Question

Calculate the coefficient of correlation between $x$ and $y$ for the data

x	1	2	3	4	5	6	7	8	9	10
y	3	10	5	1	2	9	4	8	7	6

A) 0.12
B) 0.19
C) 0.22
D) 0.62

Answer 1

To calculate the coefficient to correlation, we will be using Karl Pearson’s coefficient of correlation which is given by ${r_{xy}} = \dfrac{{n\sum {{x_i}{y_i}} - \sum {{x_i}} \sum {{y_i}} }}{{\sqrt {n\sum {{x_i}^2} - {{\left( {\sum {{x_i}} } \right)}^2}} \sqrt {n\sum {{y_i}^2} - {{\left( {\sum {{y_i}} } \right)}^2}} }}$. Since there are ten terms, $n = 10$. A table is drawn to find the required terms to be substituted in Karl Pearson’s coefficient of correlation equation and do further simplification.

Complete step-by-step answer:
We need to find the correlation coefficient between $x$ and $y$.
Karl Pearson’s coefficient of correlation is given as
${r_{xy}} = \dfrac{{n\sum {{x_i}{y_i}} - \sum {{x_i}} \sum {{y_i}} }}{{\sqrt {n\sum {{x_i}^2} - {{\left( {\sum {{x_i}} } \right)}^2}} \sqrt {n\sum {{y_i}^2} - {{\left( {\sum {{y_i}} } \right)}^2}} }}$
Where $n$ is the number of terms.
${x_i}$ is the sum of values of $x$.
${y_i}$ is the sum of values of $y$.
Now, find the values,

$x$	$y$	${x^2}$	${y^2}$	$xy$
1	3	1	9	3
2	10	4	100	20
3	5	9	25	15
4	1	16	1	4
5	2	25	4	10
6	9	36	81	54
7	4	49	16	28
8	8	64	64	64
9	7	81	49	63
10	6	100	36	60
$\sum {{x_i}} = 55$	$\sum {{y_i}} = 55$	$\sum {{x_i}^2} = 385$	$\sum {{y_i}^2} = 385$	$\sum {{x_i}{y_i}} = 321$

Substitute the values in the formula,
$\Rightarrow {r_{xy}} = \dfrac{{10 \times 321 - 55 \times 55}}{{\sqrt {10 \times 385 - {{\left( {55} \right)}^2}} \sqrt {10 \times 385 - {{\left( {55} \right)}^2}} }}$
Simplify the terms,
$\Rightarrow {r_{xy}} = \dfrac{{3210 - 3025}}{{\sqrt {3850 - 3025} \times \sqrt {3850 - 3025} }}$
Subtract the values,
$\Rightarrow {r_{xy}} = \dfrac{{185}}{{\sqrt {825} \times \sqrt {825} }}$
We know that,
$\sqrt a \times \sqrt a = {\left( {\sqrt a } \right)^2} = a$
Use this formula in the denominator,
$\Rightarrow {r_{xy}} = \dfrac{{185}}{{825}}$
Divide numerator by the denominator,
$\therefore {r_{xy}} = 0.22$

Hence, option (C) is correct.

Note: The degree of correlation can be determined as follows:
There will be a perfect correlation between two variables if the value of the correlation coefficient is near $\pm 1$, that is, as one variable increases, the other variable tends to also increase (if positive) or decrease (if negative).
There will be a high degree of correlation or strong correlation between two variables if the value of the correlation coefficient lies between $\pm 0.5$ and $\pm 1$.
There will be a moderate degree of correlation or medium correlation between two variables if the value of the correlation coefficient lies between $\pm 0.3$ and $\pm 0.49$.
There will be a low degree of correlation or a small correlation between two variables if the value of the correlation coefficient lies below $\pm 0.29$.
There will be no correlation between the two variables if the value of the correlation coefficient is zero.