2uData ⮞Regression & Correlation ⮞Correlation

Correlation

As we already mentioned, there are linear relationship and nonlinear one. In this page, we investigate only linear relationship.

Consider two paired variables `x_i` and `y_j`, correlation coefficient `r` between these variables is defined by:

`r=(sum (x_i-bar x)(y_i-bar y))/sqrt(sum (x_i-bar x)^2xxsum (y_i-bar y)^2)`

(1)

There are some characteristics of correlation coefficient:

• value of `r` is in interval (- 1, 1),
• `r>0` represent positive correlation, `r< 0` represent negative correlation,
• the higher `|r|` is, the stronger the relationship is.

Fig. 1 illustrates these characteristics of `r`.

Fig. 1 Main cases of correlation coefficient r (Source : Wikipedia)

Example : Heights and weights of 10 students are shown on Table 1.

Table 1 Heights and weights of 10 students

	1	2	3	4	5	6	7	8	9	10
Height (m)	1,57	1,62	1,58	1,64	1,74	1,68	1,71	1,60	1,66	1,64
Weight (kg)	55	72	52	65	82	79	78	66	78	71

Using (7.1) : `r = 0,8932`.

• `r>0` : this is a positive correlation, it means that the higher the student is, the heavier he weighs.
• the value of `|r|` rather high, so there is a strong correlation between height and weight of students.