218 
From this is derived the probable error of 
estimate if any given year in the future is pre¬ 
dicted to have a rainfall equal to the mean 
of the whole period. 
P. E. = 0.6745 (20.3) = 13.7 per cent 
This means that the odds are equal, that 
there is a fifty-fifty likelihood that the rain¬ 
fall in any future year will not differ from 
the mean by more than 13.7 per cent. This 
is a basis of prediction that is sound and 
easily arrived at. Now we must inquire 
whether, by use of cycles shown in the 
record, we can reduce this probable error of 
estimate or prediction. 
Taking the 16-year cycle as most promis¬ 
ing, we say that the rainfall of any given 
year is most likely to be similar to that of a 
year 16, or 32, or 48 years earlier. For such 
a group of years, the best representative 
value is the average, and for each year, the 
deviation from that group average is the 
error that would have resulted from using 
the average. Hence we take the whole series 
of deviations of the rainfall of each year 
from the average of the years separated from 
it by multiples of 16. The root-mean-square 
of this series is determined. This is the 
standard deviation of the estimates based on 
use of the mean of the 16-year cycle. For any 
single series there are only 4 years, and it is 
granted that the average can be much dis¬ 
torted by a very exceptional year. However, 
by including the whole series in the root- 
mean-square calculation we derive a stand¬ 
ard deviation comparable to the standard 
deviation about a single mean. This turns 
out to be 14.4 per cent. The corresponding 
probable error is 9.7 per cent. This suggests 
that by using the mean of the 16-year cycle, 
rather than the mean of the whole series, the 
fifty-fifty likelihood is that the estimate will 
not be in error by more than 9.7 per cent. 
Here we have an apparent improvement of 
about one third of the expected error in¬ 
volved in using the over-all mean. This is 
PACIFIC SCIENCE, Vol. 1, October, 1947 
an improvement which can be used, even 
though it is not startling. 
The following table shows the standard 
deviations and probable errors of variations 
from the over-all mean and from the group 
means for a few of the best and a few of 
the worst cycles. In offering these data, the 
writer makes no claim to having discovered 
a cycle that will have general value or that 
will be found useful for any other series 
than the Honolulu Rainfall Index. For any 
other series a similar procedure will indicate 
whether any cycle appears to have at least 
short-term prediction value. 
TABLE 3 
Measures of Estimate 
CYCLE 
STANDARD 
DEVIATION 
PROBABLE 
ERROR 
Variations from arithmetic 
mean of 56 years, no 
Per cent 
Per cent 
cycle ... 
Variations from group 
20.3 
13.7 
mean on 16-year cycle.... 
Variations from group 
14.4 
9.7 
mean on 20-year cycle.... 
Variations from group 
15.5 
10.5 
mean on 10-year cycle.... 
Variations from group 
19.2 
12.9 
mean on 18-year cycle.... 
18.6 
12.5 
The above values indicate the probable 
error of any given estimate based on the use 
of the cycle named, assuming that the cycle 
average is correct. However, this is an un¬ 
warranted assumption since the cycle aver¬ 
ages (such as the mean of the years 1890, 
1906, 1922, and 1938) are at most based on 
four values for the 16-year cycle and only 
three for the 20-year cycle. We therefore 
must examine the source of these averages. 
For the 16 groups of years spaced by 16 
years, we find that the probable errors of 
the means range from the maximum of 8.6 
per cent to 1.6 per cent, with a mean of 4.95 
per cent. For the 20 groups of years spaced 
by 20 years, these same values are 11.0 per 
cent, 1.1 per cent, and 5.45 per cent. 
