Probable Error
The probable Error is the part of the coefficient of correlation which helps in interpreting its value. By using probable error it is possible to determine the reliability of the value of the coefficient as it depends on the conditions of random sampling. Probable Error is obtained as follows:
P. E. r = 0.6745 (1 - r2 )/N
Where r is the coefficient of correlation and N is the number of pairs of observations
If the value of r is less than the probable error then there is no evidence of correlation, the value of r is not at all significant.
If the value of r is greater than six times the probable error, the coefficient of correlation is practically certain, and then the value of r is significant.
By adding and subtracting the value of probable error from the coefficient of correlation we get respectively the upper and lower limits within which the coefficient of correlation in the population can be expected to lie symbolically,
P = r ± P.E.
Where p (rho) represents correlation in the population,
Let us compute the probable error, suppose the coefficient of correlation of 0.80 and a sample of pairs of items. We have
P.E = 0.6745 {1- (.8)2}/16 = .06
The limits of the correlation in the population would be r ± P.E... 0.8 ± 0.06 or 0.74 - 0.86.
The instances are quite common wherein a correlation coefficient of 0.5 or even 0.4 is obviously taken to be a fairly high degree of correlation by a writer or research worker. Till now a correlation coefficient of 0.5 means that only 25 percent of the variation is explained, A correlation coefficient of 0.4 means that only 16 per cent of the variation is explained.
The conditions for the use of probable error are:
The measurement of the probable error can be properly used only when the following 3 conditions exist
1. The data should approximate a normal frequency curve (bell - shaped curve).
2. The statistical measure for which the P.E is computed should have been calculated from a sample.
3. The sample must be selected even in an unbiased manner and the individual items must be independent.
4. However these conditions are normally satisfied and as such the reliability of the correlation coefficient is determined largely on the basis of exterior tests of reasonableness which are often of a statistical character.
Illustration:
If r = 0.6 and N = 64, then find out the probable error of the coefficient of correlation and determine the limits for population r.
Solution
P.E r = 0.675 (1 - r2)/N, r = 0.6 and N = 64
P.E r= 0.6745 {1 - (0.6)2}/(64) = 0.6745 x (0.64/8) = 0.054
Limits of population correlation = 0.6 ± 0.054 = 0.546 to 0.654.