The stiffness of the arterial wall, which is modified by many cardiovascular diseases such as atherosclerosis, is known to be an indicator of vulnerability. This work focuses on the in vivo quantification of the stiffness of the common carotid artery (CCA) by applying the Magnitude Based Finite Element Model Updating (MB-FEMU) method to 13 healthy and diseased volunteers aged from 24 to 76 years old. The MB-FEMU method is based on the minimisation of the deviation between the image of a deformed artery and a registered image of this artery deformed by means of a finite elements analysis. Cross sections of the neck of each subject at different times of the cardiac cycle are recorded using a Phase Contrast cine-MRI. Applanation tonometry is then performed to obtain the blood pressure variations in the CCA throughout a heart beat. First, a time averaged elastic modulus of each CCA between diastole and systole is identified and a stiffening of the artery with age and disease is observed. Second, four elastic moduli are identified during a single heart beat for each artery, highlighting the nonlinear mechanical behaviour of the artery. A stiffening of the artery is observed and quantified at systole in comparison to diastole.