CORRELATIONBy Ryan, Joy, and NitsCorrelationkey concepts:Types of correlationa) Scatter diagram b) Karl Pearsons coefficient of correlation c) Spearmans Rank correlation coefficient

CorrelationCorrelation is a statistical tool that helps to measure and analyze the degree of relationship between two or more ariables! CorrelationThe degree of relationship between the ariables under consideration is measured through the correlation analysis! The measure of correlation called the correlation coefficient !The degree of relationship is e"pressed by coefficient which range from correlation ( -1r!"1#The direction of change is indicated by a sign!Types of Correlation TypeICorrelationPositive Correlation Negative CorrelationTypes o$ Correlation Type#%ositi&e Correlation: the alues of two ariables change with same direction!$"! %eight & weight!Ne'ati&e Correlation: the alues of ariables changewith opposite direction! $"! Price & 'ty! demanded! (irection o$ t)e Correlation%ositi&e relations)ip ( )ariables change in the same direction!*s + is increasing, - is increasing*s + is decreasing, - is decreasing$!g!, *s height increases, so does weight!Ne'ati&e relations)ip ( )ariables change in opposite directions!*s + is increasing, - is decreasing*s + is decreasing, - is increasing$!g!, *s T) time increases, grades decreaseIndicated bysign; (+) or (-).*ore e+a,ples%ositi&e %ositi&e relations)ipsrelations)ipswater consumption and temperature!study time and grades!Ne'ati&e Ne'ati&e relations)ipsrelations)ips:alcohol consumption and driing ability!Price & 'uantity demandedTypes of Correlation TypeIICorrelationSimple MultiplePartial TotalTypes o$ Correlation Type##-i,ple correlation: .nder simple correlation problem there are only t.o &aria/les are studied!*0ltiple Correlation: .nder /ultiple Correlation t)ree or ,ore t)an t)ree &aria/les are studied! %artial correlation: analysis recognizes more than two ariables but considers only two ariables keeping the other constant!Total correlation: is based on all the releant ariables, which is normally not feasible!Types of Correlation Type IIICorrelationLINEAR NON LINEAR Types o$ Correlation Type IIILinear correlation: Correlation is said to be linear when the amount of change in one ariable tends to bear a constant ratio to the amount of change in the other! The graph of the ariables haing a linear relationship will form a straight line! $"+ 0 1, 2, 3, 4, 5, 6, 7, 8, - 0 5, 7, 9,11, 13, 15, 17, 19, - 0 3 : 2"Non Linear correlation: The correlation would be non linear if the amount of change in one ariable does not /ear a constant ratio to the amount of change in the other ariable! *et)ods o$ -t0dyin' CorrelationScatter ;iagram /ethodKarl Pearsons Coefficient of CorrelationSpearmans Rank Correlation Coefficient-catter (ia'ra, *et)odScatter ;iagram isa graph of obsered plotted points where each points represents the alues of+ & - as a coordinate!)ery similar as plotting points ins the Cartesian Coordinate PlaneA per$ect positi&ecorrelationHeighteightHeightof AWeightof AHeightof BWeightof BA linearrelationship1i') (e'ree o$ positi&e correlationPositie relationshipHeightWeightr = +.80(e'ree o$ correlation*oderate%ositi&e CorrelationWeightShoe Sizer ! " #$%(e'ree o$ correlation%er$ectNe'ati&e CorrelationExam scoreTV watching perweekr ! &'$# (e'ree o$ correlation1i') (e'ree Ne'ati&e CorrelationExam scoreTV watching perweekr = -.80 (e'ree o$ correlation2eak ne'ati&e CorrelationWeightShoe Sizer ! & #$( (e'ree o$ correlationNo Correlation ()ori3ontal line#HeightIQr ! #$#(e'ree o$ correlation (r#r ! "$)# r ! "$*#r ! "$%#r ! "$(#4# (irection o$ t)e Relations)ip%ositi&e relations)ip ( )ariables change in the same direction!*s + is increasing, - is increasing*s + is decreasing, - is decreasing$!g!, *s height increases, so does weight!Ne'ati&e relations)ip ( )ariables change in opposite directions!*s + is increasing, - is decreasing*s + is decreasing, - is increasing$!g!, *s T) time increases, grades decreaseIndicated bysign; (+) or (-).Ad&anta'es o$ -catter (ia'ra,Simple & r The coefficient of correlation >r measure the degree of linear relationshipbetween two ariables say " & y!5arl %earson6s Coe$$icient o$ Correlation Karl earson!s "oefficient of "orrelation #enote# $%& r&' (r ) *'+egree of "orrelation is expresse# $% a ,al-e of "oefficient+irection of change is In#icate# $% sign . & ,e/ or . * ,e/5arl %earson6s Coe$$icient o$ Correlation @hen deiation taken from actual meanA r(+, y#7 8+y 9: 8+; 8y; @hen deiation taken from an assumed meanA r 7 N 8d+dy -8d+ 8dy +N ,-./&0,-.1/0N ,-y/&0,-y1/%roced0re $or co,p0tin' t)e correlation coe$$icientCalculate the mean of the two series >" &yCalculate the deiations >" &y in two series from their respectie mean! S'uare each deiation of >" &y then obtain the sum of the s'uared deiation i!e!B"2 & !By2 /ultiply each deiation under " with each deiation under y & obtain the product of >"y!Then obtain the sum of the product of " , y i!e! B"y Substitute the alue in the formula!Interpretation o$ Correlation Coe$$icient (r#The alue of correlation coefficient >r ranges from ?1 to :1 #f r 0 :1, then the correlation between the two ariables is said to be perfect and positie #f r 0 ?1, then the correlation between the two ariables is said to be perfect and negatie #f r 0 C, then there e"ists no correlation between the ariables %roperties o$ Correlation coe$$icientThe correlation coefficient lies between ?1 & :1 symbolicallyD ? 1E r F 1 ) The correlation coefficientis independent of the change of origin & scale! The coefficient of correlation is the geometric mean of two regression coefficient! r 0 Gb"y H by"The one regression coefficient is D:e)other regression coefficient is also D:e) correlation coefficientis D:e)Ass0,ptions o$ %earson(4 # 9 N (N4 ? 1#R 0 Rank correlation coefficient ; 0 ;ifference of rank between paired item in two series!< 0 Total number of obseration!Interpretation o$ Rank Correlation Coe$$icient (R#The alue of rank correlation coefficient, R ranges from ?1 to :1 #f R 0 :1, then there is complete agreement in the order of the ranks and the ranks are in the same direction #f R 0 ?1, then there is complete agreement in the order of the ranks and the ranks are in the opposite direction #f R 0 C, then there is no correlationRank Correlation Coe$$icient (R#a# %ro/le,s .)ere act0al rank are 'i&en@1) Calculate the difference >; of two Ranks i!e! DR1 ( R2)!2) S'uare the difference & calculate the sum of the difference i!e! B;2 3) Substitute the alues obtained in the formula! Rank Correlation Coe$$icient/# %ro/le,s .)ere Ranks are not 'i&en :#f the ranks are not gien, then we need to assign ranks to the data series! The lowest alue in the series can be assigned rank 1 or the highest alue in the series can be assigned rank 1! @e need to follow the same scheme of ranking for the other series! Then calculate the rank correlation coefficient in similar way as we do when the ranks are gien!Rank Correlation Coe$$icient (R#EA0al Ranks or tie in Ranks: #n such cases aerage ranks should be assignedto each indiidual! R 7 1- (= >(4 # " AB 9 N (N4 ? 1#AB 7 1914(,1C ? ,1# " 1914(,4C ? ,4# "D@ 1914(,4C ? ,4# m 0 The number of time an item is repeated*erits -pear,ana & >b By 0 na : bB" B"y 0 aB" : bB"2Re'ression EA0ation o$ + on yE 7 c " dy #n order to obtain the alues of >c & >d B" 0 nc : dBy B"y 0 cBy : dBy2 Re'ression EA0ation 9 Line .)en(e&iation taken $ro, Arit),etic *ean Re'ression EA0ation o$ y on +: F 7 a " /+ #n order to obtain the alues of >a & >ba 0 - ( b+b 0 B"y J B"2 Re'ression EA0ation o$ + on y: E 7 c " dyc 0 + ( d-d 0 B"y J By2 Re'ression EA0ation 9 Line .)en(e&iation taken $ro, Arit),etic *eanRe'ression EA0ation o$ y on +: F ? F 7 /y+ (E ?E#/y+ 7 >+y 9 >+4

/y+ 7 r (Iy 9 I+ # Re'ression EA0ation o$ + on y: E ? E 7 /+y (F ?F#/+y 7 >+y 9 >y4/+y 7 r (I+ 9 Iy # %roperties o$ t)e Re'ression Coe$$icientsThe coefficient of correlation is geometric mean of the two regression coefficients! r 7 : /y+ J /+y #f by" is positie than b"y should also be positie & ice ersa!#f one regression coefficient is greater than one the other must be less than one!The coefficient of correlation will hae the same sign as that our regression coefficient!*rithmetic mean of by" & b"y is e'ual to or greater than coefficient of correlation! by" : b"y J 2 F r Regression coefficient are independent of origin but not of scale! -tandard Error o$ Esti,ate!Standard $rror of $stimate is the measure of ariation around the computed regression line! Standard error of estimate DS$) of - measure the ariability of the obsered alues of - around the regression line! Standard error of estimate gies us a measure about the line of regression! of the scatter of the obserations about the line of regression! -tandard Error o$ Esti,ate!-tandard Error o$ Esti,ate o$ F on E is:-@E@ o$ Fon E (-E+y# 7 :>(F ? Fe #4 9 n-4Y = Observed value of y Ye = Estimated values from the estimated equation that correspond to each y value

e = The error term (Y Ye)

n = Number of observation in sample!T)e con&enient $or,0la: (-E+y# 7 :>F4K a>F K />FE 9 n ? 4X = Value of independent variableY = Value of dependent variablea = Yinterceptb = !lope of estimatin" equationn = Number of data points Correlation analysis &s@Re'ression analysis@Regression is the aerage relationship between two ariablesCorrelation need not imply cause & effect relationship between the ariables understudy!? R *clearly indicate the cause and effect relation ship between the ariables!There may be non?sense correlation between two ariables!? There is no such thing like non?sense regression!Correlation analysis s! Regression analysis! Regression is the aerage relationship between two ariablesR *!2)at is re'ressionL=itting a line to the data using an e'uation in order to describe and predict data-i,ple Re'ression.ses Lust 2 ariables D+ and -)MtherA /ultiple Regression Done - and many +s)Linear Re'ression=its data to a straight lineMtherA Curilinear Regression Dcured line)Were doing:i!ple" #inear $egressionBro, Meo,etry:*ny line can be described by an e'uation=or any point on a line for +, there will be a corresponding -the e'uation for this is y 0 m" : bm is the slope, b is the -?intercept Dwhen + 0 C)-lope 0 change in - per unit change in +F-intercept 0 where the line crosses the - a"is Dwhen + 0 C)Re'ression eA0ation=ind a line that fits the data the best, 0 find a line that minimizes the distance from all the data points to that lineRegression $'uationA -D-?hat) 0 b+ : a-Dhat) is the predicted alue of - gien a certain +b is the slopea is the y?intercept%Re'ression EA0ation:We can predict a Y score from an X by plugging a value for X into the equation and calculating YWhat would we expect a person to get on quiz #4 if they got a 1!" on quiz ##$& = .8'() + -*.'(+& = .8'((,'.-) + -*.'(+ = ..0*+Ad&anta'es o$ Correlation st0diesShow the amount Dstrength) of relationship presentCan be used to make predictions about the ariables studiedCan be used in many places, including natural settings, libraries, etc!$asier to collect correlational data