The parameters of the theoretical distribution are now calculated for the date of Table 8.
They are shown in Table 9. below.
Table 9: Estimation of Parameters
Peak Flow:
Maximum of average
Daily discharges
N= Number of observations of largest value 18
x Sample mean (thousand cusecs) 257.05
Sx= Sample standard deviation (thousand cusecs) 127.05
an 1.0493
Yn 0.5202
1/a thousand cusecs)= Sx/ an 121.08
u (thousand cusecs)= x-Yn/ a 194.04
Having calculated the constants, we can now proceed to derive the observed and
theoretical distributions of the largest value for each of the two types of peak-flow data
considered. These are shown in Table 11
Column 1 of Table 11 shows values of the reduced variety at intervals of 0.25, the length
of the class interval being necessarily arbitrary. Column 2 gives the corresponding values
of y as calculated from Becker’s tables [3]; it gives the cumulative probability of y for
each value of y tabulated in column 1. Column 3 is derived by multiplying each item in
column 2 by the sample size; it gives the theoretical cumulative frequency. Column 4
gives the first differences of successive entries in column 3 and represents the theoretical
frequencies in each class interval.
The peak flows x, equal to the yearly maximum of average daily discharges obtained by
letting y take on the values shown in column 1, are given in column
5(x=Y/a+u;a, u being found from Table 9) The cumulative observed frequencies
corresponding to these values of x are derived from Table 10 which gives the peak- flow
data of Table 8 ranked in ascending order. K These are given in column 6. Column 7 is
obtained by taking first differences of the figures in cloumn6. These shown the observed
frequencies for the same class intervals for which the theoretical frequencies were
calculated.