I know the average saving rate in the US is ~5%, but I'm curious about the distribution. An ideal answer would be a histogram with the data broken down by what proportion of population are saving at what percentage.
How much do people save?
About 90% of the population has very little or negative savings. The savings is driven by the upper middle class which typically saves 10% or so of pretax income. You can look at the net worth percentiles here to get a rough idea: