Section 1 : Correlation
 

[correlation][product moment correlation coefficient]

 

 

 

Correlation

Correlation is simply the relationship between one set of data and the other. It is expressed as a number called the correlation coefficient(r) and has a range of values between -1 and +1 .

There are a number of different correlation coefficients derived from different statistical principles. Here are a few of the more important ones:

product moment correlation coefficient ( rpm)

Spearman's rank correlation coefficient ( rS)

We shall deal with the Product Moment* correlation coefficient here. Spearman's rank correlation is dealt with at the end of Statistics - Section 3.

*sometimes called Pearson's Product Moment

 

 

Product Moment Correlation Coefficient (rpm)

By definition, the product moment correlation coefficient is the covariance of two variables divided by the product of their standard deviations.

product mean correlation coefficient equation

The more familiar form is:

product mean correlation coefficient equation

where,

covariance of x and y

the variance of x

the variance of y

 

Properties

Product moment correlation coefficient has no units.

It is symmetric. The coefficient for x versus y is the same as for y versus x.

It is i ndependent of change in origin and scale.

 

Example

The table describes the % exam results of 10 students taking Physics and French. Calculate the Product Moment correlation coefficient and comment on the result.

physics
45
76
71
68
54
59
74
60
48
43
French
57
84
69
72
83
65
75
72
56
38

The first step is to make a table to calculate x2 y2 xy

physics(x)
French(y)
x2
y2
xy
45
57
2,025
3,249
2,565
76
84
5,776
7,056
6,384
71
69
5,041
4,761
4,899
68
72
4,624
5,184
4,896
54
83
2,916
6,889
4,482
59
65
3,481
4,225
3,835
74
75
5,476
5,625
5,550
60
72
3,600
5,184
4,320
48
56
2,304
3,136
2,688
43
38
1,849
1,444
1,634
598
671
37,092
46,753
41,253
(Σx)
(Σy)
(Σx2)
(Σy2)
(Σxy)

 

 

 

 

 

 

 

 

 

 

 

covariance of x and y

 

the variance of x

 

the variance of y

 

remembering that,

product mean correlation coefficient equation

substituting for Sxy Sxx and Syy

From the table of relationship strengths, a rpm of 0.743 means that there is a strong relation between performance in physics and French. Students do equally well in both.

rpm relationship
-1.0 to –0.5 or 1.0 to 0.5 Strong
-0.5 to –0.3 or 0.3 to 0.5 Moderate
-0.3 to –0.1 or 0.1 to 0.3 Weak
–0.1 to 0.1 None or very weak


 

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