I have a maybe really silly question about t-test. I know that t-test is a test for comparing means between two sets of data. But I also was taught that when 2 SD cross over with each between the data (ie 2 (mean +/- 1 SD) ranges overlap), it means there is no statistical difference in between two groups. But my result was different than what I was expected. Two mean +/- 1 SD overlap with each other (ex mean1= 28.01, mean2=28.96 and the SDs are about 10 for each mean) but has p-value at 0.003. Can someone tell me how did this happen and what should I pay attention to?
ps. N>30K and the data is heteroscedastic.