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.

x_i	y_i	x_i ·y_i	x_i²	y_i²
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

1º Find the arithmetic means.

2º Calculate the covariance.

3º Calculate the standard deviations.

4º Apply the formula for the 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.

x_i	y_i	f_i	x_i · f_i	x_i² · f_i	y_i · f_i	y_i² · f_i	x_i · y_i· f_i
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

The correlation is negative.

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

x_i	y_i	x_i ·y_i	x_i²	y_i²
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

x_i	y_i	f_i	x_i · f_i	x_i² · f_i	y_i · f_i	y_i² · f_i	x_i · y_i· f_i
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

x_i	y_i	x_i ·y_i	x_i²	y_i²
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

x_i	y_i	f_i	x_i · f_i	x_i² · f_i	y_i · f_i	y_i² · f_i	x_i · y_i· f_i
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

x_i	y_i	x_i ·y_i	x_i²	y_i²
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

x_i	y_i	f_i	x_i · f_i	x_i² · f_i	y_i · f_i	y_i² · f_i	x_i · y_i· f_i
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