Pearson Correlation Coefficient Examples
Example 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
Example 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
Example 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
Example 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
Example 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
Example 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 576 4624 3 26 85 2210 676 7225 4 35 67 2345 1225 4489 ∑ 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
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Pearson Correlation Coefficient
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 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.
The Pearson correlation coefficient, often symbolized as (r), is a widely used metric for assessing linear relationships between two variables. It yields a value ranging from –1 to 1, indicating both the magnitude and direction of the correlation. A change in one variable is mirrored by a corresponding change in the other variable in the same direction.
This article provides detailed information on the Pearson Correlation Coefficient, its meaning, formula, interpretation, examples, and FAQs.
Table of Content
- What is the Pearson Correlation Coefficient?
- Pearson’s Correlation Coefficient Formula
- Pearson Correlation Coefficient Table
- Pearson Correlation Coefficient Origin
- Types of Pearson Correlation Coefficient
- Adjusted Correlation Coefficient
- Weighted Correlation Coefficient
- Reflective Correlation Coefficient
- Scaled Correlation Coefficient
- Pearson’s Distance
- Circular Correlation Coefficient
- Partial Correlation
- Pearson Correlation Coefficient Interpretation
- Finding the Correlation Coefficient with Pearson Correlation Coefficient Formula
- Assumptions of Pearson Correlation Coefficient
- Correlation Coefficient Properties
- Pearson Correlation Coefficient Interpretation
- Bivariate Correlation
- Correlation Matrix
- Pearson Correlation Coefficient Examples
- Pearson Correlation Coefficient Practice Problems
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