Analyzing the daily returns of NASDAQ Composite Index by using MF-DFA method has led to findings that the return series does not fit the normal distribution and its leptokurtic indicates that a single-scale index is insufficient to describe the stock price fluctuation. Furthermore, it is found that the long-term memory characteristics are a main source of multifractality in time series. Based on the main reason causing multifractality, a contrast of the original return series and the reordered return series is made to demonstrate the stock price index fluctuation, suggesting that the both return series have multifractality. In addition, the empirical results verify the validity of the measures which illustrates that the stock market fails to reach the weak form efficiency.