Correlation coefficient formula: Explanation and Solved Question

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The Correlation Coefficient formula in statistics is used to mathematically measure the relationship between two variables. The correlation coefficient can be calculated by various formulas, the most commonly used one is the Pearson’s correlation for linear regression. The other correlation coefficient formulas include Linear Correlation Coefficient Formula, Sample Correlation Coefficient Formula, and Population Correlation Coefficient Formula.

In this article, we will discuss about correlation coefficient formula and solve examples based on the correlation coefficient.

Table of Contents

What is coefficient of correlation?
Pearson’s correlation coefficient formula
Sample correlation coefficient formula
Population correlation coefficient formula
Relationship between covariance and correlation coefficient
Things to Remember
Sample Questions

Keyterms: Correlation, Coefficient, Statistics, Linear Correlation, Sample Correlation, Population Correlation, Linear regression, Variables

What is coefficient of correlation?

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The coefficient of correlation gives us a general idea about the relationship between two variables, whether or not they are correlated with each other. As we look into the correlation coefficient formula in the next sections you will notice that the returned value for the coefficient of correlation is always between -1 to +1.

Click here to learn other formulas of statistics.

The value of the correlation coefficient is positive (i.e. r > 0) when the two values in question are positively correlated to each other. In other words, with an increase in the value of one variable, the value of the other variable will increase as well.
Whereas, a negative correlation coefficient (r < 0) shows that the two values in question are negatively correlated to each other. In simple words, as the value of one variable increases the value of the other variable decreases.
When the correlation coefficient is equal to zero (r = 0), it shows that the variables are not correlated to each other at all.

Graph showing positive, negative, and no correlation

Graph showing positive, negative, and no correlation

Pearson’s correlation coefficient formula

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Pearson’s correlation coefficient is the most commonly used correlation coefficient formula. The value of correlation coefficient is denoted by ‘r’ which lies between -1 to +1.

The Pearson’s correlation coefficient formula is also known as the linear correlation coefficient formula.

Pearson’s correlation coefficient formula –

\(r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2] [n\sum y^2 - (\sum y)^2]}}\)

Where,

n = Quantity of Information

Σx = sum of values of x

Σy = sum of values of y

Σxy = Sum of the Product of first & Second Value

Σx² = Sum of the Squares of the First Value

Σy² = Sum of the Squares of the Second Value

Sample correlation coefficient formula

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The Sample correlation coefficient formula is given below –

\(r_{xy} = \frac{S_{xy}}{S_xS_y}\)

Where,

S_x = sample standard deviation of x

S_y = sample standard deviation of y

S_xy = sample covariance

Population correlation coefficient formula

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The population correlation coefficient is denoted by ρ (rho). The population correlation coefficient formula is –

\(\rho _{xy} = \frac{\sigma _{xy}}{\sigma _x \sigma_y}\)

Where,

σ_x and ρ_y = population standard deviations

σ_xy = population covariance

Relationship between covariance and correlation coefficient formula

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Both covariance and correlation coefficient are methods to measure the relationship between two variables. The relation between correlation and covariance is given below:

Correlation = \(\frac{Cov (x,y)}{\sigma x * \sigma y}\)

Where,

σ_x and σ_y = standard deviations of x and y respectively

cov(x,y) = covariance

Learn more about variance and covariance here.

Things to remember

Correlation coefficient is typically used to measure the relationship between two variables.
The most commonly used correlation coefficient formula is the Pearson’s correlation coefficient formula also known as the linear correlation coefficient. The other two formulas are the sample correlation coefficient and the population correlation coefficient.
The value of correlation coefficient lies between +1 to -1. Positive value of the correlation coefficient indicates the two variables are positively correlated whereas, a negative value of the correlation coefficient indicates the variables are negatively correlated.
When the correlation coefficient is equal to zero, it indicates that the two variables are not correlated to each other.
The Pearson’s correlation coefficient is given by

\(r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2] [n\sum y^2 - (\sum y)^2]}}\)

Sample correlation coefficient formula is given by-

\(r_{xy} = \frac{S_{xy}}{S_xS_y}\)

The population correlation coefficient formula is –

\(\rho _{xy} = \frac{\sigma _{xy}}{\sigma _x \sigma_y}\)

The relation between correlation and covariance is given below:

Correlation = \(\frac{Cov (x,y)}{\sigma x * \sigma y}\)

Solved questions

Ques 1. Calculate the correlation coefficient for the following data. X = 4, 8 ,12, 16 and Y = 5, 10, 15, 20. (5 marks)

Ans. Given variables are,

X = 4, 8 ,12, 16 and Y = 5, 10, 15, 20

Table for the required values.

x	y	x²	y²	xy
4	5	16	25	20
8	10	64	100	80
12	15	144	225	180
16	20	256	400	320
Σ x = 40	Σ y =50	480	750	600

According to the formula of Pearson’s correlation we have,

r(xy)= [(4×600)−(40×50)] / [√4(480)−40²√4(750)−50²]

Or, r(xy)=(2400−2000) / [√(1920−1600) x √(3000−2500)]

Or, r(xy)=400 / √320√500

Or, r(xy)=400 / 17.89×22.36

Or, r(xy)=400 / 400=1

Therefore, r(xy) = 1

Ques 2: Find the value of the correlation coefficient from the data given in the following table: (5 marks)

Ans. To calculate the correlation coefficient of the data given in the table, we need to construct a table with the values we need to put in the correlation coefficient formula:

\(r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2] [n\sum y^2 - (\sum y)^2]}}\)

SUBJECT	AGE (x)	GLUCOSE LEVEL (y)	xy	x²	y²
1	43	99	4257	1849	9801
2	21	65	1365	441	4225
3	25	79	1975	625	6241
4	42	75	3150	1764	5625
5	57	87	4959	3249	7569
6	59	81	4779	3481	6561
Σ	247	486	20485	11409	40022

So, from the table we have,

n = 6
Σx = 247
Σy = 486
Σxy = 20,485
Σx² = 11,409
Σy² = 40,022

So, r(xy) = 6(20,485) – (247 × 486) / [√[[6(11,409) – (247²)] × [6(40,022) – 486²]]]

Or, = 2868 / 5413.36

Or, = 0.5298

Ques 3. The correlation coefficient of a set of data is found to be 0.8. The standard deviation of data set x (σ_x) = 1 , and standard deviation of data set y (σ_y) = 1.4. Find out the covariance of the data. (4 marks)

Ans. The relationship between correlation and covariance is given by –

Correlation = \(\frac{Cov (x,y)}{\sigma x * \sigma y}\)

Values given in the question are,

Correlation = 0.8

σ_x = 1

σ_y = 1.4

Therefore, cov(x,y) = 0.8 x 1 x 1.4

= 1.12

Ques 4. Calculate the correlation coefficient for the following data. X = 4, 10, 11, 15 and Y = 5, 9, 13, 14. (5 marks)

Ans. Given variables are,

X = 4, 10, 11, 15 and Y = 5, 9, 13, 14.

Table for the required values.

x	y	x²	y²	xy
4	5	16	25	20
10	9	100	81	90
11	13	121	169	143
15	24	225	576	360
Σ x = 40	Σ y =41	462	851	613

Putting the values in the correlation coefficient formula,

\(r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2] [n\sum y^2 - (\sum y)^2]}}\)

Or, r = [(4 x 613) - (40 x 41)] / [√(4 x 46² - 40²)(4 x 851 - 41²)]

Or, r = 812 / 248 x 1723

Or, r = 812 / 427304

Or, r = 0.002

Ques 5. What is the significance of coefficient of correlation? (5 marks)

Ans. The coefficient of correlation tells us about how the two variables under observation are correlated to each other. The Pearson’s correlation coefficient is one of the most commonly used correlation coefficients. The value of the coefficient of correlation lies between -1 to +1.

When the value of the correlation coefficient is near -1, the two variables are said to be negatively correlated. Meaning with an increase in the value of one variable, the value of the second variable decreases.

Correlation coefficient equal to zero or near zero means that the variables are not correlated or weakly correlated.

The correlation coefficient value near 1 indicates that the variables are positively correlated. Meaning they are directly proportional to each other.

Ques 6. What does an r-value between – 0.2 to + 0.2 indicate? (5 marks)

Ans. The value of coefficient of correlation near 0 implies that the values are very weakly correlated with each other. When a dataset gives an r in the range -0.2 to +0.2, it indicates that they are weakly correlated to each other.

The correlation coefficient of -0.8 and 0.8 indicates that the dependent and independent variables are strongly correlated to each other.

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Correlation coefficient formula: Explanation and Solved Question

What is coefficient of correlation?

Pearson’s correlation coefficient formula

Sample correlation coefficient formula

Population correlation coefficient formula

Relationship between covariance and correlation coefficient formula

Things to remember

Solved questions

CBSE CLASS XII Related Questions

1.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

Similar Mathematics Concepts

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Correlation coefficient formula: Explanation and Solved Question

What is coefficient of correlation?

Pearson’s correlation coefficient formula

Sample correlation coefficient formula

Population correlation coefficient formula

Relationship between covariance and correlation coefficient formula

Things to remember

Solved questions

CBSE CLASS XII Related Questions

1. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

2.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

5.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).