Ultrarelativistic heavy-ion collisions produce a state of hot, dense, strongly interacting QCD matter known as quark-gluon plasma (QGP). On an event-by-event basis, the volume of the QGP in ultra-central collisions is mostly constant, while its total entropy can vary significantly, leading to variations in the temperature of the system. Exploiting this unique feature of ultra-central collisions allows us to interpret the correlation between the mean transverse momentum $(\langle p_{\mathrm{T}} \rangle)$ of produced charged hadrons and the number of charged hadrons as a measure of the the speed of sound, $c_{s}$. The speed of sound, $c_{s}$, which relates to the speed at which compression waves travel in a medium (in this case the QGP), is determined by fitting the relative increase of $\langle p_{\mathrm{T}} \rangle$ with respect to the relative change of the average charged-particle density $(\langle \mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \rangle)$ measured at midrapidity. This study reports the $\langle p_{\mathrm{T}} \rangle$ of charged particles in ultra-central Pb-Pb collisions at a center-of-mass energy of 5.02 TeV per nucleon pair, using the ALICE detector. Different centrality estimators based on charged-particle multiplicity or the transverse energy of the event are employed to select ultra-central collisions. By ensuring a pseudorapidity gap between the region used to define the centrality and the region used for measurement, the influence of biases and their potential effects on the rise of the mean transverse momentum are tested. The measured $c_{s}^{2}$ is found to strongly depend on the type of the centrality estimator, ranging from $0.113 \pm 0.003 \, \mathrm{(stat)} \pm 0.007 \, \mathrm{(syst)}$ to $0.438 \pm 0.001 \, \mathrm{(stat)} \pm 0.019 \, \mathrm{(syst)}$ in natural units.