  We study the multi-band variability and correlations of the TeV blazar Mrk
421 on year time scales, which can bring additional insight on the processes
responsible for its broadband emission. We observed Mrk 421 in the very high
energy (VHE) gamma-ray range with the Cherenkov telescope MAGIC-I from March
2007 to June 2009 for a total of 96 hours of effective time after quality cuts.
The VHE flux variability is quantified with several methods, including the
Bayesian Block algorithm, which is applied to data from Cherenkov telescopes
for the first time. The 2.3 year long MAGIC light curve is complemented with
data from the Swift/BAT and RXTE/ASM satellites and the KVA, GASP-WEBT, OVRO,
and Mets\"ahovi telescopes from February 2007 to July 2009, allowing for an
excellent characterisation of the multi-band variability and correlations over
year time scales. Mrk 421 was found in different gamma-ray emission states
during the 2.3 year long observation period. Flares and different levels of
variability in the gamma-ray light curve could be identified with the Bayesian
Block algorithm. The same behaviour of a quiet and active emission was found in
the X-ray light curves measured by Swift/BAT and the RXTE/ASM, with a direct
correlation in time. The behaviour of the optical light curve of GASP-WEBT and
the radio light curves by OVRO and Mets\"ahovi are different as they show no
coincident features with the higher energetic light curves and a less variable
emission. The fractional variability is overall increasing with energy. The
comparable variability in the X-ray and VHE bands and their direct correlation
during both high- and low-activity periods spanning many months show that the
electron populations radiating the X-ray and gamma-ray photons are either the
same, as expected in the Synchrotron-Self-Compton mechanism, or at least
strongly correlated, as expected in electromagnetic cascades.
