# How to calculate the precision of the mean of univariate time series data, based on the precision of the discrete measurements it consists of?

Assume that the precision of the measurements is fixed and independent – each discrete measurement has the same degree of random error, and is independent of all other measurements.

For example, suppose you have an instrument that measures some variable every minute, and the mean of 7 days of data (with no gaps – 10,080 equally spaced measurements) is 144.2. If the precision of the instrument has been previously established, and the SD is 12.5, how would you calculate the precision of that mean?

Alternatively, if you compare the measurement instrument to a trusted reference instrument that takes individual measurements of the same variable (not modeled or replicated, but actually the same physical quantity) at random intervals a dozen times per day, and you calculate a MAPE of 7.3% based on 84 comparisons throughout the interval, how would you calculate the precision of that 7 day mean of 144.2?

If those are not the most useful precision metrics for this purpose, please let me know what would be better. Also, I’m not fixated on any specific metric to represent the precision of the mean. Whether that should be expressed as SE, 95% CI, or something else is part of what I’m asking.