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Using the command uptime, I can find that the average number of processes using my Linux system over the past 1, 5, and 15 minutes is 1.55, 1.66, and 1.93 respectively.

But I want to know what the average is over the last 10 minutes.

How can I calculate that with only the above information? Is it even possible?

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You can't but you can figure that:

Between 0-1: there were 1.55 on average and 1.55 total.

Between 0-5: there were 1.66 on average so a total of 8.3.

So between 1-5: there were 8.3 - 1.55 = 6.75 total and an average of 1.6875 per minute.

Between 0-15: there were 1.93 on average so a total of 28.95.

So between 5-15: there were 28.95 - 8.3 = 20.65 or an average of 2.065 per minute.

That's all we can know for certain. We can guess as the average seems to be slowing down we can assume a little less than half of 20.65 occurred in 5-10: say about 10... so between 0-10 the could have been about 8.3 + 10 = 18.3 so about an average of 1.83.

But that is just a guess. It could be anything between .83 (if nothing happened between 5-10) or 2.895 (if nothing happened between 10-15).

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