Summary: Many situations, especially in Bayesian statistical inference, call for the use of a Markov chain Monte Carlo (MCMC) method as a way to draw approximate samples from an intractable probability distribution. With the use of any MCMC algorithm comes the question of how long the algorithm must run before it can be used to draw an approximate sample from the target distribution. A common method of answering this question involves verifying that the Markov chain satisfies a drift condition and an associated minorization condition [J. S. Rosenthal, J. Am. Stat. Assoc. 90, No. 430, 558–566 (1995; Zbl 0824.60077); G. L. Jones and J. P. Hobert, Stat. Sci. 16, No. 4, 312–334 (2001; Zbl 1127.60309)]. This is often difficult to do analytically, so as an alternative, it is typical to rely on output-based methods of assessing convergence. The work presented here gives a computational method of approximately verifying a drift condition and a minorization condition specifically for the symmetric random-scan Metropolis algorithm. Two examples of the use of the method described in this article are provided, and output-based methods of convergence assessment are presented in each example for comparison with the upper bound on the convergence rate obtained via the simulation-based approach.