COEFFICIENT OF CORRELATION AND PROBABLE ERROR

The probable error of the coefficient of correlation helps in interpreting its value. With the help of probable error it is possible to determine the reliability of the value of the coefficient in so far as it depends on the conditions of random sampling. The probable error of the coefficient of correlation is obtained as follows:

                                                        P. E_r . =  _0.6745  1- r²
                                                                                   √N

where r is the coefficient of correlation and N the number of pairs of observations.

1. If the value of r is less than the probable error there is no evidence of correlation, i.e., the value of   r is not at all significant.

2. If the value of r is more than six times the probable error the coefficient of correlation is
practically certain, i.e., the value of r is significant.

3. By adding and subtracting the value of probable error from coefficient of correlation
we get respectively the upper and lower limits within which coefficient of correlation in the population can be expected to lie. Symbolically,

                 ρ = r + P. E.

where ρ (rho) denotes correlation in the population.

Let us compute probable error, assuming a coefficient of correlation of 0.80 and a sample of
16 pairs of items. We will have

P. E. _r =  _0.6745   1- 0.8²    = 0.06
                             √16

The limits of the correlation in population would be   r + P.E., i.e., 0.8 +  0.06 or 0.74 --- 0.86.

Instances are quite common wherein a correlation coefficient of 0.5 or even 0.4 is obviously
considered to be a fairly high degree of correlation by a writer or research worker. Yet a correlation
coefficient of 0.5 means that only 25 per cent of the variation is explained. A correlation coefficient of 0.4 means that only 16 percent of the variation is explained.

Conditions for the Use of Probable Error:-

The measure of probable error can be properly used only when the following three conditions exist:

1. The data must approximate a normal frequency curve (bell – shaped curve).

2. The statistical measure for which the P.E is computed must have been calculated from a sample.

3. The sample must have been selected in an unbiased manner and the individual items must
     be independent.

However, these conditions are generally not 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.

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