Chapters

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. 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.

xiyixi ·yixi²y
21241
33999
428164
44161616
54202516
64243616
66363636
74284916
76424936
87566449
1099010081
1010100100100
7260431504380

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/X024
1213
2142
3250

Calculate the correlation coefficient.

Turn the double entry table into a single table.

xiyifixi · fixi² · fiyi · fiy · fixi · yi · fi
01200220
02100240
032006180
21124112
22481681616
2351020154530
41312483312
4228324816
2040120419776     The correlation is negative.

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

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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