The infrared sky quality is an important parameter to take into account for the evaluation of astronomical sites. The traditional idea of considering higher altitude sites as better for infrared astronomical observations than sites at lower altitudes is not in agreement with observational data. It has been shown that the observational infrared spectrum at the Observatorio del Teide (OT) at an altitude of 2400m on the island of Tenerife (Spain) is similar to that expected for a site at the altitude of Mauna Kea (4100m) in Hawaii (USA). This result suggests that other parameters besides site altitude is playing an important role in determining the quality of a particular location for infrared astronomical observations. In this paper, we propose the troposphere thickness as one of the parameters that determine the suitability and quality of an astronomical site for infrared observations. The tropopause altitude defines the tropospheric thickness and hence, we present in this paper a statistical study of the tropopause layer altitude for four different astronomical sites. The results presented in this work suggest that the infrared quality at La Palma, La Silla and Mauna Kea could be similar in some epochs of the year, although they are located at different altitudes above the sea level. Mauna Kea presents the thinnest troposphere during Summer and Autumn among the four studied sites, whereas La Palma exhibits the lowest altitude of the tropopause in Winter and Spring. Paranal presents most of the time the thickest troposphere, suggesting worse infrared conditions for astronomical observations (based only in the thickness parameter) at this site, when compared to the other three in study.