D :


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1. It ignores algebraic signs whiletaking deviations of the items.

2. It is not much used for further

algebraic treatment.3. It can not be computed for data with

open end classes.

4. Calculation of M.D becomes tediouswhen the values of Mean, median,mode are in decimals.

Variance


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Where the mean is a measure of the centre of agroup of numbers, the variance is the measure of the

spread.

It involves measuring the distance between each ofthe values and the mean.

To calculate the variance :

1. calculate the mean

2. for each value in the distribution subtract the

mean and then square the result (the squareddifference)

3. calculate the average of those squareddifferences.

Variance


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= Sum of (observed value

mean score)2

Total number of scores -1

The larger the variance value the further the observedvalues of the data set are dispersed from the mean.

A variance value of zero means all observed values arethe same as the mean.

1

2

2

N

XXs

i

4. Standard Deviation (S.D)


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( )

The square root of variance isknown as standard deviation.


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Central Tendency (Stats)