Rank Correlation Coefficient

CORRELATION -Rank Correlation

Correlation Definition:

In a distribution if the change in one variable eff ffe ects a change in th the e oth the er vari ria able, the th e variable are said to be corr rre elated(or there is a co corr rre ela lati tion on bet etw wee een n th the e vari riab able les) s) Let X and Y measure some characteristics of  a part rtiicular syste tem m .T .To o stu tud dy th the e overa ralll measure of the system it is necessary to measure the interdependence of X and Y.

Correlation If the quantities(X,Y) vary in such a way that cha ch ang nge e in one var aria iabl ble e co corr rres espo pond ndss to change in the oth the er variable then th the e variables X and Y are correlated. Types of Correlation:

The im The impo port rtan antt ways of cl cla ass ssiify fyiing th the e correl cor relati ation on are are:: 1. Positive and Negative 2. Simple , Partial and Multiple 3. Linear and non-Linear.

Correlation Methods

  

of Studying Correlation:

The following are the important methods of   asce as certa rtain inin ing g be betw twee een n tw two o va vari riab able les. s. Scatter diagram method Karl Pearson¶s Co-efficient Spearman¶s Rank Correlation Co-Efficient

 Scatter Diagram Method: Th The e si simp mplles estt dev evic ice e fo for  r  studying correlation between two variables is a special type of dot chart.

Correlation Karl Pearson¶s Co-Efficient of Correlation: r

= ( xy)

/ (N x y )

where x x = (x-¯) y= (y (y- ¯y ) x =Standard Deviation of Series X.  y =Standard Deviation of Series Y. r= The correlation Coefficient N=Number of Pairs of Observation.

Demerits  By applying Scatter diagram method we can get an idea about th the e dire reccti tio on of correlati tio on and also whether it is high or low. But we cannot establish the exact degr gre ee of corre rellati tio on between the variables as it is poss po ssib ible le by ap appl plyi ying ng ma math them emat atic ical al me meth thod ods. s.  The Karl Pearson¶s method is based on the ass ssum umpt ptio ion n th tha at th the e po popu pullat atio ion n be beiing st stud udiied is nor orma mallllyy distributed. Wh Whe en it is kn kno own that the population is not normal or when he shape the distribution is not kn kno own, there is need fo forr a measure of corre rellati tio on th tha at involves no as assu sump mpti tio on ab abo out th the e pa para rame mete terr of th the e po popu pullat atio ion. n.

So Why Rank Correlation???? It is possible to avoid making any assumptions about the populations being studied by ranking rank ing the observations according to size and basing the calculations on the ranks rather than upon the original observations. It does not matter which way the items are ranked, item number one may be the largest or it may be the smallest. Using ranks rather than actual observations gives the coefficient of rank correlations. This method of finding out co variability or the lack of it between two variables was developed by the British Psychologist Charles Edward Spearman in 1904.

Ranking A ranking is a relationship between a set of items such that, fo forr any two items, the fi firrst is either 'ranked higher than', 'ranked lower than' or 'ranked equal to' the second. In mathematics, this is known as a weak order or total preorder of objects. It is not necessarily a total order of objects because two different objects can have the same ranki kin ng. The rankings themselves are totally ordered. For example, materials are totally preordered by hardness, while degrees of hardness are ar e to tota tallllyy or orde dere red d.

Rank Correlation Rank  correlation´   is Rank

the stu tud dy of relationships between diff ffe erent ranki kin ngs on the same set of items. It deal de alss wit ith h me meas asur urin ing g co corr rres espo pond nden ence ce be betw twee een n tw two o rank ra nkiing ngs, s, an and d ass sses essi sin ng th the e si sign gnif ific ica anc nce e of th thiis corre co rresp spon onde denc nce. e. Sp Spea earma rman¶ n¶ss co corre rrela lati tion on co coeffi effici cien entt is defin de fined ed as as:: r = 1-((6D2)/(N(N-1)2)) Whe Wh ere r , denotes rank coeff ffiicient of correlation and D refe fers rs to the diff ffe ere ren nce of rank relation betw twe een paired I tems in two series. ³  

Features The rank method has principal uses:  The su sum m of the the diffe differe renc nces es be betw twee een n two two varia variabl bles es is zero.  Spea Spearma rman¶ n¶ss rank rank cor corre rela lati tion on coe coeffic fficie ient nt  is is the Pearsonian correlation coefficient between the ranks.  The ra rank nk cor corre rela lati tion on can can be be inter interpr pret eted ed in in the the same same way as Karl Pearson¶s correlation coefficient.

Features  Kar Karll Pear Pearson son cor correl relati ation on coe coeffici fficient ent ass assume umess that that the sample observations are drawn from a normal population. Rank correlation coefficient is a distributionfree measure since no strict assumption is made about the population from which it is drawn.  The va valu lues es obta obtain ined ed for for two two formul formulae ae are are differ differen entt due due to the fact that when ranking r anking is used some information is hidden.  Spe Spearm arman¶ an¶ss formu formula la is the the onl onlyy formul formulae ae av avail ailabl able e to find the correlation between qualitative characters.

Types of Rank Methods In the rank correlation we may have two types of  problems:  Wh Whe ere ranks ar are gi given  Wh Whe ere ranks are not given  Wh Whe ere re repe pea ate ted d ra rank nkss occ occur  ur  Note: If r = 1 then there is a perfect perf ect Positive correlation If r = 0 then the variables are uncorrelated If r=-1 then there is a perfect Negative Correlation

Where

ranks are given :

Where actual ranks are given to us the steps required r equired for  computing rank correlation are:  Tak Take e the the diff differ eren ence ce of the the two two ran ranks, ks, i. i.e. e.,, (R1(R1-R2 R2)) and denotes these differences by D.  Squ Squar are e the these se diff differ eren ence ce an and d obtai obtain n the the tota totall  Apply the formula

Where ranks are not given  Whe When n we ar are e give given n the the act actua uall dat data a and and no nott the the ranks, it will be necessary to assign the ranks. r anks. Ranks can be assigned by taking either highest values as 1 or  the lowest value as 1. But whether we start with the lowest value or the highest value we must follow the same method in case of both the variables.

Steps To Find RC  Step 1:  ± Dra Draw w the the tab table le like like

 Step 2:  ± Fill the the data data field field with with the given data data

 Step 3:  Giv Give e the Ran Rankk for for the the data data

 Step 4  ± Find the diff differen erence ce d & d2

 Step 5:  ± App Apply ly the the formu formula: la:

r=

Where d= differen difference, ce, n=no.o n=no.off data

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