A new North Atlantic Oscillation (NAO) index, the NAOI, is defined as the differences of normalized sea level pressures regionally zonal-averaged over a broad range of longitudes 80°W-30°E. A comprehensive comparison of six NAO indices indicates that the new NAOI provides a more faithful representation of the spatial-temporal variability associated with the NAO on all timescales. A very high signal-to-noise ratio for the NAOI exists for all seasons, and the life cycle represented by the NAOI describes well the seasonal migration for action centers of the NAO. The NAOI captures a larger fraction of the variance of sea level pressure over the North Atlantic sector (20°-90°N, 80°W-30°E), on average 10% more than any other NAO index. There are quite different relationships between the NAOI and surface air temperature during winter and summer. A novel feature, however, is that the NAOI is significantly negative correlated with surface air temperature over the North Atlantic Ocean between 10°-25°N and 70°-30°W, whether in winter or summer. From 1873, the NAOI exhibits strong interannual and decadal variability. Its interannual variability of the twelve calendar months is obviously phase-locked with the seasonal cycle.Moreover, the annual NAOI exhibits a clearer decadal variability in amplitude than the winter NAOI. An upward trend is found in the annual NAOI between the 1870s and 1910s, while the other winter NAOindices fail to show this tendency. The annual NAOI exhibits a strongly positive epoch of 50 years between 1896 and 1950. After 1950, the variability of the annual NAOI is very similar to that of the winter NAO indices.