Tag: spearman's rank correlation method

Questions Related to spearman's rank correlation method

Correlation rank coefficient for the tied rank is 

business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $1-\dfrac{6\sum D^2}{n(n^2-1)}$

  2. $\dfrac{1}{n}\sum(x-\overline x)(y-\overline y)$

  3. $\dfrac{\dfrac{1}{n}\sum(x-\overline x)(y-\overline y)}{\sigma _x\sigma _y}$

  4. $1-\dfrac{6[\sum D^2+\dfrac{1}{12}(m _1^3-m _1)+frac{1}{12}(m-2^3-m _2)+....]}{n(n^2-1)}$

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Correct Option: D

Rank correlation depends on________________.

business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. a specific distribution

  2. the ranks of observations

  3. the ranks of unknown value

  4. the ranks of known value

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Correct Option: B
Explanation:

Rank correlation is the measure of association or strength between the ranked variables. For example: the rank of this  numerical data 65, 25, 75, 69 would be 3, 4, 1, 2 respectively.

The value of Spearman's rank coefficient lies between 

business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $2$ and $3$

  2. $1$ and $2$

  3. $0$ and $1$

  4. $-1$ and $1$

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Correct Option: D
Explanation:

Spearman's rank cofficient : $R=1-\dfrac { 6\sum { { d } _{ i }^{ 2 } }  }{ n({ n }^{ 2 }-1) } $

Its values lies between $-1$ and $1$
So option $D$ is correct.

If x, y are independent variable, then

business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $Cov\left ( x, y \right )=1$

  2. $r _{xy}=0$

  3. $r _{xy}=1$

  4. $Cov\left ( x, y \right )=0$

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Correct Option: B,D
Explanation:

Fact. If the variables are uncorrelated or independent then covariance
and coefficient of correlation between the variable both are equal to 0
i.e. $r _{xy}=Cov\left ( x, y \right )=0$ 

If $n=10, \sum x=4,\sum y=3, \sum x^2=8,\sum y^2=9$ and $\sum xy=3,$ then the coefficient of $r _{x,y}$ is

business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $\frac{3}{4}$

  2. $\frac{1}{5}$

  3. $\frac{1}{6}$

  4. $\frac{1}{4}$

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Correct Option: D
Explanation:

Correlation coefficient 
${ r } _{ x,y }=\dfrac { n\sum { xy } -\sum { x } \sum { y }  }{ \sqrt { \left[ n\sum { { x }^{ 2 }-{ \left( \sum { x }  \right)  }^{ 2 } }  \right] \left[ n\sum { { y }^{ 2 }-{ \left( \sum { y }  \right)  }^{ 2 } }  \right]  }  } $

$=\displaystyle\frac { 30-12 }{ \sqrt { 64\times 81 }  } $
$\Rightarrow r _{x,y}=\dfrac{1}{4}$

FInd the rank correlation from the following data:

S. No. 1 2 3 4 5 6 7 8 9 10
Rank Differences -2 -4 -1 3 2 0 -2 3 3 -2
business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. 0.64

  2. 0.50

  3. 0.45

  4. 0.34

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Correct Option: A
Explanation:

Rank Difference $(d)$ | $d^2$ | | --- | --- | --- | | 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. | -2 -4 -1 3 2 0 -2 3 3 -2 | 4 16 1 9 4 0 4 9 9 4 |

 $\sum d^2=60,\quad n=10$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}$

$r=1-\cfrac{6(60)}{10(10^2-1)}$

$r=1-\cfrac{360}{990}$

$r=0.6363....\approx 0.64$

The marks obtained by nine students in physics and Mathematics are given below:

Physics 48 60 72 62 56 40 39 52 30
Mathematics 62 78 65 70 38 54 60 32 31

calculate spearman's coefficient.

business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $r=0.66$

  2. $r=0.32$

  3. $r=0.53$

  4. $r =0.28$

Show answer
Correct Option: A
Explanation:

Descending order arranged data will be as follows:

Physics: $72,62,60,56,52,48,40,39,30$
MAthematics: $78,70,65,62,60,54,38,32,31$
Thus data will be

Mathematics $(M)$ | Rank $(P)$ | Rank $(P)$ | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 48 60 72 62 56 40 39 52 30 | 62 78 65 70 38 54 60 32 31 | 6 3 1 2 4 7 8 5 9 | 4 1 3 2 7 6 5 8 9 | 2 2 2 0 3 1 3 3 0 | 4 4 4 0 9 1 9 9 0 |

$n=9,\quad \sum d^2=40$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{40\times 6}{9(9^2-1)}=1-\cfrac{240}{720}=0.66$

Find the spearman's rank coefficient of correlation from the following data:

X 48 33 40 9 16 16 65 25 16 57
Y 13 13 24 6 15 4 20 9 6 19
business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $0.76$

  2. $0.52$

  3. $0.61$

  4. $0.85$

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Correct Option: A
Explanation:

Rank | $Y$ | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 48 33 40 9 16 16 65 25 16 57 | 3 5 4 10 7 7 1 6 7 2 | 13 13 24 6 15 4 20 9 6 19 | 5 5 1 8 4 10 2 7 8 3 | 2 0 3 2 3 3 1 1 1 1 | 4 0 9 4 9 9 1 1 1 1 |

$n=10,\quad \sum d^2=39$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 39}{10(10^2-1)}=1-\cfrac{234}{990}=0.76$

The final position of twelve clubs in a football league and the average attendance at their home matches were as follows. Calculate a coefficient of correlation by ranks.

Club A B C D E F G H I J K L
Position 1 2 3 4 5 6 7 8 9 10 11 12
Attendance (thousands) 27 30 18 25 32 12 19 11 32 12 12 15
business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. 0.34

  2. 0.56

  3. 0.32

  4. 0.48

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Correct Option: D
Explanation:

Attendance | Rank | Position | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | A B C D E F G H I J K L | 27 30 18 25 32 12 19 11 32 12 12 15 | 4 3 7 5 1 9 6 12 1 9 9 8 | 1 2 3 4 5 6 7 8 9 10 11 12 | 3 1 4 1 4 3 1 4 8 1 2 4 | 9 1 16 1 16 9 1 16 64 1 4 16 |

$n=12,\quad \sum d^2=154$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 154}{12(12^2-1)}=1-\cfrac{924}{1716}=0.48$

Find the rank correlation coefficient between the heights of fathers and sons from the following data:

Heights of fathers in inches  65 66 67 67 68 69 70 72
Height of sons in inches 67 68 65 68 72 72 69 71
business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $0.67$

  2. $0.58$

  3. $0.42$

  4. $0.92$

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Correct Option: B
Explanation:

Rank | Height(Son) | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 65 66 67 67 68 69 70 72 | 8 7 5 5 4 3 2 1 | 67 68 65 68 72 72 69 71 | 7 5 8 5 1 1 4 3 | 1 2 3 0 3 2 2 2   | 1 4 9 0 9 4 4 4 |

$n=08,\quad \sum d^2=35$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 35}{8(8^2-1)}=1-\cfrac{210}{504}=0.58$