Context. Previous simulations predicted the activity-induced radial-velocity (RV) variations of M dwarfs to range from 1 cm s-1 to 1 km s-1, depending on various stellar and activity parameters. Aims: We investigate the observed relations between RVs, stellar activity, and stellar parameters of M dwarfs by analyzing CARMENES high-resolution visual-channel spectra (0.5-1μm), which were taken within the CARMENES RV planet survey during its first 20 months of operation. Methods: During this time, 287 of the CARMENES-sample stars were observed at least five times. From each spectrum we derived a relative RV and a measure of chromospheric Hα emission. In addition, we estimated the chromatic index (CRX) of each spectrum, which is a measure of the RV wavelength dependence. Results: Despite having a median number of only 11 measurements per star, we show that the RV variations of the stars with RV scatter of >10 m s-1 and a projected rotation velocity v sin i > 2 km s-1 are caused mainly by activity. We name these stars "active RV-loud stars" and find their occurrence to increase with spectral type: from 3% for early-type M dwarfs (M0.0-2.5 V) through 30% for mid-type M dwarfs (M3.0-5.5 V) to >50% for late-type M dwarfs (M6.0-9.0 V). Their RV-scatter amplitude is found to be correlated mainly with v sin i. For about half of the stars, we also find a linear RV-CRX anticorrelation, which indicates that their activity-induced RV scatter is lower at longer wavelengths. For most of them we can exclude a linear correlation between RV and Hα emission. Conclusions: Our results are in agreement with simulated activity-induced RV variations in M dwarfs. The RV variations of most active RV-loud M dwarfs are likely to be caused by dark spots on their surfaces, which move in and out of view as the stars rotate. The data presented in Figs. 5 and A.1 are only available in electronic form at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz- bin/qcat?J/A+A/614/A122