Mean and standard deviation are most useful if data are normally distributed, or at least symmetric. If your data isn't normally distributed, using a five number summary (minimum, 25th percentile, median, 75th percentile, maximum) will be more useful. This can be shown in a box plot as well.
Likewise, for data that are not bi-normally distributed, using Spearman or Kendall correlation will be better than Pearson correlation.