The average transverse momentum $\left\langle p_T\right\rangle$ of final particles is an important observable in high-energy heavy-ion collision experiments. It reflects the properties of soft hadrons and thermonuclear matter, and it can be used to deduce the information on the evolution of collision systems. Four normalized physical quantities, i.e., the collision centrality, the average number of binary collisions per participant pair $\frac{2 N_{\text {coll }}}{N_{\text {part }}}$, the average pseudorapidity density of charged particles per participant pair $\frac{2}{N_{\text {part }}} \frac{d N_{c h}}{d \eta}$ and the average pseudorapidity density of charged particles per binary collision $\frac{1}{N_{c o l l}} \frac{d N_{c h}}{d \eta}$, dependencies of the average transverse momentum $\left\langle p_T\right\rangle$ at midrapidity in Au + Au and Pb + Pb collisions from the STAR, PHENIX and ALICE Collaborations are studied with the phenomenological linear and power-law functions. The results show that the average transverse momentum $\left\langle p_T\right\rangle$ of identified particles exhibits a great linear relationship with collision centrality, whereas it follows a nice power-law relationship with the average number of binary collisions per participant pair$\frac{2 N_{\text {coll }}}{N_{\text {part }}}$, the average pseudorapidity density of charged particles per participant pair $\frac{2}{N_{\text {part }}} \frac{d N_{c h}}{d \eta}$ and the average pseudorapidity density of charged particles per binary collision $\frac{1}{N_{\text {coll }}} \frac{d N_{c h}}{d \eta}$. It is also found that the fitting parameters of the proposed phenomenological functions for the average transverse momentum $leftlangle p_Tightangle$ with collision centrality and the average number of binary collisions per participant pair follow a power-law function with the collision energy which enable the prediction power to the phenomenological approach. Therefore, the collision centrality and the average number of binary collisions per participant pair are good physical quantities for studying the average transverse momentum of identified particles in high-energy heavy-ion collisions. The results discovered in this study can be used to predict the average transverse momentum of identified particles at other collision energies which the experimental data are not available so far. The mass ordering of the average transverse momentum of identified particles, i.e., π ^-， K ^-and $\bar{p}$, is also discussed and explained by the particle production time, relevant to energy conservation, at a given collision centrality and energy.