Page 309 - DMTH404_STATISTICS

Unit 22: Correlation

22.5 Probable Error of r Notes

It is an old measure to test the significance of a particular value of r without the knowledge of
test of hypothesis. Probable error of r, denoted by P.E.(r) is 0.6745 times its standard error. The

value 0.6745 is obtained from the fact that in a normal distribution r  0.6745 S.E. covers 50%
of the total distribution.
According to Horace Secrist "The probable error of correlation coefficient is an amount which if
added to and subtracted from the mean correlation coefficient, gives limits within which the
chances are even that a coefficient of correlation from a series selected at random will fall.”

1 r 2 1 r 2


Since standard error of r, i.e., S.E.  ,  P.E.   r  0.6745
r
n n
22.5.1 Uses of P.E.(r)
(i) It can be used to specify the limits of population correlation coefficient  (rho) which are
defined as r – P.E.(r)  r  r + P.E.(r), where  denotes correlation coefficient in population
and r denotes correlation coefficient in sample.
(ii) It can be used to test the significance of an observed value of r without the knowledge of
test of hypothesis. By convention, the rules are:
(a) If |r| < 6 P.E.(r), then correlation is not significant and this may be treated as a situation of
no correlation between the two variables.
(b) If |r|> 6 P.E.(r), then correlation is significant and this implies presence of a strong
correlation between the two variables.

(c) If correlation coefficient is greater than 0.3 and probable error is relatively small, the
correlation coefficient should be considered as significant.

Example 11: Find out correlation between age and playing habit from the following
information and also its probable error.

Age : 15 16 17 18 19 20
No. of Students : 250 200 150 120 100 80
Regular Players : 200 150 90 48 30 12
Solution.

Let X denote age, p the number of regular players and q the number of students. Playing habit,
denoted by Y, is measured as a percentage of regular players in an age group, i.e., Y = (p/q)×100.
Table for calculation of r

2 2
X q p Y u = X - 17 v =Y - 40 uv u v
15 250 200 80 - 2 40 - 80 4 1600
16 200 150 75 - 1 35 - 35 1 1225
17 150 90 60 0 20 0 0 400
18 120 48 40 1 0 0 1 0
19 100 30 30 2 - 10 - 20 4 100
20 80 12 15 3 - 25 - 75 9 625
Total 3 60 - 210 19 3950

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