# More efficient formulas for efficiency correction of cumulants and effect of using averaged efficiency

## TL;DR

This paper introduces simplified formulas for efficiency correction of cumulants with multiple efficiency bins, demonstrating that averaging efficiencies can lead to inaccuracies, especially for higher order cumulants.

## Contribution

It presents a more straightforward derivation of efficiency correction formulas that reduces computational cost and highlights the pitfalls of using averaged efficiencies.

## Key findings

- Simpler formulas reduce numerical cost compared to previous methods.
- Using averaged efficiency can cause significant errors in corrected cumulants.
- Higher order cumulants are more affected by efficiency averaging errors.

## Abstract

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.