Linear Correlation Coefficient Formula

Correlation coefficients are used to measure how strong a relationship is between two variables. There are different types of formulas to get a correlation coefficient, one of the most popular is Pearson’s correlation (also known as Pearson’s R) which is commonly used for linear regression. Pearson’s correlation coefficient is denoted with the symbol “R”. The correlation coefficient formula returns a value between 1 and -1. Here,

• 1 indicates strong positive relationships
• -1 indicates strong negative relationships
• A result of zero indicates no relationship at all

Linear Correlation Coefficient Formula

The linear correlation coefficient is known as Pearson’s r or Pearson’s correlation coefficient. Which reflects the direction and strength of the linear relationship between the two variables x and y. It returns a value between -1 and +1. In this -1 indicates a strong negative correlation and +1 indicates a strong positive correlation. If it lies 0 then there is no correlation. This is also known as zero correlation.

The “crude estimates” for interpreting the strengths of correlations using Pearson’s Correlation:

The formula used to get the linear correlation coefficient of the data is :

R = n(∑xy) – (∑x)(∑y) / √[n∑x²-(∑x)²][n∑y²-(∑y)²

Types of Linear Correlation Coefficients

The linear correlation coefficient is reflected by Pearson’s r. So, the value of r can be range between +1 and -1.

There are three types of linear correlation coefficient as follows:

• Positive values indicate a Positive Correlation (0 < r < 1)
• Negative values indicate a Negative Correlation (-1 < r < 1)
• A Value of 0 indicates No Correlation (r=0)

Positive correlation: In positive correlation both the variables move in the same direction. If one increases the other also increases and if one decreases the other also decreases. Whenever the r indicates a positive value it shows a positive relationship

Negative correlation: In negative correlation both the variables move in different directions. If one increases the other decreases and if one decreases the other increases. Whenever the r indicates a negative value it shows a negative relationship

No correlation: when there is no statistical association between the variables. They are said to have no correlation. In this case, their correlation coefficient (also known as r) is 0.

Sample Problems – Linear Correlation Coefficient Formula

Problem 1: Calculate the correlation coefficient for the following data:

X = 5, 9,14, 16

and

Y = 6, 10, 16, 20

Solution:

Given variables are,

X = 12,16 ,4, 8

and

Y = 15, 20, 55, 10

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula also add all the values in the columns to get the values used in the formula

X	Y	XY	X²	Y²
5	6	180	144	225
9	10	320	256	400
14	16	20	16	20
16	20	80	56	100
∑40	∑50	∑600	∑480	∑750

∑xy = 600

∑x = 40

∑y = 50

∑x² = 470

∑y² = 750

n = 4

Put all the values in the Pearson’s correlation coefficient formula:-

R = n(∑xy) – (∑x)(∑y) / √[n∑x²-(∑x)²][n∑y²-(∑y)² 

R = 4(600) – (40)(50) / √[4(470)-(40)²][4(750)-(50)²]

R = 400 / √[320][500]

R = 400/400

R =1

It shows that the relationship between the variables of the data is a very strong positive relationship.

Problem 2: Find the value of the correlation coefficient from the following table:

Solution:

Make a table from the given data and add three more columns of XY, X², and Y² also add all the values in the columns to get ∑xy, ∑x, ∑y, ∑x², and ∑y² and n =6.

SUBJECT	AGE X	GLUCOSE  LEVEL Y	XY	X²	Y²
1	42	98	4116	1764	9604
2	23	68	1564	529	4624
3	22	73	1606	484	5329
4	47	79	3713	2209	6241
5	50	88	4400	2500	7744
6	60	82	4980	3600	6724
∑	244	488	20379	11086	40266

∑xy= 20379

∑x=244

∑y=488

∑x² =11086

∑y² =40266

n =6.

Put all the values in the Pearson’s correlation coefficient formula:-

R = n(∑xy) – (∑x)(∑y) / √ [n∑x²-(∑x)²][n∑y²-(∑y)² 

R = 6(20379) – (244)(488) / √ [6(11086)-(244)²][6(40266)-(488)² 

R = 3202 / √ [6980][3452]

R = 3202/4972.238

R = 0.6439

It shows that the relationship between the variables of the data is a strong positive relationship.

Problem 3: Calculate the correlation coefficient for the following data:

X = 21,31,25,40,47,38

and

Y = 70,55,60,78,66,80

Solution:

Given variables are,

X = 21,31,25,40,47,38

and

Y = 70,55,60,78,66,80

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula also add all the values in the columns to get the values used in the formula

X	Y	XY	X²	Y²
21	70	1470	441	4900
31	55	1705	961	3025
25	60	1400	625	3600
40	78	3120	1600	6084
47	66	3102	2209	4356
38	80	3040	1444	6400
∑202	∑409	∑13937	∑7280	∑28265

∑xy= 13937

∑x=202

∑y=409

∑x² =7280

∑y² =28265

n =6

Put all the values in the Pearson’s correlation coefficient formula:-

R=  n(∑xy) – (∑x)(∑y) / √ [n∑x²-(∑x)²][n∑y²-(∑y)² 

R= 6(13937) – (202)(409) / √ [6(7280)-(202)²][6(28265)-(409)²]

R= 1004 / √[2876][2909]

R=1004 / 2892.452938

R=-0.3471

It shows that the relationship between the variables of the data is a moderate positive relationship.

Problem 4: Calculate the correlation coefficient for the following data:

X= 12, 10, 42, 27,35,56

and

Y = 13, 15, 56, 34,65,26

Solution:

Given variables are,

X= 12, 10, 42, 27,35,56

and

Y = 13, 15, 56, 34,65,26

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula also add all the values in the columns to get the values used in the formula

X	Y	XY	X²	Y²
12	13	156	144	169
10	15	150	100	225
42	56	2353	1764	3136
27	34	918	729	1156
35	65	2275	1225	4225
56	26	1456	3136	676
∑182	∑209	∑7307	∑7098	∑9587

∑xy= 7307

∑x=182

∑y=209

∑x² =7098

∑y² =9587

n =6

Put all the values in the Pearson’s correlation coefficient formula:-

R=  n(∑xy) – (∑x)(∑y) / √ [n∑x²-(∑x)²][n∑y²-(∑y)²

R=  6(7307) – (182)(209)  / √ [6(7098)-(182)²][6(9587)-(209)²] 

R= 5804 / √[9464][13841]

R= 5804/11445.139

R=0.5071

It shows that the relationship between the variables of the data is a strong positive relationship.

Problem 5: There is some correlation coefficient that was given to tell whether the variables are positive or negative?

0.69

0.42

-0.23

-0.99

Solution:

The given correlation coefficient is as follows:

0.64

0.46

-0.29

-0.95

Tell whether the relationship is negative or positive

0.64

The relationship between the variables is a strong positive relationship

0.46

The relationship between the variables is a strong positive relationship

-0.29

The relationship between the variables is a weak negative relationship

-0.95

The relationship between the variables is a very strong negative relationship.

Problem 6: Calculate the correlation coefficient for the following data:

X = 10, 13, 15 ,17 ,19

and

Y = 5,10,15,20,25.

Solution:

Given variables are,

X = 10, 13, 15 ,17 ,19

and

Y = 5,10,15,20,25.

To, find the correlation coefficient of the following variables Firstly a table is to be constructed as follows, to get the values required in the formula also add all the values in the columns to get the values used in the formula.

X	Y	XY	X²	Y²
10	5	50	100	25
13	10	130	169	100
15	15	225	225	225
17	20	340	289	400
19	25	475	361	625
∑74	∑75	∑1103	∑1144	∑1375

∑xy= 1103

∑x=74

∑y=75

∑x² =1144

∑y² =1375

n =5

Put all the values in the Pearson’s correlation coefficient formula:-

R=  n(∑xy) – (∑x)(∑y) / √ [n∑x²-(∑x)²][n∑y²-(∑y)²

R=  5(1103) – (74)(75)  / √ [5(1144)-(74)²][5(1375)-(75)²] 

R= -35 / √[244][1250]

 R= -35/552.26

 R=0.0633

It shows that the relationship between the variables of the data is a negligible relationship.

Problem 7: Find the value of the correlation coefficient from the following table:

Solution:

SUBJECT	AGE X	Weight Y	XY	X²	Y²
1	40	99	3960	1600	9801
2	25	79	1975	625	6241
3	22	69	1518	484	4761
4	54	89	4806	2916	7921
∑	151	336	12259	5625	28724

∑xy= 12258

∑x=151

∑y=336

∑x² =5625

∑y² 28724

n =4

Put all the values in the Pearson’s correlation coefficient formula:-

R=  n(∑xy) – (∑x)(∑y) / √ [n∑x²-(∑x)²][n∑y²-(∑y)²

R= 4(12258) – (151)(336) / √ [4(5625)-(151)²][4(28724)-(336)²]

R= -1704 / √ [-301][-2000]

 R=-1704/775.886

R=-2.1961

It shows that the relationship between the variables of the data is a very strong negative relationship.

Practice Problems on Linear Correlation Coefficient Formula

1. Calculate r for the data points: (1,2), (2,3), (3,5), (4,4), (5,6)

2. Find the correlation coefficient for: (0,1), (1,3), (2,4), (3,5), (4,8)

3. Determine r for: (-2,4), (-1,1), (0,0), (1,1), (2,4)

4. Calculate the correlation coefficient for: (1,10), (2,8), (3,6), (4,4), (5,2)

5. Find r for: (0,0), (1,2), (2,4), (3,6), (4,8)

6. Determine the correlation coefficient for: (1,1), (2,4), (3,9), (4,16), (5,25)

7. Calculate r for: (10,11), (20,22), (30,31), (40,43), (50,52)

8. Find the correlation coefficient for: (1,5), (2,7), (3,6), (4,8), (5,9)

9. Determine r for: (100,102), (200,199), (300,301), (400,398), (500,503)

10. Calculate the correlation coefficient for: (1,1), (2,1), (3,2), (4,3), (5,5)

Summary

The Linear Correlation Coefficient, also known as Pearson’s correlation coefficient, is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship. The formula involves calculating the covariance of the two variables and dividing it by the product of their standard deviations. This coefficient is widely used in statistics, data analysis, and various scientific fields to quantify the degree of linear dependence between pairs of observations.