Question1
Draw a scatter diagram and indicate the nature of correlation.
X 
10 
20 
30 
40 
50 
60 
70 
80 
Y 
5 
10 
15 
20 
25 
30 
35 
40 
Solution:
Comment : The diagram indicates that there is perfect negative correlation between the values of the two variables X and Y.
Question2
Draw a scatter diagram and interpret whether the correlation is positive or negative.
X 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
Y 
78 
72 
66 
60 
54 
48 
42 
36 
30 
24 
18 
12 
Solution:
Comment : The diagram indicates that there is perfect negative correlation between the values of the two variables X and Y.
Interpretation : The diagram indicates that there is high degree of positive correlation because the plotted points are near to each other and the trend of the points is upward
Question3
Compute karl pearson's coefficient of correlation and interpret the result :
Marks in Mathematics 
15 
18 
21 
24 
27 
Marks in Accountancy 
25 
25 
27 
31 
32 
Solution:
Let X and Y denote marks in mathematics and accountancy
X 

(x â€“ x)^{2} 
Y 

(y â€“ y)^{2} 
(x â€“ x) (y â€“ y) 
15 18 21 24 27 
6 3 0 +3 +6 
36 9 0 9 36 
25 25 27 31 32 
3 3 1 +3 +4 
9 9 1 9 16 
18 9 0 9 24 







It indicates that there is high degree of positive correlation between marks in Mathematics and Accountancy.
Question4
Xseries 
Yseries 

Mean 
15 
28 
Sum of squares deviations from mean 
144 
225 
Sum of products of deviations of X and Y series from their respective mean is 20. Number of pairs of observations is 10.
Solution:
Given :
Question5
Calculate coefficient of correlation of the following data by the product Moment Method :
X 
8 
6 
4 
3 
4 
Y 
9 
7 
4 
4 
6 
Solution:
X 
X^{2} 
Y 
Y^{2} 
XY 
8 6 4 3 4 
64 36 16 9 16 
9 7 4 4 6 
81 49 16 16 39 
72 42 16 12 24 





Using product Moment Method
Question6
X 
65 
66 
67 
68 
69 
70 
71 
Y 
67 
68 
66 
69 
72 
72 
69 
Solution:
X 


Y 



65 66 67 68 69 70 71 
3 2 1 0 +1 +2 +3 
9 4 1 0 1 4 9 
67 68 66 69 72 72 69 
2 1 3 0 +3 +3 0 
4 1 9 0 9 9 0 
6 2 3 0 3 6 0 







Question7
Calculate the correlation coefficient between X and Y and comment on the relationship :
X 
3 
2 
1 
1 
2 
3 
Y 
9 
4 
1 
1 
4 
9 
Solution:
In this question the mean of X and Y comes zero or in fractions. It will create a problem in computing deviations. So here product Moment Method will be used.
X 
X^{2} 
Y 
Y^{2} 
XY 
3 2 1 1 2 3 
9 4 1 1 4 9 
9 4 1 1 4 9 
81 16 1 1 16 81 
27 8 1 1 8 27 





Comment : r=0 shows that there is absence of correlation between the variables X and Y. But, we observe that there remains a nonlinear correlation between the two variables, ie., Y=X^{2}. So in this question, the correlation coefficient fails to indicate the correct correlation between these two variables.
Question8
Calculate Karl pearson's coefficient of correlation between ages of husband and wife from the following data :
Age of husband (in yrs.) 
21 
22 
23 
24 
25 
26 
27 
Age of wife (in yrs.) 
16 
15 
17 
18 
19 
20 
21 
Solution:
Let X and Y denote ages of husband and wife
X 

X^{2} 
Y 

y^{2} 
Xy 
21 22 23 24 25 26 27 
3 2 1 0 1 2 3 
9 4 1 0 1 4 9 
16 15 17 18 19 20 21 
2 3 1 0 +1 +2 +3 
4 9 1 0 1 4 9 
6 6 1 0 1 4 9 







Question9
X 
14 
15 
18 
20 
25 
30 
Y 
40 
45 
65 
28 
30 
40 
Take 20 and 40 as assumed mean for X and Y series.
Solution:
X 


Y 



14 15 18 20 A 25 30 
6 5 2 0 +5 +10 
36 25 4 0 25 100 
40 A 45 65 28 30 40 
0 5 25 12 10 0 
0 25 625 144 100 0 
0 25 50 0 50 0 







Since actual means are not in whole numbers, we take 20 as assumed mean for X and 40 as assumed mean for Y.
It shows that there is weak negative correlation between X and Y.
Question10
Calculate coefficient of correlation between age group and rate of mortality from the following data :
Age group 
020 
2040 
4060 
6080 
80100 
Rate of Mortality 
350 
280 
540 
760 
900 
Solution:
Since class intervals are given for age, their and values should be used for the calculation of r.
Age group 
M.V. 



Rate of Mor. (Y) 




020 2040 4060 6080 80100 
10 30 50 A 70 90 
40 20 0 +20 +40 
2 1 0 +1 +2 
4 1 0 1 4 
350 280 540 A 760 900 
190 260 0 +220 +360 
19 26 0 +22 +36 
361 676 0 484 1296 
38 26 0 22 72

N = 5 






Question11
From the data given below, calculate Karl pearson's coefficient of correlation between density of population and death rate :
Region 
Area in Sq.Km. 
Population 
Deaths 
A B C D 
200 150 120 80 
40,000 75,000 72,000 20,000 
480 1200 1080 280 
Solution:
First of all, we shall compute density of population, i.e., population per sq.km and death rate per 1000.
Density of population =
Region 
Density (X) 
dx 


Death Rate (Y) 
dy 
dy' 
dy'^{2} 
dx'.dy' 
A B C D 
200 500 A 600 250 
300 0 +100 250 
6 0 +2 5 
36 0 4 25 
12 16 A 15 14 
4 0 1 2 
4 0 1 2 
16 0 1 4 
+24 0 2 +10 







Question12
A 
3 
5 
8 
4 
7 
10 
2 
1 
6 
9 
B 
6 
4 
9 
8 
1 
2 
3 
10 
5 
7 
What is the coefficient of rank correlation ?
Solution:
R_{1} 
R_{2} 


3 5 8 4 7 10 2 1 6 9 
6 4 9 8 1 2 3 10 5 7 
3 +1 1 4 +6 +8 1 9 +1 +2 
9 1 1 16 36 64 1 81 1 4 
N = 10 


Question13
Rank by Judge A 
1 
2 
3 
4 
5 
Rank by Judge B 
2 
4 
1 
5 
3 
Rank by Judge C 
1 
3 
5 
2 
4 
Using rank correlation coefficient, determine which pair of judges has the nearest approach to common tastes in beauty.
Solution:
R_{1} 
R_{2} 
R_{3} 






1 2 3 4 5 
2 4 1 5 3 
1 3 5 2 4 
1 2 +2 1 +2 
+1 +1 4 +3 1 
0 1 2 +2 +1 
1 4 4 1 4 
1 1 16 9 1 
0 1 4 4 1 
N= 5 






Applying the formulae,
Since the coefficient of rank correlation is positive and highest in the judgement of the judeges a and c, we conclude that they have the similar tastes in beauty. Judges B and C have very different tastes.
Question14
Entries 
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
K 
L 
XJudge 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
YJudge 
12 
9 
6 
10 
3 
5 
4 
7 
8 
2 
11 
1 
What degree of agreement is there between the judges ?
Solution:
Entry 
Rank by X (R_{1}) 
Rank by Y (R_{2}) 
D=R_{1}R_{2} 
D^{2} 
A B C D E F G H I J K L 
1 2 3 4 5 6 7 8 9 10 11 12 
12 9 6 10 3 5 4 7 8 2 11 1 
11 7 3 6 +2 +1 +3 +1 +1 +8 0 +11 
121 49 9 36 4 1 9 1 1 64 0 121 
N = 12 


It indicates that the judges X and Y have fairly strong divergent likes and dislikes so far as ranking of the babies is concerned.
Question15
Student 
A 
B 
C 
D 
E 
Marks in Economis 
60 
48 
49 
50 
55 
Marks in statistics 
85 
60 
55 
65 
75 
Calculate the coefficient of rank correlation.
Solution:
Marks in Eco. (X) 
R_{1} 
Marks in Stat. (Y) 
R_{2} 
D=R_{1}R_{2} 
D^{2} 
60 48 49 50 55 
1 5 4 3 2 
85 60 55 65 75 
1 4 5 3 2 
0 +1 1 0 0 
0 1 1 0 0 
N = 5 


It indicates that there is high degree of relationship between the marks in Economics and statistics.
Question16
X 
15 
17 
14 
13 
11 
12 
16 
18 
10 
9 
Y 
15 
12 
4 
6 
7 
9 
3 
10 
2 
5 
Calculate the coefficient of rank correlation. [KVS 2004]
Solution:
X 
R_{1} 
Y 
R_{2} 
D=R_{1}=R_{2} 
D^{2} 
15 17 14 13 11 12 16 18 10 9 
4 2 5 6 8 7 3 1 9 10 
15 12 4 6 7 9 3 10 2 5 
1 2 8 6 5 4 9 3 10 7 
+3 0 3 0 +3 +3 6 2 1 +3 
9 0 9 0 9 9 36 4 1 9 
N = 10 


Question17
Solution:
Correlation studies and measures the direction and intensity of relationship among variables.
Question18
Solution:
Question19
Solution:
Correlation is said to be positive when the variables move together in the same direction, i.e., when X rises. Y also rises and when X falls. Y also falls.
Question20
Solution:
Positive correlation
Question21
Solution:
The maximum value of coefficient of correlation (r) = + 1 and
The minimum value of coefficient of correlation (r) =  1
Question22
Solution:
When the values of both the variables under study change at a constant ratio irrespective of its direction. It is a case of perfect correlation.
Question23
Solution:
Perfect correlation.
Question24
Solution:
Rank correlation is preferred to Pearsonian coefficient of correlation when extreme values are present.
Question25
Solution:
When relationship among three or more than three variables is studied simultaneously, then such correlation is called multiple correlation.
Question26
Solution:
When the relationship between two variables is studied, then such correlation is called simple correlation.
Question27
Solution:
When the values of r = + 1, there is perfect positive correlation.
Question28
Solution:
Question29
Solution:
Coefficient of correlation (r) cannot be 1.98 as its value is either equal to 1 or less than.
Question30
Solution:
Karl Pearson's coefficient of correlation measures the degree of relationship between the two variables X and Y. It is denoted by 'r'.
Question31
Solution:
A scatter diagram is graphic method of measuring correlation between the two variables. In scatter diagram, we plot the values of two variables as a set of points on a graph paper. The cluster of points is called scatter diagram. Scatter diagram does not give us the degree of correlation between two variables. It simply indicates the direction of correlation.
Question32
Solution:
Question33
Solution:
Two properties of correlation coefficient are :
(i) Correlation coefficient always remains between 1 and +1. Symbolically
(ii) The values of coefficient of correlation (r) is unaffected by change of origin and scale.
Question34
Solution:
Under Spearman's rank correlation method, correlation is measured on the basis of ranks rather than the original values of the variables.
Question35
Solution:
The various methods of studying correlation are :
(i) Scatter Diagram,
(ii) Karl Pearson's coefficient of correlation and
(iii) Spearman's rank correlation.
Question36
Solution:
No, correlation does not imply causation. It implies covariation. It should never be interpreted as implying cause and effect.
Question37
Solution:
r is preferred to covariance as measure of association because it studies and measures the direction and intensity of relationship among variables.
Question38
Solution:
The values of coefficient of correlation always lies between 1 and +1. Symbolically
Question39
Solution:
Question40
Solution:
(i) (where ranks are not repeated)
(ii) (where ranks are repeated).
Question41
Solution:
Covariance is defined and given by:
Cov
Question42
Solution:
Correlation coefficient between U and V is the same as correlation coefficient betwen X and Y, i.e., since the coefficient of correlation is unaffected by change of origin and change scale.
Question43
Solution:
r = 0 imply that there is no linear relationship between X and Y.
Question44
Solution:
r = +1 imply that there is perfect positive correlation and r = 1 imply that there is perfect negative correlation.
Question45
Solution:
When the variables cannot be measured meaningfully, then in that case rank correlation is more precise than simple correlation coefficient. Ranking may be a better alternative to quantitative fraction of quantities.
Question46
Solution:
No. but there is possibility of dependence among the variables. A zero value of r simply indicates that there is no linear relationship between the variables. The variables may have quadratic relationship among the variables. I.e., Y=X^{2}.
Question47
Solution:
No, it simply indicates that there is no linear relationship between the variables.
Question48
Solution:
Honesty, beauty, judgement, fragment secularism etc. are some examples of variables where accurate measurement is difficult.
Question49
Solution:
Because rank correlation coefficient provides a measure of linear association between the ranks assigned to the values of the variables and not their values.