Linear Correlation Coefficient

The linear correlation coefficient is the ratio between the covariance and the product of standard deviations of both variables.

The linear correlation coefficient is denoted by the letter r.

Linear Correlation Coefficient Formula

Properties of the Correlation Coefficient

1. The correlation coefficient does not change the measurement scale.

That is, if the height is expressed in meters or feet, the correlation coefficient does not change.

2. The sign of the correlation coefficient is the same as the covariance.

3. The linear correlation coefficient is a real number between −1 and 1.

−1 ≤ r ≤ 1

4. If the linear correlation coefficient takes values closer to −1, the correlation is strong and negative, and will become stronger the closer r approaches −1.

5. If the linear correlation coefficient takes values close to 1 the correlation is strong and positive, and will become stronger the closer r approaches 1

6. If the linear correlation coefficient takes values close to 0, the correlation is weak.

7. If r = 1 or r = −1, there is perfect correlation and the line on the scatter plot is increasing or decreasing respectively.

8. If r = 0, there is no linear correlation.

Example

The scores of 12 students in their mathematics and physics classes are:

Mathematics 2 3 4 4 5 6 6 7 7 8 10 10
Physics 1 3 2 4 4 4 6 4 6 7 9 10

Find the correlation coefficient distribution and interpret it.

xi yi xi ·yi xi2 yi2
2 1 2 4 1
3 3 9 9 9
4 2 8 16 4
4 4 16 16 16
5 4 20 25 16
6 4 24 36 16
6 6 36 36 36
7 4 28 49 16
7 6 42 49 36
8 7 56 64 49
10 9 90 100 81
10 10 100 100 100
72 60 431 504 380

Find the arithmetic means.

Arithmetic Means Operations

Calculate the covariance.

Covariance Calculations

Calculate the standard deviations.

Standard Deviation Operations

Apply the formula for the linear correlation coefficient.

Linear Correlation Coefficient

The correlation is positive.

As the correlation coefficient is very close to 1, the correlation is very strong.


The values of the two variables X and Y are distributed according to the following table:

Y/X 0 2 4
1 2 1 3
2 1 4 2
3 2 5 0

Calculate the correlation coefficient.

Turn the double entry table into a single table.

xi yi fi xi · fi xi2 · fi yi · fi yi2 · fi xi · yi · fi
0 1 2 0 0 2 2 0
0 2 1 0 0 2 4 0
0 3 2 0 0 6 18 0
2 1 1 2 4 1 1 2
2 2 4 8 16 8 16 16
2 3 5 10 20 15 45 30
4 1 3 12 48 3 3 12
4 2 2 8 32 4 8 16
    20 40 120 41 97 76

Correlation Coefficient Calculations

Correlation Coefficient Calculations

Correlation Coefficient Calculations

Correlation Coefficient Calculations

Correlation Coefficient Solution

The correlation is negative.

As the correlation coefficient is very close to 0, the correlation is very weak.