This data set contains two WLAN Received Signal Strengths (RSS) databases suitable for fingerprinting positioning. One database contains training data (Training_rss.csv, Training_coordinates.csv), the radio map, the second database contains test data (Test_rss.csv, Test_coordinates.csv), RSS measurements on a path and the coordinates of that path. The data was collected in a three-floor building at Tampere University of Technology. The files Training_rss.csv and Test_rss.csv represent a matrix, with a reference point per row and an access point per column. The radio map consists of 446 reference points and 489 access points. Empty RSS values are set to 100. The files Training_coordinates.csv and Test_coordinates.csv contain the reference positions, 3D coordinates in a metric local reference frame. The data format allows to use previously published software (https://doi.org/10.5281/zenodo.889797) to analyze the data. The data is postprocessed: The reference positions of each floor are mapped onto a regular grid with 5 meter grid point spacing and the RSS values at each reference position are spatial averages of the RSS values in the resulting cells. The test data is mapped as well, but to a grid of 1 meter grid point spacing, from which only every third value was selected.