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Measures of Dispersion - NCERT

As the summer heat rises, hillstations, are crowded with more andmore visitors. Ice-cream sales becomemore brisk. Thus, the temperature isrelated to number of visitors and saleof ice-creams. Similarly, as the supplyof tomatoes increases in your localmandi, its price drops. When the localharvest starts reaching the market,the price of tomatoes drops from Rs 40per kg to Rs 4 per kg or even less. Thussupply is related to price. Correlationanalysis is a means for examining suchrelationships systematically. It dealswith questions such as: Is there any relationship betweentwo variables?Correlation1. INTRODUCTIONIn previous chapters you have learnthow to construct summary measuresout of a mass of data and changesamong similar variables. Now you willlearn how to examine the relationshipbetween two this chapter shouldStudying this chapter shouldStudying this chapter shouldStudying this chapter shouldStudying this chapter shouldenable you to:enable you to:enable you to:enable you to:enable you to: understand the meaning of theterm correlation; understand the nature ofrelationship between twovariables; calculate the different measuresof correlation; analyse the degree and directionof the 2023-24 CORRELATION75 It the value of one variable changes,does the value of the other alsochange?

variation, there are certain measures namely: (i) Range (ii) Quartile Deviation (iii) Mean Deviation (iv) Standard Deviation Apart from these measures which give a numerical value, there is a graphic method for estimating dispersion. Range and quartile deviation measure the dispersion by calculating the spread within which the values lie.

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Transcription of Measures of Dispersion - NCERT

1 As the summer heat rises, hillstations, are crowded with more andmore visitors. Ice-cream sales becomemore brisk. Thus, the temperature isrelated to number of visitors and saleof ice-creams. Similarly, as the supplyof tomatoes increases in your localmandi, its price drops. When the localharvest starts reaching the market,the price of tomatoes drops from Rs 40per kg to Rs 4 per kg or even less. Thussupply is related to price. Correlationanalysis is a means for examining suchrelationships systematically. It dealswith questions such as: Is there any relationship betweentwo variables?Correlation1. INTRODUCTIONIn previous chapters you have learnthow to construct summary measuresout of a mass of data and changesamong similar variables. Now you willlearn how to examine the relationshipbetween two this chapter shouldStudying this chapter shouldStudying this chapter shouldStudying this chapter shouldStudying this chapter shouldenable you to:enable you to:enable you to:enable you to:enable you to: understand the meaning of theterm correlation; understand the nature ofrelationship between twovariables; calculate the different measuresof correlation; analyse the degree and directionof the 2023-24 CORRELATION75 It the value of one variable changes,does the value of the other alsochange?

2 Given a cause and effect may be just coincidence. Therelation between the arrival ofmigratory birds in a sanctuary and thebirth rates in the locality cannot begiven any cause and effectinterpretation. The relationships aresimple coincidence. The relationshipbetween size of the shoes and moneyin your pocket is another suchexample. Even if relationships exist,they are difficult to explain another instance a thirdvariable s impact on two variablesmay give rise to a relation between thetwo variables. Brisk sale of ice-creamsmay be related to higher number ofdeaths due to drowning. The victimsare not drowned due to eating of ice-creams. Rising temperature leads tobrisk sale of ice-creams. Moreover, largenumber of people start going toswimming pools to beat the heat. Thismight have raised the number of deathsby drowning. Thus, temperature isbehind the high correlation betweenthe sale of ice-creams and deaths dueto Does Correlation measure ?

3 Correlation studies and measuresthe direction and intensity ofrelationship among Measures covariation, notcausation. Correlation should never beinterpreted as implying cause andeffect relation. The presence ofcorrelation between two variables Xand Y simply means that when thevalue of one variable is found to changein one direction, the value of the other Do both the variables move in thesame direction? How strong is the relationship?2. TYPES OF RELATIONSHIPLet us look at various types ofrelationship. The relation betweenmovements in quantity demanded andthe price of a commodity is an integralpart of the theory of demand, which youwill study in Class XII. Low agriculturalproductivity is related to low examples of relationship may beRationalised 2023-2476 STATISTICS FOR ECONOMICS variable is found to change either in thesame direction ( positive change) orin the opposite direction ( negativechange), but in a definite way.

4 Forsimplicity we assume here that thecorrelation, if it exists, is linear, therelative movement of the two variablescan be represented by drawing astraight line on graph of CorrelationCorrelation is commonly classifiedinto negative and positivecorrelation. The correlation is said tobe positive when the variables movetogether in the same direction. Whenthe income rises, consumption alsorises. When income falls,consumption also falls. Sale of ice-cream and temperature move in thesame direction. The correlation isnegative when they move in oppositedirections. When the price of applesfalls its demand increases. When theprices rise its demand you spend more time instudying, chances of your failingdecline. When you spend less hoursin your studies, chances of scoringlow marks/grades increase. Theseare instances of negative variables move in FOR MEASURINGCORRELATIONT hree important tools used to studycorrelation are scatter diagrams, KarlPearson s coefficient of correlation andSpearman s rank scatter diagram visually presentsthe nature of association without givingany specific numerical value.

5 Anumerical measure of linearrelationship between two variables isgiven by Karl Pearson s coefficient ofcorrelation. A relationship is said tobe linear if it can be representedby a straight line. Spearman scoefficient of correlation Measures thelinear association between ranksassigned to indiviual items accordingto their attributes. Attributes are thosevariables which cannot be numericallymeasured such as intelligence ofpeople, physical appearance, honesty, DiagramA scatter diagram is a usefultechnique for visually examining theform of relationship, withoutcalculating any numerical value. Inthis technique, the values of the twovariables are plotted as points on agraph paper. From a scatter diagram,one can get a fairly good idea of thenature of relationship. In a scatterdiagram the degree of closeness of thescatter points and their overall directionenable us to examine the relation-ship. If all the points lie on a line, thecorrelation is perfect and is said to bein unity.

6 If the scatter points are widelydispersed around the line, thecorrelation is low. The correlation issaid to be linear if the scatter points lienear a line or on a diagrams spanning overFig. to Fig. give us an idea ofRationalised 2023-24 CORRELATION77the relationship between two shows a scatter around anupward rising line indicating themovement of the variables in the samedirection. When X rises Y will also is positive correlation. In Fig. points are found to be scatteredaround a downward sloping line. Thistime the variables move in oppositedirections. When X rises Y falls and viceversa. This is negative correlation. there is no upward rising ordownward sloping line around whichthe points are scattered. This is anexample of no correlation. In Fig. Fig. , the points are no longerscattered around an upward rising ordownward falling line. The pointsthemselves are on the lines. This isreferred to as perfect positive correlationand perfect negative Collect data on height, weightand marks scored by studentsin your class in any two subjectsin class X.

7 Draw the scatterdiagram of these variables takingtwo at a time. What type ofrelationship do you find?A careful observation of the scatterdiagram gives an idea of the natureand intensity of the Pearson s Coefficient ofCorrelationThis is also known as product momentcorrelation coefficient or simplecorrelation coefficient. It gives a precisenumerical value of the degree of linearrelationship between two variables Xand is important to note that KarlPearson s coefficient of correlationshould be used only when there is alinear relation between the there is a non-linear relationbetween X and Y, then calculating theKarl Pearson s coefficient of correlationcan be misleading. Thus, if the truerelation is of the linear type as shownby the scatter diagrams in figures , , and , then the KarlPearson s coefficient of correlationshould be calculated and it will tell usthe direction and intensity of therelation between the variables.

8 But ifthe true relation is of the type shown inthe scatter diagrams in Figures , then it means there is a non-linearrelation between X and Y and we shouldnot try to use the Karl Pearson scoefficient of is, therefore, advisable to firstexamine the scatter diagram of therelation between the variables beforecalculating the Karl Pearson scorrelation X1, X2, .., XN be N values of Xand Y1, Y2 ,.., YN be the correspondingvalues of Y. In the subsequentpresentations, the subscripts indicatingthe unit are dropped for the sake ofsimplicity. The arithmetic means of Xand Y are defined asXYX; YNN ==Rationalised 2023-2478 STATISTICS FOR ECONOMICSFig. : Positive CorrelationFig. : Negative CorrelationFig. : Perfect Positive CorrelationFig. : Positive non-linear relationFig. : Negative non-linear relationFig. : Perfect Negative CorrelationFig. : No CorrelationRationalised 2023-24 CORRELATION79and their variances are as follows2222( XX )XxXNN- s ==-and 2222( Y Y)YyYNN- s ==-The standard deviations of X andY, respectively, are the positive squareroots of their variances.

9 Covariance ofX and Y is defined asCov(X,Y) (X X )( Y Y)xyNN-- ==Wherex X X= - and y Y Y= - are thedeviations of the ith value of X and Yfrom their mean values sign of covariance between Xand Y determines the sign of thecorrelation coefficient. The standarddeviations are always positive. If thecovariance is zero, the correlationcoefficient is always zero. The productmoment correlation or the KarlPearson s measure of correlation isgiven byxyx yr/ N =s (1)or22(X X)(Y Y)r(X X(Y Y))-- =-- ..(2)or2222XY ( X )( Y)Nr( X )( Y)XYNN- = -- ..(3)or2222N XY ( X)( Y)rN X ( X)N Y ( Y) = ..(4)Properties of Correlation CoefficientLet us now discuss the properties of thecorrelation coefficient r has no unit. It is a pure means units of measurement arenot part of r. r between height in feetand weight in kilograms, forinstance, could be say A negative value of r indicates aninverse relation.

10 A change in onevariable is associated with changein the other variable in theopposite direction. When price ofa commodity rises, its demandfalls. When the rate of interestrises the demand for funds alsofalls. It is because now funds havebecome costlier. If r is positive the two variablesmove in the same direction. Whenthe price of coffee, a substitute oftea, rises the demand for tea alsorises. Improvement in irrigationfacilities is associated with higheryield. When temperature rises thesale of ice-creams becomes 2023-2480 STATISTICS FOR ECONOMICS The value of the correlationcoefficient lies between minus oneand plus one, 1 r 1. If, in anyexercise, the value of r is outsidethis range it indicates error incalculation. The magnitude of r is unaffected bythe change of origin and change ofscale. Given two variables X and Ylet us define two new =X AB; V =Y CDwhere A and C are assumed meansof X and Y respectively.


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