  We derive formulas for the efficiency correction of cumulants with many
efficiency bins. The derivation of the formulas is simpler than the previously
suggested method, but the numerical cost is drastically reduced from the naive
method. From analytical and numerical analyses in simple toy models, we show
that the use of the averaged efficiency in the efficiency correction can lead
to wrong corrected values, which have larger deviation for higher order
cumulants. These analyses show the importance of carrying out the efficiency
correction without taking the average.
