Karl Pearson Coefficient
Of the various mathematical methods of measuring correlation the Karl person's method, popularly termed as Pearson's coefficient of correlation, is most widely used in practice. In Pearson's the coefficient of correlation is represented by the symbol r. It is one of the very few symbols that are used universally for describing the degree of correlation between the two series. The formula for computing Pearson's r is:
R = (∑xy)/(Nσxσy)
Here, x = (‾x - x); y = (‾y - y)
σx = standard deviation of series X
σY = standard deviation of series Y
N = number of pairs of observations
R = the product moment correlation coefficient
This method is to be applied only where the deviations of items are taken from actual mean and not from assumed mean.
The value of the coefficient of correlation as obtained by the above formula should always lie between ± 1. When r = + 1, it means that there is perfect positive correlation between the variables. When r = - 1, that means there is perfect negative correlation between the variables. However, in practice such values of r as + 1, - 1 and 0 are rare. We generally get values which lie between + 1 and - 1 like + 0.8, - 0.26, etc. the coefficient of correlation describes not only the magnitude 0f correlation but also its direction. Thus = 0.8 would mean that the correlation is positive as the sigh of e is + and the magnitude of correlation is 0.8 similar - 0.26 means low degree of negative correlation.
The above formula for computing the person's coefficient of correlation can be transformed to the following form which is easier to apply.
R = (∑xy)/(√∑x2)(∑y2)
Where, x = (x - x) and y = (y - y)
It is quite obvious that while applying this formula we have not to calculate separately the standard deviation of X and Y series as is required by formula
1) This simplifies greatly the task of calculating the correlation coefficient.
2) Take the deviations of X series from the mean of X and represent these deviations by X.
3) Square these deviations and obtain the total i.e. ∑x2.
4) Take the deviations of Y series from the mean of Y and represent these deviations by Y.
5) Square these deviations and obtain the total i.e. ∑y2.
6) Multiply the deviations of X and Y series and obtain the total i.e. ∑xy.
7) Substitute the values of ∑xy, ∑x2 and ∑y2 in the above formula.
Illustration:
Calculate Karl Pearson's coefficient from the following data and interpret its value:
|
Roll no. of students
|
1
|
2
|
3
|
4
|
5
|
|
Marks in Accountancy
|
48
|
35
|
17
|
23
|
47
|
|
Marks in Statistics
|
45
|
20
|
40
|
25
|
45
|
Solution: let marks in Accountancy be denoted by X and marks in Statistics by Y.
|
Roll no.
|
X
|
(X - 34)x
|
X2
|
Y
|
(Y - 35)y
|
y2
|
xy
|
|
1
|
48
|
+14
|
196
|
45
|
+10
|
100
|
+140
|
|
2
|
35
|
+1
|
1
|
20
|
-15
|
225
|
-15
|
|
3
|
17
|
-17
|
289
|
40
|
+5
|
25
|
-85
|
|
4
|
23
|
-11
|
121
|
25
|
-10
|
100
|
+110
|
|
5
|
47
|
+13
|
169
|
45
|
+10
|
100
|
+130
|
|
|
ΣX = 170
|
Σx = 0
|
Σx2 = 776
|
Σy = 0
|
Σy = 0
|
Σy2 = 550
|
Σxy = 280
|
r = Σxy/√Σ x2 × Σy2
x = (X - ‾X), y = (Y - ‾Y)
‾X = Σ(X/N) = 170/5 = 34; ‾Y = Σ‾(Y/N) = 175/5 = 35
Σxy = 280, Σx2 = 776, Σy2 = 550
r = 280/√776 ×550 = 280/653 - 299 = 0.429