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Karl Pearson’s ProductMoment Correlation Coefficient or simply Pearson’s Correlation Coefficient for short, is one of the important methods used in Statistics to measure Correlation between two variables.
A few words about Karl Pearson. Karl Pearson was a British mathematician, statistician, lawyer and a eugenicist. He established the discipline of mathematical statistics. He founded the world’s first statistics department In the University of London in the year 1911. He along with his colleagues Weldon and Galton founded the journal “Biometrika” whose object was the development of statistical theory.
The Correlation between two variables X and Y, which are measured using Pearson’s Coefficient, give the values between +1 and 1. When measured in population the Pearson’s Coefficient is designated the value of Greek letter rho (ρ). But, when studying a sample, it is designated the letter r. It is therefore sometimes called Pearson’s r. Pearson’s coefficient reflects the linear relationship between two variables. As mentioned above if the correlation coefficient is +1 then there is a perfect positive linear relationship between variables, and if it is 1 then there is a perfect negative linear relationship between the variables. And 0 denotes that there is no relationship between the two variables.
The degrees 1, +1 and 0 are theoretical results and are not generally found in normal circumstances. That means the results cannot be more than 1, +1. These are the upper and the lower limits.
Pearson’s Coefficient computational formula
Sample question: compute the value of the correlation coefficient from the following table:
Subject

Age x

Weight Level y

1

43

99

2

21

65

3

25

79

4

42

75

5

57

87

6

59

81

Step 1: Make a chart. Use the given data, and add three more columns: xy, x2, and y2.
Subject

Age x

Weight Level y

xy

x^{2}

y^{2}

1

43

99


2

21

65


3

25

79


4

42

75


5

57

87


6

59

81

Step 2: Multiply x and y together to fill the xy column. For example, row 1 would be 43 × 99 = 4,257
Step 3: Take the square of the numbers in the x column, and put the result in the x^{2} column
Subject

Age x

Weight Level y

xy

x^{2}

y^{2}

1

43

99

4257

1849


2

21

65

1365

441


3

25

79

1975

625


4

42

75

3150

1764


5

57

87

4959

3249


6

59

81

4779

3481

Step 4: Take the square of the numbers in the y column, and put the result in the y^{2} column.
Step 5: Add up all of the numbers in the columns and put the result at the bottom.2 column. The Greek letter sigma (Σ) is a short way of saying “sum of.”
Subject

Age x

Weight Level y

xy

x^{2}

y^{2}

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

Step 6: Use the following formula to work out the correlation coefficient.
The answer is: 1.3787 × 10^{4}
The range of the correlation coefficient is from 1 to 1. Since our result is 1.3787 × 10^{4}, a tiny positive amount, we can’t draw any conclusions one way or another.
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