Tag: spearman's coefficient of correlation

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

Correlation rank coefficient for the tied rank is 

  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. $1-\dfrac{6[\sum D^2+\dfrac{1}{12}(m _1^3-m _1)+frac{1}{12}(m-2^3-m _2)+....]}{n(n^2-1)}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

When ranks are tied, Spearman's rank correlation formula requires a correction term: 1 - [6ΣD² + Σ(m³-m)/12]/[n(n²-1)], where m is the number of tied observations in each group. This adjusts for the fact that tied ranks affect the correlation calculation. Option D shows this correction (though there's a typo in 'frac').

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

Rank correlation depends on________________.

  1. a specific distribution

  2. the ranks of observations

  3. the ranks of unknown value

  4. the ranks of known value
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

The value of Spearman's rank coefficient lies between 

  1. $2$ and $3$
  2. $1$ and $2$
  3. $0$ and $1$
  4. $-1$ and $1$
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

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

If x, y are independent variable, then

  1. $Cov\left ( x, y \right )=1$
  2. $r _{xy}=0$
  3. $r _{xy}=1$
  4. $Cov\left ( x, y \right )=0$
Reveal answer Fill a bubble to check yourself
B,D Correct answer
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. $r _{xy}=Cov\left ( x, y \right )=0$ 

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

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

  1. $\frac{3}{4}$
  2. $\frac{1}{5}$
  3. $\frac{1}{6}$
  4. $\frac{1}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 }  } $
$\Rightarrow r _{x,y}=\dfrac{1}{4}$

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

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

  1. 0.64

  2. 0.50

  3. 0.45

  4. 0.34
Reveal answer Fill a bubble to check yourself
A Correct answer
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 |

 $\sum d^2=60,\quad n=10$

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

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

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

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

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

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.

  1. $r=0.66$
  2. $r=0.32$
  3. $r=0.53$
  4. $r =0.28$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 |

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

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

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

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
  1. $0.76$
  2. $0.52$
  3. $0.61$
  4. $0.85$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 |

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

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

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

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

  1. 0.34

  2. 0.56

  3. 0.32

  4. 0.48
Reveal answer Fill a bubble to check yourself
D Correct answer
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 |

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

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

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

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
  1. $0.67$
  2. $0.58$
  3. $0.42$
  4. $0.92$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 |

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

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