Is there a value in keeping 'history of statistics' (how statistics over a set of data changed over time)?

Question

I currently have statistics calculated (mean, average, mode, range, variance) for my dataset (timestamped data), but it's overwritten every time - so I can access only latest measurements.
My question is - is it useful to gather stats and store history of them, for example, "at 10 a.m. 10.06.2016 the mean was x1, the average=y1, variance=z1, at 10-30 the mean was x2, avg=y2, variance=z2"? May be volatility (speed of change) can be measured this way?
Same question for new data (only portion of data that arrived now, e.g. current stock prices) - is this kind of history useful?
I'm sorry if I'm being vague here

Answer 1

It depends what the statistic history is, it becomes itself just another statistic. Let me clarify, Suppose a company like google wanted to keep track of average search per time. Then a simple way to get the wanted statistic is to keep a history of searches at time 10 Am, history at time 10:15, and so on so it very much depends.

Answer 2

You can use this to see if the distribution is changing with time. For example, you could accumulate the statistics for data taken in one hour blocks. You can then see whether the distribution is different at different times of day, or changing over a span of days. If you just store the hourly statistics along with the count of the data for each hour, you can recover the overall count, mean, and variance without going back to the raw data.