Simple differentially private mechanisms such as the Laplace or geometric, and even more complicated methods such as the matrix mechanism have closed-formed expressions for the variance of the output. These expressions are not affected by changing the input dataset. However, closed-form expressions for more complex differentially private algorithms such as the 2020 Decennial Census Disclosure Avoidance System (DAS) that utilize post-processing techniques such as non-negativity and integer solutions are not available and depend on their input dataset. Spending additional privacy-loss budget, one could estimate the error of a particular realization of the mechanism or use Monte Carlo simulations repeatedly applying the mechanism to the input data to estimate the variance. However, either option is costly in terms of the privacy-loss budget. Instead, we propose a bootstrap-inspired method using the single production run of the algorithm to estimate the variance without spending additional privacy-loss budget. Through simulations utilizing the 1940 publicly available Census data, we illustrate the utility of this methodology.