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.