NEXY-EX.EY ey Fametnod [NEX®— (EX)? x [nzv2 - (yy 4, fe NZ dxdy ~ Sdx x Edy cut Method YNEdx?= (Bax? x JNEdy?~ Gay)? 4 ner = oe N&dx’dy’ — Sdx’ x Edy’ ovation etl INEdx’?— (Sdx’? x [NEdy?— (Zdy)® ‘ ‘Spearman's Rank Correlation Coefficient ar Py Lulezpe igs are not Eq! S NS=N when Ran! 4 1 6] 2D?+—(m—m) +—(m?-m) + a 12 12 7 ranks are Equal fate when Ne-N s Used sbrevition” i] Pearson’s Coefficient of Correlation. re Bixdy Zdx’ Edy’ dx? Lay? Liddy! qh me onudn none Number of pair of observations. a Deviation of X series from mean (X - X). Deviation of Y series from mean (Y -Y). =x? Standard deviation of X series, ie., J F-- 6 ee al Standard deviation of ¥ series, ie, | -- Sum of deviations of X values from assumed mean. Sum of deviations of Y values from assumed mean. Sum of squared deviations of X values from assumed mean. Sum of squared deviations of Y values from assumed mean. Sum of the products of deviations dx and dy. Sum of step deviations of X values from assumed mean. Sum of step deviations of Y values from assumed mean. Sum of squared step deviations of X values from assumed mean. Sum of squared step deviations of Y values from assumed mean. Sum of the products of step deviations dx’ and dy’. Coefficient of rank correlation. Sum of squares of rank differences. Number of times an item is assigned equal rank.‘SUMMARY OF KARL PEARSON'S COEFFICIENT OF CORRELATION ~~ Example: Calculate the Coefficient of Coraiation (?) from the folowing data by different methods. Wexae 170x630 210 ‘34 Method: Short-Cut Method or Assumed 2 4 6 8 40. 4*' Method: Actual Mean Method 24 Method: Direct Methog X-Series ¥ Series, X-Series SI ¥Series Ti =| Lf ¥ yy x xe Y “ex ¥-¥ Peed an mes 2] 4 | 6 ats s/s | 7 foe oe AE ‘o) “| 6 | 8 | ie 6-1 -3 | 9 3 a | 64 | 24 8 3] 9 | 3 10 100 | 30 7 9) os) 2 wz | a | 36 7 | 25 9S we. mae B=42 DO =304 FY= 1262 Xe | zy? = 690 | Exy= 20 || SPE SON EY = 1282" = 9,278 | exy = 1 069 Hae alin fae 1 aeRO DY es aR Y -2 126.94 INDE (EXP NEVE Nn 6 & x 1092 - 42 x 126 wee temas eg by 210210 VERSES OP xox SPIER 6582-5292 V2q04= 1,764 x 119,650= TRF 1.260 1,260 ¥420x3,780 1.260, 4 Methos ‘Mean Method ] Step Deviation Method X Series __¥Seres | i xSeries ] Ta X/ax=| oe ax? | pecalen =O) 9G lay = 97 | ror Heroes oe 2i-6| -3 | 9 9) 8 4 ca] 2 | 4 4|4 ej-2| 4 14 a! 4 8 0| ° o| 0 10) 2) 1 1] 4 12) 4 4 4 | Zdx’ Ed? | Eay’ |Eay? Zoxay’ i i 219) | #3 [=19) ae NEw dy! ~ Zax’ x By" INEG= aay NEG BOE (6228) ~(-6 x18) Y6xT76~ CBP x10x 684 = (TEE 1,988 - 108 Va56~36 x V4,104~ 304 1,260 ¥420x 3,780 1,260 7,260 AINZdx’?= (Zdx’)? x INZay?= (Eay'? {6 x 19} -{-3 x-3}elation res of Corre east! 40 f pte = inna OF SPEARMAN’S RANK CORRELATION ranks are Given onl ona 24 Case: When Ranks are NOT | Given 57 snkthe 5 contestants|| Example 2. Calculate . 23.2 B pearman's Rank correl twojudat coeticient rom the folowing data: ees x e7 | 22 | 33 [ 7s | 37 3 petition X | Ranks | y | Ranks | o= & By) (Fig) | Py=R | 7 4 29 | 4 16 22 | 8 63 4 16 a3 | 4 2 2 2 3 ° Siena ak te ita kee. T Vi Ranks (R)__| T 18 8 1 2 Je, | 7 2 . 34 5 225 : |iakiie'94 = aes 5 225, : | 5 ° u é Peeps! 2 6 mp | ¢ | 8 3 16 | Beer A eee ar Leet eter a 3 | ai cee a a | 3D?= 150.5 | umber 75 is tepeated twice in series X and 34 is repeated thrice in series. Therefore, in X, m=2 andinY, m= 3. | 1 en? 1 . 8 | BD? + (en? mm) + mem) hate Ne-N 1 6] 1595+ — (2-2) +— 3-3 ie pe erI+Gye-9 8 2 1 ~ 811595405 +2) ex 162 ee 512-8 504)er Statsties Econ, Mies, REVISION OF KEY POINTS + Correlation indicates the relationship between two variables of a series so that c of one variable are associated with changes in the values of the other variable, * Correlation and Causation: A correlation between two variables may be due to fol (i) Effect of Third Variable (i) Mutual Dependence (ii) Pure Chance + Importance or Significance of Correlation ()_Ithelps in measuring the extent of relationship between two variables in one fguy, re. (i) facilitates understanding of economic behaviour and helps in locating the ¢ i variables on which others depend. (il) When two variables are correlated, then value of one variable can be estimated, gue, | Given the vay, 8 haNgES in the ina es OWI reasons "Healy impcran of another. (iv) Correlation facilitates the deci * ‘Types of Correlation: Correlation can be classified into following main categories. () Positive and Negative Correlation (i) Linear and Non-Linear (Curviinear) Con, (ii) Simple, Muttiple and Partial Correlation elation * Positive and Negative Correlation (Positive Correlation: When two variables move in the same directionie. when oneinereasesiy other also increases and when one decreases the other also decreases, then sucha relaisrs called positive correlation. is (@)_ Negative Correlation: When two variables move in opposite directions, ie. when one increases the other decreases and when one decreases the other increases, then such arelationiscaiss negative correlation. + Linear and Non-Linear (Curvilinear) Correlation (i) Linear Correlation: Linear correlation is said to existif the amount of change in one variable ends to bear a constant ratio to the amount of change in the other variable. (i) Non-Linear (Curviinear) Correlation: in non-linear or curvilinear correlation, the amount ofchange in one variable does not bear a constant ratio to the amount of change in the other related variable n-making in the business world * Simple, Multiple and Partial Correlation (i) Simple Correlation: When only two variables are studied. (i) Muttipte Correlation: When relationship among three or more than three variables is studied simultaneously. (ii) Partial Correlation: Relationship between two variables is examined keeping other variables as constant. * Degrees of Correlation (i) Perfect Correlation: If relationship between two variables is such that values of two variables change (increase or decrease) in the same proportion, correlation between them is said tobe perfect. (ii) Zero Correlation: When there is no relationship between the two variables, we say that there is Zero correlation (or absence of correlation). (ii) Limited Degree of Correlation: In real life, economic data do not indicate perfect positive oF negative correlation. At the same time, the cases of zero correlation are also very limited.of Correlation «Measure ierd nt of Correlation: Following eottoas rome '9 methods are used for measuring correlation 10% riables: ethos oval wee! jagram vedi cattor DIAS efficient of Correlation i) on's 7 atl fe Rank Correlation Coefficient " year™: ca) SPORT grams a simple and altractive method of diagrammatic representation of a bivariate cate’ Pig determine the nature of correlation between the variables. . i oso ¢seatter Diagram erits Or © Simple and a non-mathematical method wise easily understood and interpreted @ wear vratuenced by size of the extreme values gi tis ist step in investigating the relationship between two variables, ee rscatter Diagram mathematical method broad and rough idea of the degree and nature of correlation le for large observations merits of SC jrisanon yes only @ not suitabl ° 5 not imply causation "s Coefficient of Correlation: According to Karl Pearson, Coefficient of Correlation (1) dividing the sum of products of deviations from their respective means by the product irs and their standard deviations. 0 fo 9 ao tis {w) Ieee « Karl Pearson igdetermined BY ot number of Pall smalation of Karl Pearson’s Coefficient of Correlation: It can be calculated by the folowing methods: ; {Actual Mean Method (ii) Short-Cut Method {iy Direct Method (iv) Step Deviation Method umptions of Coefficient of Correlation {) tisassumed that there isa linear telationship between the variables. (i) There is no cause and effect relationship between the two variables under study. {i) The two variables under study are affected by a large number of independent causes of such a nature as to produce normal distribution. (wv) itis more reliable if the error of measurement is reduced to the minimum. + properties of Coefficient of Correlation (i) Coefficient of Correlation lies between - 1 and + 1. (ij Itis independent of the change of origin and scale of measurements. (ii) tis a measure of the linear relationship. (wv) Ittwo variables X and are independent, coefficient of correlation between them will be zero. * Merits of Coefficient of Correlation () ‘tisthe most popular and most widely used mathematical method of studying correlation between ‘wo variables. (i) summarises in one figure not only the degree of correlation but also the direction. * Demerits of Coefficient of Correlation a a aes ot correlation are unduly affected by the value of extreme items suming Method. + Asstey ae Statistics for Econom, ies (ii) The correlation coetticient always assume linear relationship regardless of the fact whey assumption is correct or not. (iv). Possibility of wrong interpretation. + Spearman's Rank Correlation: In this method, various items are assigned ranks accord | characteristics and a correlation is computed between these ranks. ther that 9 0 the + Merits of Rank Correlation (i) This methodis easy to calculate and simple to understand as compared to Karl Pearsons (i) tis very useful when the data is qualitative in nature like honesty, beauty, intelligence, i (ii) It can be used where we are given the ranks but not the actual data. * (iv) When actual values are given (instead of ranks), then this method can be used to get rou about the degree of correlation. * Demerits of Rank Correlation (i). Itcannot be used for finding out correlation in a bivariate frequency distribution, (i) Ifthe number of values is quite large, it becomes a difficult task to ascertain the ranks and th differences. er (iil) It lacks precision as compared to Karl Pearson's method. i « Karl Pearson's Method Vs Spearman's Rank Method: The coefficient of correlation by both th ti methods ranges between —1 and +1. Still, there exist the following differences: 7 (i) Karl Pearson's Method of Correlation measures correlation for quantitative data, whereas ‘Spearman's Method of rank correlation measures coefficient of correlation for qualitative data (i) Karl Pearson's Method calculates deviations from actual or assumed mean, whereas Spearman’ Method calculates the rank differences. i) Rank Correlation gives less importance to the extreme values because it gives them rank, However, Karl Pearson's Method of Correlation gives more importance to extreme values as it is based on actual values. Method, ete, '9hidea etek gu Pee] pale) TS Q.1. Give two examples each of ‘Positive Correlation’ and ‘Negative Correlation’ ‘Ans. Examples of Positive Correlation: (') Relationship between age of husband and age of wife; (i) Relationship between Income and Expenditure, Examples of Negative Correlation: (i) Relationship between Price and Demand; (ii) Relationship between day temperature and sale of woollen garments. Q.2, What will be the correlation in each of the following individual cases, if as per Scatter Diagram: () All the points cluster around a straight line going upwards from left to right. (i) Allthe points are scattered in a haphazard manner. (ii) All the points fall on a straight line with positive slope. All the points cluster around a straight line with negative slope. (v) Allthe points fall on a straight line with negative slope. Ans. (i) Positive Correlation; (ji) Zero or No Correlation; (i) Perfect Positive Correlation; (iv) Negati (v) Perfect Negative Correlation. Q.3, ‘Both Karl Pearson’s Method and Spearman's Rank Method measure correlation for quam! data’ Comment. Ans. The given statements incorrect. Karl Pearson's method of Correlation measures correlation for a whereas Spearman's method of rank correlation measures coefficient of correlation for ive Correlation: tive quantitative quaitative