# 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 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 strong positive relationships
- -1 indicates strong negative relationships
- And 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 strengths of correlations using Pearson’s Correlation:

r value | crude estimates |

+.70 or higher | A very strong positive relationship |

+.40 to +.69 | Strong positive relationship |

+.30 to +.39 | Moderate positive relationship |

+.20 to +.29 | weak positive relationship |

+.01 to +.19 | No or negligible relationship |

0 | No relationship [zero correlation] |

-.01 to -.19 | No or negligible relationship |

-.20 to -.29 | weak negative relationship |

-.30 to -.39 | Moderate negative relationship |

-.40 to -.69 | Strong negative relationship |

-.70 or higher | The very strong negative relationship |

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

**Explain the 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<r1)

Negative values indicate a Negative Correlation (-1r<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**

**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:**

SUBJECT | AGE X | GLUCOSE LEVEL Y |

1 | 42 | 98 |

2 | 23 | 68 |

3 | 22 | 73 |

4 | 47 | 79 |

5 | 50 | 88 |

6 | 60 | 82 |

**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:**

SUBJECT | AGE X | Weight Y |

1 | 40 | 99 |

2 | 25 | 79 |

3 | 22 | 69 |

4 | 54 | 89 |

**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.