Aims: The observation of propagating magneto-hydrodynamic kink waves in magnetic structures and measurement of their properties (amplitude, phase speed) can be used to diagnose the plasma conditions in the neighbourhood of the magnetic structure via magneto-seismology. We aim to reveal properties of the chromosphere/transition region above the sunspot penumbra using this technique. Methods: Hinode SOT observed a sunspot as it was crossing over the limb, providing a unique side on view of the atmosphere above a sunspot. The presence of large spicule-like jets is evident in Ca ii H images. The jets are found to support transverse wave motions that displace the central axis of the spicules, which can be interpreted as the kink wave. The properties of a specific wave event are measured and used to determine the magnetic and density stratification along the structure. In addition, we measure the width of the spicule and the intensity profile along the structure in order to provide a test for the magneto-seismological results. Results: The measurements of the wave properties reveal an initial rapid increase in amplitude with height above the solar surface, followed by a decrease in amplitude. The magneto-seismological inversions suggests this initial increase corresponds to large changes in density and magnetic field strength. In addition, we provide the first measurements of spicule width with height, which confirm that the spicule under goes rapid expansion. The measured rates of expansion show good agreement with the results from the magneto-seismology. The observed rapid variations in plasma parameters are suggested to be partly due to the presence of a gravitational stratified, ambient atmosphere. Combining width measurements with phase speed measurements implies the observed decrease in wave amplitude at greater heights can be explained by wave damping. Hence, we provide the first direct evidence of wave damping in chromospheric spicules and the quality factor of the damping is found to be significantly smaller than estimated coronal values.