Pearson Correlation Coefficient

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

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

Pearson’s Correlation Coefficient Formula

The Pearson’s correlation coefficient formula is the most commonly used and the most popular formula to get the correlation coefficient. It is denoted with the capital “R”. The formula for Pearson’s correlation coefficient is shown below,

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

The full name for Pearson’s correlation coefficient formula is Pearson’s Product Moment correlation (PPMC). It helps in displaying the Linear relationship between the two sets of the data.

The Pearson’s correlation helps in measuring the strength(it’s given by coefficient r-value between -1 and +1) and the existence (given by p-value )of a linear relationship between the two variables and if the outcome is significant we conclude that the correlation exists.

Cohen (1988) says that an absolute value of r of 0.5 is classified as large, an absolute value of 0.3 is classified as medium and an absolute value of 0.1 is classified as small.

The interpretation of the Pearson’s correlation coefficient is as follows:-

• A correlation coefficient of  1 means there is a positive increase of a fixed proportion of others, for every positive increase in one variable. Like, the size of the shoe goes up in perfect correlation with foot length.
• If the correlation coefficient is 0, it indicates that there is no relationship between the variables.
• A correlation coefficient of -1 means there is a negative decrease of a fixed proportion, for every positive increase in one variable. Like, the amount of water in a tank will decrease in a perfect correlation with the flow of a water tap.

Steps to find the correlation coefficient with Pearson’s correlation coefficient formula:

Step 1: Firstly make a chart with the given data like subject, x, and y and add three more columns in it xy,x² and y².

Step 2: Now multiply the x and y columns to fill the xy column. For example:- in x we have 24 and in y we have 65 so xy will be 24×65=1560.

Step 3: Now, take the square of the numbers in the x column and fill the x² column.

Step 4: Now, take the square of the numbers in the y column and fill the y² column.

Step 5: Now, add up all the values in the columns and put the result at the bottom. Greek letter sigma (Σ) is the short way of saying summation.

Step 6: Now, use the formula for Pearson’s correlation coefficient:-

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

To know which type of variable we have either positive or negative.

Sample Problems

Problem 1: 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:

0.69, 0.42, -0.23, -0.99

Tell whether the relationship is negative or positive

0.69: The relationship between the variables is a strong positive relationship

0.42: The relationship between the variables is a strong positive relationship

-0.23: The relationship between the variables is a weak negative relationship

-0.99: The relationship between the variables is a very strong negative relationship

Problem 2: Calculate the correlation coefficient for the following data by the help of Pearson’s correlation coefficient formula:

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	362	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

The correlation coefficient is 0.064

Problem 3: Calculate the correlation coefficient for the following table with the help of Pearson’s correlation coefficient formula:

SUBJECT	AGE X	Weight Y
1	40	99
2	25	79
3	22	69
4	54	89

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 =4.

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

The correlation coefficient is -2.196

Problem 4: Calculate the correlation coefficient for the following data with the help of Pearson’s correlation coefficient formula:

X = 5 ,9 ,14, 16

and

Y = 6, 10, 16, 20 .

Solution:

Given variables are,

X = 5 ,9 ,14, 16

and

Y = 6, 10, 16, 20 .

To, find the correlation coefficient of the following variables Firstly a table 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	30	25	36
9	10	90	81	100
14	16	224	196	256
16	20	320	256	400
∑ 44	∑ 52	∑ 664	∑ 558	∑ 792

∑xy= 664

∑x=44

∑y=52

∑x² =558

∑y² =792

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(664) – (44)(52) / √ [4(558)-(44)²][4(792)-(52)²]

R= 368 / √[296][464]        

R=368/370.599

R=0.994

The correlation coefficient is 0.994

Problem 5: Calculate the correlation coefficient for the following data by the help of Pearson’s correlation coefficient formula:

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	1500	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

The correlation coefficient is -0.3471

Problem 6: Calculate the correlation coefficient for the following data by the help of Pearson’s correlation coefficient formula:

SUBJECT	Height X	Weight Y
1	43	78
2	24	68
3	26	85
4	35	67

Solution:

Make a table from the given data and add three more columns of XY , X² and Y² and add all the values in the columns to get ∑xy, ∑x, ∑y, ∑x² and ∑y² and n =4.

SUBJECT	Height X	Weight Y	XY	X²	Y²
1	43	78	3354	1849	6084
2	24	68	1632	567	4624
3	26	85	2210	676	7225
4	35	67	2345	1225	4889
∑	128	298	9541	4317	22422

∑xy= 9541

∑x=128

∑y=298

∑x² =4317

∑y² 22422

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(9541) – (128)(298) / √ [4(4317)-(128)²][4(22422)-(298)²]

R= 20 / √ [884][884]

R=20/884

R=0.02262

The correlation coefficient is 0.02262