In this thesis, three kinds of fractal dimensions, correlation dimension, Hausdorff dimension and box-counting dimension were used to examine time series. To demonstrate the universality of the method, ECG (Electrocardiogram) time series were chosen. The ECG signals consisted of ECGs of three persons in four states for two applications. States are normal, walk, rapid walk and run. These three people are selected from the same age, and height group to minimize variations. First application was made for approximately 1000 samples of size of ECG signals and the second for the whole of the measured ECG signals. Fractal dimension measurements under different conditions were carried out to find out whether these dimensions could discriminate the states under question. A total of 24 ECG signals were measured to determine their corresponding fractal dimensions through the above-mentioned methods. It was expected that fractal dimension values would indicate the states related to the different activities of the persons. Results show that no direct link was found connecting a certain dimension to a certain activity in a consistent manner. Furthermore, no congruence was also found among the three dimensions that were employed. According to these results, it can be stated that fractal dimension values on their own may not be sufficient to identify distinct cases hidden in time series. Time series analysis may be facilitated when additional tools and methods are utilized as well as fractal dimensions at detecting telltale signs in signals of different states.