Nm Shah 2018 Solutions for Class 11 Commerce Economics Chapter 8 Measures Of Correlation are provided here with simple step-by-step explanations. These solutions for Measures Of Correlation are extremely popular among Class 11 Commerce students for Economics Measures Of Correlation Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Nm Shah 2018 Book of Class 11 Commerce Economics Chapter 8 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Nm Shah 2018 Solutions. All Nm Shah 2018 Solutions for class Class 11 Commerce Economics are prepared by experts and are 100% accurate.

#### Question 1:

The following pairs give the value of variables of capital employed and profit.

 Capital employed (in crores of â‚¹ ) (X) : 2 3 5 6 8 9 Profit (in lacs of â‚¹ ) (Y) : 6 5 7 8 12 11
(a) Make a scatter diagram.
(b) Do you think that there is any correlation between profit and capital employed? Is it positive or negative ? Is it high or low?
(c) By graphic inspection , draw an estimating line.

a) Scatter Diagram

b) The points obtained on the scatter diagram lie close to each other and reflect an upward trend. Thus, there exists a high degree of positive correlation between capital employed and profits earned.

c)

#### Question 2:

Plot the following data as a scatter diagram and comment on the result obtained:

 X : 11 10 15 13 10 16 13 8 17 14 Y : 6 7 9 9 7 11 9 6 12 11

Scatter Diagram

Thus, there exists a positive correlation of moderate degree between X and Y.

#### Question 3:

Following are the heights and weights of 10 students in a class. Draw a scatter diagram and indicate whether the correlation is positive or negative.

 Height (in inches) : 72 60 63 66 70 75 58 78 72 62 Weight (in kg) : 65 54 55 61 60 54 50 63 65 50

Thus, there exists a very low degree of positive correlation between height and weight of students.

#### Question 4:

Draw a scatter diagram for the data given below and interpret it:

 X : 10 20 30 40 50 60 70 80 Y : 32 20 24 36 40 28 48 44

Thus, there exists a moderate degree of correlation between X and Y.

#### Question 5:

Draw a scatter diagram of the following data:

 X : 15 18 30 27 25 23 30 Y : 7 10 17 16 12 13 9

Thus, there exists a moderate degree of positive correlation between X and Y.

#### Question 6:

From the following data compute the product moment correlation between X and Y.

 X series Y series Arithmetic Mean 25 18 Sum of Square of deviations from Arithmetic Mean 136 138
Summation of products of deviations of X and Y series from their respective means = 122
Number of points of values = 15

Thus, the product moment correlation between X and Y is 0.89.

#### Question 7:

Calculate Karl Pearson's Coefficient of Correlation on the following data:

 X : 15 18 21 24 27 30 36 39 42 48 Y : 25 25 27 27 31 33 35 41 41 45

 X Y $\left(\mathbit{X}\mathbit{-}\overline{)\mathit{X}}\right)$(x) x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$(y) y2 xy 15 25 −15 225 −8 64 120 18 25 −12 144 −8 64 96 21 27 −9 81 −6 36 54 24 27 −6 36 −6 36 36 27 31 −3 9 −2 4 6 30 33 0 0 0 0 0 36 35 6 36 2 4 12 39 41 9 81 8 64 72 42 41 12 144 8 64 96 48 45 18 324 12 144 216 Æ©X = 300 Æ©Y =330 Æ©x2 =1080 Æ©y2 =480 Æ©xy =708

N = 10

Thus, the value of correlation coefficient is 0.983.

#### Question 8:

Calculate Karl Pearson's Coefficient of Correlation between the sales and expenses of the following 10 firms:

 Firms : 1 2 3 4 5 6 7 8 9 10 Sales(in â‚¹ '000) : 50 50 55 60 65 65 65 60 60 50 Expenses (in â‚¹ '000) : 11 13 14 16 16 15 15 14 13 13

 Sales (X) Expenses (Y) $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y y2 xy 50 11 −8 64 −3 9 24 50 13 −8 64 −1 1 8 55 14 −3 9 0 0 0 60 16 2 4 2 4 4 65 16 7 49 2 4 14 65 15 7 49 1 1 7 65 15 7 49 1 1 7 60 14 2 4 0 0 0 60 13 2 4 −1 1 −2 50 13 −8 64 −1 1 8 580 140 360 22 70

N = 10

Note: As per the textbook, coefficient of correlation is 0.67. However, as per the above solution coefficient of correlation should be 0.786.

#### Question 9:

Calculate correlation coefficient between X, the number of rainy days per month and Y, the number of rain coats sold in that month in a certain shop for 12 months. Interpret the results.

 X : 14 8 18 10 22 9 3 5 6 11 13 13 Y : 15 11 20 12 15 7 3 4 7 10 11 29

 X Y $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$(x) x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$(y) y2 xy 14 15 3 9 3 9 9 8 11 −3 9 −1 1 3 18 20 7 49 8 64 56 10 12 −1 1 0 0 0 22 15 11 121 3 9 33 9 7 −2 4 −5 25 10 3 3 −8 64 −9 81 72 5 4 −6 36 −8 64 48 6 7 −5 25 −5 25 25 11 10 0 0 −2 4 0 13 11 2 4 −1 1 −2 13 29 2 4 17 289 34 Æ©X =132 Æ©Y =144 Æ©x2 =326 Æ©y2 =572 Æ©xy =288

N = 12

There is a moderate degree of (+) correlation between the number of rainy days and the number of rain coats sold. In other words, as the number of rainy days increases in a month, the number of rain coats sold in that month increases moderately.

Note: As per the textbook, coefficient of correlation is $-$0.67. However, as per the above solution coefficient of correlation should be +0.67.

#### Question 10:

The deviations from their means of two series (X and Y) are given below:

 X : −4 −3 −2 −1 0 +1 +2 +3 +4 Y : +3 −3 −4 0 +4 +1 +2 −2 −1

Calculate the Karl Pearson's coefficient of correlation and interpret the result .

 x y x2 y2 xy −4 3 16 9 −12 −3 −3 9 9 9 −2 −4 4 16 8 −1 0 1 0 0 0 4 0 16 0 1 1 1 1 1 2 2 4 4 4 3 −2 9 4 −6 4 −1 16 1 −4 Æ©x2 =60 Æ©y2 =60 Æ©xy =0

Thus, there is no correlation between series X and series Y.

#### Question 11:

Find the product moment correlation of the following data:

 X : 1 2 3 4 5 Y : 9 8 10 12 11

 X Y $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y x2 y2 xy 1 9 −2 −1 4 1 2 2 8 −1 −2 1 4 2 3 10 0 0 0 0 0 4 12 1 2 1 4 2 5 11 2 1 4 1 2 Æ©X =15 Æ©Y =50 Æ©x2 = 10 Æ©y2 = 10 Æ©xy = 8

N = 5

Thus, the product moment correlation is 0.80.

#### Question 12:

Calculate the correlation coefficient of the marks obtained by 12 students in Mathematics and Statistics and interpret it.

 Students : A B C D E F G H I J K L Marks (in Maths) : 50 54 56 59 60 62 61 65 67 71 71 74 Marks ( in Statis.) : 22 25 34 28 26 30 32 30 28 34 36 40

 Maths (X) Statis. (Y) $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)Y}\right)$(y) y2 xy 50 22 −12.5 156.25 −8.41 70.72 105.12 54 25 −8.5 72.25 −5.41 29.26 45.98 56 34 −6.5 42.25 3.59 12.88 −23.33 59 28 −3.5 12.25 −2.41 5.80 8.43 60 26 −2.5 6.25 −4.41 19.44 11.02 62 30 −0.5 0.25 −0.41 0.168 0.205 61 32 −1.5 2.25 1.59 2.52 −2.38 65 30 2.5 6.25 −.41 0.168 −1.02 67 28 4.5 20.25 −2.41 5.80 −10.84 71 34 8.5 72.25 3.59 12.88 30.51 71 36 8.5 72.25 5.59 31.24 47.51 74 40 11.5 132.25 9.59 91.96 110.28 Æ©X = 750 Æ©Y= 365 Æ©x2 = 595 Æ©y2 = 282.836 Æ©xy = 321.485

N = 12

Thus, there exists sufficiently high degree of positive correlation marks in Mathematics and marks in Statistics.

#### Question 13:

The height of fathers and sons are given below:

 Height of fathers (in inches) : 65 66 67 67 68 69 71 73 Height of sons (in inches) : 67 68 64 68 72 70 69 70
Calculate Karl Pearson's  coefficient of correlation .

 Father's Height (X) Son's Height (Y) $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y y2 xy 65 67 −3.25 10.56 −1.5 2.25 4.875 66 68 −2.25 5.06 −0.5 0.25 1.125 67 64 −1.25 1.56 −4.5 20.25 5.625 67 68 −1.25 1.56 −0.5 0.25 0.625 68 72 −0.25 0.06 3.5 12.25 −0.875 69 70 0 .75 0.56 1.5 2.25 1.125 74 69 2.75 7.56 0.5 0.25 1.375 73 70 4.75 22.56 1.5 2.25 7.125 Æ©X = 546 Æ©Y = 548 Æ©x2 = 49.48 Æ©y2 = 40 Æ©xy = 21

N = 8

#### Question 14:

Find Karl Pearson's coefficient of correlation from the following index numbers and interpret it.

 Wages (â‚¹) : 100 101 103 102 100 99 97 98 96 95 Cost of living : 98 99 99 97 95 92 95 94 90 91

 Wages (X) Cost of Living (Y) $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y y2 xy 100 98 0.9 0.81 3 9 27 101 99 1.9 3.61 4 16 7.6 103 99 3.9 15.21 4 16 5.6 102 97 2.9 8.41 2 4 5.8 100 95 0.9 0.81 0 0 0 99 92 −0.1 0.01 −3 9 0.3 97 95 −2.1 4.41 0 0 0 98 94 −1.1 1.21 −1 1 1.1 96 90 −3.1 9.61 −5 25 15.5 95 91 −4.1 16.81 −4 16 16.6 Æ©X = 991 Æ©Y = 950 Æ©x2 = 60.9 Æ©y2 = 96 Æ©xy = 65.2

N = 10

Wages and cost of living has high positive correlation. If wages increase by 1 unit, the cost of living increases by 0.85 units.

#### Question 15:

Find the product moment correlation between sales and expenses of the following 10 firms.

 Firms : 1 2 3 4 5 6 7 8 9 10 Sales : 50 50 55 60 65 65 65 60 60 50 Expenses : 11 13 14 16 16 15 15 14 13 13

 Sales (X) Expenses (Y) $\left(\mathbit{X}\mathbf{-}\overline{)\mathbit{X}}\right)$x x2 $\left(\mathbit{Y}\mathbf{-}\overline{)\mathbit{Y}}\right)$y y2 xy 50 11 −8 64 −3 9 24 50 13 −8 64 −1 1 8 55 14 −3 9 0 0 0 60 16 2 4 2 4 4 65 16 7 49 2 4 14 65 15 7 49 1 1 7 65 15 7 49 1 1 7 60 14 2 4 0 0 0 60 13 2 4 −1 1 −2 50 13 −8 64 −1 1 8 Æ©X = 580 Æ©X = 140 Æ©x2 = 360 Æ©y2 = 22 Æ©xy =  70

N = 10

Thus, there exists high positive correlation between sales of firm and expenses.

#### Question 16:

Calculate the coefficient of correlation for the following ages of husbands and wives in years at the time of their marriage.

 Age of Husbands : 23 27 28 28 29 30 31 33 35 36 Age of Wives : 18 20 22 27 21 29 27 29 28 29

 Husband (X) Wife (Y) $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y y2 xy 23 18 −7 49 −7 49 49 27 20 −3 9 −5 25 15 28 22 −2 4 −3 9 6 28 27 −2 4 2 4 −4 29 21 −1 1 −4 16 4 30 29 0 0 4 16 0 31 27 1 1 2 4 2 33 29 3 9 4 16 12 35 28 5 25 3 9 15 36 29 6 36 4 16 24 Æ©X = 300 Æ©Y = 250 Æ©x2 =138 Æ©y2 =164 Æ©xy =123

N = 10

Thus, there exists high positive correlation between age of husband and age of wife.

#### Question 17:

Find Karl Pearson's coefficient of correlation for the following data:

 Fertilizers used ( in tons ) : 15 18 20 24 30 35 40 50 Productivity ( in tons) : 85 90 95 105 120 130 150 160

 Fertilizers (X) Productivity (Y) $\left(\mathbit{X}\mathbit{-}\overline{)X}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y y2 xy 15 85 −14 196 −32.25 1040.06 451.5 18 93 −11 121 −24.25 588.06 266.75 20 95 −9 81 −22.25 495.06 200.25 24 105 −5 25 −12.25 150.06 61.25 30 120 1 1 2.75 7.56 2.75 35 130 6 36 12.75 162.56 76.5 40 150 11 121 32.75 1072.56 360.25 50 160 21 441 42.75 1827.56 897.75 Æ©X =232 Æ©Y = 938 Æ©x2 =1022 Æ©y2 =5343.48 Æ©xy =2317

N = 8

Thus, there exists high positive correlation between the amount of fertilizers used and the productivity.

#### Question 18:

Calculate product moment correlation  between age of cars and annual maintenance cost and comment.

 Age of cars (years) : 2 4 6 7 8 10 12 Annual maintenance â€‹Cost (in â‚¹) : 1600 1500 1800 1900 1700 2100 2000

 Age of Cars (X) Maintenance (Y) $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$x $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y x2 y2 xy 2 1600 −5 −200 25 40000 1000 4 1500 −3 −30 9 90000 900 6 1800 −1 0 1 0 0 7 1900 0 100 0 10000 0 8 1700 1 −100 1 10000 −100 10 2100 3 300 9 90000 900 12 2000 5 200 25 40000 1000 Æ©X = 49 Æ©Y =12600 Æ©x2 =70 Æ©y2 =280000 Æ©xy =3700

N = 7

Thus, there exists high positive correlation between age of car and maintenance cost.

#### Question 19:

Calculate coefficient of  correlation by Pearson's method between the density of population and death rate.

 Cities : A B C D E F Density : 200 500 400 700 600 300 Death rate : 10 16 14 20 17 13

 Density (X) Death rate (Y) $\left(\mathit{X}\mathit{-}\overline{)\mathit{X}}\right)$x x2 $\left(\mathbit{Y}\mathbit{-}\overline{)\mathit{Y}}\right)$y y2 xy 200 10 −250 62500 −5 25 1250 500 16 50 2500 1 1 50 400 14 −50 2500 −1 1 50 700 20 250 62500 5 25 1250 600 17 150 22500 2 4 300 300 13 −150 22500 −2 4 300 Æ©X = 2700 Æ©Y =90 Æ©x2 =175000 Æ©y2 =60 Æ©xy =3200

N = 6

Thus, there exists high positive correlation between density of population and death rate.

#### Question 20:

The total of the multiplication deviation of X and Y = 3044
Number of pairs of observations = 10
Total of the deviation of X = −20
Total of the deviation Y =  −170
Total of squares of deviation of X = 2264
Total of the squares of deviation of Y = 8288
Find out Karl Pearson' s coefficient of correlation when assumed mean of X and Y are 82 and 68 respectively.

Given:
n = 10
Σdxdy = 3044
Σdx = −17 assumed (instead of −170 see note below)
Σdy = −20
Σdx2 = 2264
Σdy2 = 8288

Hence, Karl Pearson's coefficient of correlation is 0.7.

Note:

1. If we take Σdx equal to −170, then the calculation of Karl Pearson's coefficient of correlation is as follows:

Since, the square root of a negative number is not defined among the set of real numbers, thus we have assumed Σdx as −17.

2. The answer as per the book is 0.78 and the answer as per our calculation is 0.70. The difference in the answer may be due to the assumption (i.e. Σdx = −17) made by us.

#### Question 21:

Number of pairs of observations of X and Y series = 10 .
X series   : Arithmetic Average = 65
: Standard Deviation = 23.33
Y series   : Arithmetic Average = 66
Standard deviation = 14.9
Summation  of products of corresponding deviations of X and Y series from their respective means = 2704.
Calculate product moment correlation of X and series.

Now,

Thus, product moment correlation of X and Y series is 0.78.

#### Question 22:

Calculate Spearman's rank correlation from the following data:

 X : 10 12 8 15 20 25 40 Y : 15 10 6 25 16 12 8

 X R1 Y R2 (R1 − R2)D D2 10 2 15 5 −3 9 12 3 10 3 0 0 8 1 6 1 0 0 15 4 25 7 −3 9 20 5 16 6 −1 1 25 6 12 4 2 4 40 7 8 2 5 25 $\sum {D}^{2}=48$

Given:
N = 7
Now,

Thus, the rank coefficient of correlation is 0.143.

#### Question 23:

The following are the marks obtained (out of 100) by a group of candidates in an employment interview held by two independent judges separately. Calculate the rank coefficient of correlation.

 Candidates : A B C D E F G H I J Judge X : 20 25 18 15 12 16 11 13 14 10 Judge Y : 22 20 15 14 10 8 11 12 13 9

 Candidate JudgeX R1 JudgeY R2 (R1 − R2)D D2 A 20 9 22 10 −1 1 B 25 10 20 9 1 1 C 18 8 15 8 0 0 D 15 6 14 7 −1 1 E 12 3 10 3 0 0 F 16 7 8 1 6 36 G 11 2 11 4 −2 4 H 13 4 12 5 −1 1 I 14 5 13 6 −1 1 J 10 1 9 2 −1 1 $\sum {D}^{2}=46$

Given:
N = 10
Now,
${r}_{k}=1-\frac{6\mathrm{\Sigma }{D}^{2}}{{N}^{3}-N}\phantom{\rule{0ex}{0ex}}=1-\frac{6×46}{1000-10}=1-\frac{276}{990}=1-0.278\phantom{\rule{0ex}{0ex}}{r}_{k}=0.721$
Thus, the rank coefficient of correlation is 0.721.

#### Question 24:

Two judges in a beautry competition rank the 12 entries as follows :

 X : 1 2 3 4 5 6 7 8 9 10 11 12 Y : 12 9 6 10 3 5 4 7 8 2 11 1
Calculate rank coefficient of correlation.

 X(R1) Y(R2) (R1 − R2)D D2 1 12 −11 121 2 9 −7 49 3 6 −3 9 4 10 −6 36 5 3 2 4 6 5 1 1 7 4 3 9 8 7 1 1 9 8 1 1 10 2 8 64 11 11 0 0 12 1 11 121 $\sum {D}^{2}=416$

Given,
N = 12
Now,
${r}_{k}=1-\frac{6\mathrm{\Sigma }{D}^{2}}{{N}^{3}-N}\phantom{\rule{0ex}{0ex}}=1-\frac{6×416}{1728-12}=1-\frac{2496}{1716}\phantom{\rule{0ex}{0ex}}=1-1.4545\phantom{\rule{0ex}{0ex}}=-0.454$
Thus, the rank coefficient of correlation is $-$0.454.

#### Question 25:

Calculate rank coefficient of correlation of the following data.

 X : 80 78 75 75 68 67 60 59 Y : 12 13 14 14 14 16 15 17

 X R1 Y R2 (R1 − R2)D D2 80 78 75 75 68 67 60 59 8 7 5.5 5.5 4 3 2 1 12 13 14 14 14 16 15 17 1 2 4 4 4 7 6 8 7 5 1.5 1.5 0$-$4$-$4$-$7 49 25 2.25 2.25 0 16 16 49 $\sum {D}^{2}=159.5$

Given:
N = 8
Now, in the given case, where a rank is repeated more than once, the following formula will be used,

Thus, the rank coefficient of correlation is $-$0.93.

#### Question 26:

Twelve entries were submitted in a flower show  competition. They were ranked by two judges as under:

 Entries : 1 2 3 4 5 6 7 8 9 10 11 12 Judge A : 7 8 2 1 9 3 12 11 4 10 6 5 Judge B : 6 4 1 3 11 2 12 10 5 9 7 8

 Entries Judge A (R1) Judge B (R2) (R1 − R2)D D2 1 7 6 −1 1 2 8 4 4 16 3 2 1 1 1 4 1 3 −2 4 5 9 11 −2 4 6 3 2 1 1 7 12 12 0 0 8 11 10 1 1 9 4 5 −1 1 10 10 9 1 1 11 6 7 −1 1 12 5 8 −3 9 $\sum {D}^{2}=40$

Given,
N = 12
To calculate Spearman's Rank correlation, the following formula is used,

Thus, the Spearman's rank correlation is 0.860.

#### Question 27:

Calculate the coefficient of rank correlation from the following data.

 X : 48 33 40 9 16 16 65 25 15 57 Y : 13 13 24 6 15 4 20 9 6 19

 X R1 Y R2 (R1 − R2)D D2 48 8 13 5.5 2.5 6.25 33 6 13 5.5 0.5 0.25 40 7 24 10 −3 9 9 1 6 2.5 −1.5 2.25 16 3.5 15 7 −3.5 12.25 16 3.5 4 1 2.5 6.25 65 10 20 9 1 1 25 5 9 4 1 1 15 2 6 2.5 −0.5 0.25 57 9 19 8 1 1 $\sum {D}^{2}=39.5$

Given:
N = 10
When, ranks are repeated more than once, the following formula is used to calculate coefficient of rank correlation.

Thus, the coefficient of rank correlation is 0.751.

#### Question 28:

Calculate rank coefficient of correlation between years of service and efficiency rating.

 Persons : A B C D E F G H I J Years of Service : 24 30 12 25 29 19 16 10 11 7 Efficiency rating : 66 51 84 66 45 81 72 97 92 70

 Persons Years of Service R1 Efficiency Rating R2 (R1 − R2)D D2 A 24 7 66 3.5 3.5 12.25 B 30 10 51 2 8 64 C 12 4 84 8 −4 16 D 25 8 66 3.5 4.5 20.25 E 29 9 45 1 8 64 F 19 6 81 7 −1 1 G 16 5 72 6 −1 1 H 10 2 97 10 −8 64 I 11 3 92 9 −6 36 J 7 1 70 5 −4 16 $\sum {D}^{2}=294.5$

Given:
N = 10
When ranks are repeated more than once, the following formula is used to calculate rank coefficient of correlation.

Thus, the rank coefficient of correlation is $-$0.787.

#### Question 29:

From the following data calculate coefficient of correlation by the method of rank differences.

 X : 75 68 95 70 60 80 81 50 Y : 120 134 150 115 110 140 142 100

 X R1 Y R2 (R1 − R2)D D2 75 5 120 4 1 1 68 3 134 5 −2 4 95 8 150 8 0 0 70 4 115 3 1 1 60 2 110 2 0 0 80 6 140 6 0 0 81 7 142 7 0 0 50 1 100 1 0 0 $\sum {D}^{2}=6$