Scales & Central Tendecy

8/9/2019 Scales & Central Tendecy

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Prof G R C NairProf G R C Nair

Scales of MeasurementScales of Measurement&&

Central TendencyCentral Tendency


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ObjectivesObjectives Learn the importance andLearn the importance andapplications of Q.Tapplications of Q.T

Learn various scales ofLearn various scales ofMeasurementMeasurement

Define Central TendencyDefine Central Tendency

Learn various Measures ofLearn various Measures ofCentral TendencyCentral Tendency Learn how to calculate theseLearn how to calculate thesevaluesvalues


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Why QuantitativeWhy Quantitative

Techniques ?Techniques ?

Statistical techniquesStatistical techniques, which, which

involves collecting, organizing,involves collecting, organizing,presenting, analyzing, andpresenting, analyzing, andinterpreting numerical data,interpreting numerical data, cancanassist in making more effectiveassist in making more effective

decisions.decisions.


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Statistical techniques are usedStatistical techniques are used

extensively by marketing,extensively by marketing,accounting, quality control,accounting, quality control,consumers, professional sportsconsumers, professional sportspeople, hospital administrators,people, hospital administrators,educators, politicians, physicians,educators, politicians, physicians,etc.etc.

ApplicationsApplications


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Population and SamplePopulation and Sample

AA populationpopulation is ais a collection of allcollection of all

possible individuals, objects, orpossible individuals, objects, or

measurements of interest.measurements of interest.

AA samplesample is a portionis a portion, or part,, or part,

of the population of interestof the population of interest


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Scales of MeasurementScales of Measurement

Measuring non tangibles isMeasuring non tangibles isdifficultdifficult

4 different scales in increasing4 different scales in increasing

order of precision, and power.order of precision, and power.


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Scales in the increasing order ofScales in the increasing order of

precision are,precision are,

1.1. Nominal Scale:Nominal Scale:

Data that is classified intoData that is classified intocategories. Number does not signifycategories. Number does not signify

its mathematical characteristicsits mathematical characteristics

Eg: gender, Blood groupEg: gender, Blood group


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2. Ordinal Scale:2. Ordinal Scale:

Involves data arranged in someInvolves data arranged in some

order, but the differencesorder, but the differencesbetween data values vary.between data values vary.

eg: Rank of students (1>2>3)eg: Rank of students (1>2>3)

Mohs Hardness scale ( 2 > 1)Mohs Hardness scale ( 2 > 1)But 2But 2--1 is not equal to 51 is not equal to 5 -- 44


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3. Interval Scale:3. Interval Scale:

Similar to the ordinal level, withSimilar to the ordinal level, withthe additional property thatthe additional property that

meaningful amounts of differencesmeaningful amounts of differencesbetween data values can bebetween data values can bedetermined.determined.

eg: Temperatureeg: Temperature ( 40( 40

00

CC--3030

00

C = 20C = 20

00

CC -- 101000 C)C)

But there is no natural zero point.But there is no natural zero point.

404000C is not twice as hot as 20C is not twice as hot as 2000CC


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4. Ratio Scale:4. Ratio Scale:

Constant interval with an inherentConstant interval with an inherentzero starting point. Differenceszero starting point. Differences

and ratios are meaningful for thisand ratios are meaningful for thislevel of measurement.level of measurement.

eg:eg: Distance ( 10km is double 5km,Distance ( 10km is double 5km,

20 kg weighs double of 10 kg)20 kg weighs double of 10 kg) Use the most precise scaleUse the most precise scalepossible.possible.


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Central TendencyCentral Tendency

All natural tabulated data haveAll natural tabulated data havea tendency to cluster arounda tendency to cluster aroundsome central value.some central value.

There will be low frequency atThere will be low frequency at

the two extremes.the two extremes. This is known as CentralThis is known as CentralTendency.Tendency.


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Measures of Central ValuesMeasures of Central Values The central Value has to beThe central Value has to bemeasured for various uses.measured for various uses.

Representative Value forRepresentative Value forwhole datawhole data

For ComparisonFor Comparison

To establish relationshipsTo establish relationships To derive inferencesTo derive inferences To aid decision makingTo aid decision making


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MeanMeanThe Arithmetic mean (meanThe Arithmetic mean (mean--M) is the mostM) is the most

commonly used measurecommonly used measureIt is the sum of all the values divided byIt is the sum of all the values divided by

the total number of values:the total number of values:

where is the mean.where is the mean. NN is the total number of observations.is the total number of observations. XX is a particular value.is a particular value. 77 indicates the operation of addingindicates the operation of adding

N

X!Q


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GeometricGeometric Mean =Mean =

nnthth root of (xroot of (x11* x* x22*.x*.xnn))

HarmonicHarmonic Mean =Mean =n / (1/xn / (1/x11+1/x+1/x22+1/x+1/x331/x1/xnn))


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MedianMedian

The Median (Me) is the midpoint of theThe Median (Me) is the midpoint of thevalues after they have been arranged fromvalues after they have been arranged fromthe smallest to the largest.the smallest to the largest.

There are as many values above the median asThere are as many values above the median asbelow it in the data array.below it in the data array.

It is the (n+1)/2 th termIt is the (n+1)/2 th term

For an even set of values, the median will be theFor an even set of values, the median will be thearithmetic average of the two middle numbers.arithmetic average of the two middle numbers.


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The ages for a sample of five collegeThe ages for a sample of five collegestudents are: 21, 25, 19, 20, 22students are: 21, 25, 19, 20, 22

Arranging the data in ascending orderArranging the data in ascending ordergives: 19, 20, 21, 22, 25. Thus thegives: 19, 20, 21, 22, 25. Thus themedian is 21.median is 21.

The heights of four basketball players, inThe heights of four basketball players, ininches, are: 76, 73, 80, 75inches, are: 76, 73, 80, 75

Arranging the data in ascending orderArranging the data in ascending ordergives: 73, 75, 76, 80. Thus the median isgives: 73, 75, 76, 80. Thus the median is75.575.5


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Mean of Grouped DataMean of Grouped DataGrouping of data , FrequencyGrouping of data , FrequencyInclusiveInclusive Exclusive methodExclusive methodClass limit, width, mid pointClass limit, width, mid pointClass boundary, interval.Class boundary, interval.

The mean of a sample of dataThe mean of a sample of dataorganized in a frequencyorganized in a frequency

distributiondistributionis computed by the followingis computed by the followingformula:formula:

n =n = 77 ff

n

f7!


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Sample of ten movie theaters in a city wereSample of ten movie theaters in a city weretabulated for the total number of moviestabulated for the total number of movies

shown during a month. Compute the meanshown during a month. Compute the meannumber of movies shown.number of movies shown.

Movies

showing

frequency f

1 up to 3 1

3 up to 5 2

5 up to 7 3

7 up to 9 1

9 up to 11 3

Total 10


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Xf 6.610

66!!

7!

n

Xf

X

Moviesshowing

frequencyf

classmidpoint

X

(f)( X)

1 up to 3 1 2 2

3 up to 5 2 4 8

5 up to 7 3 6 18

7 up to 9 1 8 8

9 up to 11 3 10 30

Total 10 66


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Short Cut MethodShort Cut Method

X = XX = X00 ++ 77(d x f ) / n(d x f ) / n

XX00 = assumed mean= assumed mean

d = deviation of x from meand = deviation of x from mean nn = total number (ie= total number (ie 77 f)f)

Further, if u = d/w,Further, if u = d/w,X = XX = X00 ++ ww 77(u x f ) / n(u x f ) / n


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Short cut MethodShort cut MethodAns:Ans: 28.73128.731

ClassClass xx ff dd uu fufu00--1010 55 44 --2020 --22 --88

1010--20 1520 15 1616 --1010 --11 --16162020--30 2530 25 1515 00 00 003030--40 3540 35 2020 1010 11 20204040--50 4550 45 77 2020 22 1414

5050--60 5560 55 55 3030 33 15156767 2525

Ans 25+(25/67)x10 = 28.731Ans 25+(25/67)x10 = 28.731


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Median of Grouped DataMedian of Grouped Data The median of a sample of data organized in aThe median of a sample of data organized in afrequency distribution is computed byfrequency distribution is computed by

where Lmwhere Lm is the lower boundary of the medianis the lower boundary of the medianclass,class,F is the cumulative frequency preceding theF is the cumulative frequency preceding the

median classmedian classfmfm is the frequency of the median class,is the frequency of the median class,w is the median class interval.w is the median class interval.N is the total frequencyN is the total frequency

W

fm

(F+1)][(N+1)/2 -LmMedian +!


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AssignmentAssignment Page 64. LevinPage 64. Levin

Page 82Page 82-- SC 3.1,Page 98 SC3.8SC 3.1,Page 98 SC3.8

Scales & Central Tendecy

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