# How to quantify the uncertainty of the time series average?

Usually, we measure a fixed physical quantity, then we simply average MEAN on the measurement results and calculate the standard deviation STD, then the measurement result can be written as MEAN±STD. But for fluctuating values, such as the temperature of a day, sometimes, for example, the value at 12 o'clock is $$26$$ degrees, and the measurement accuracy of our thermometer is $$0.5$$ degrees, and we measure a result $$y_{i}$$ every minute, then how to represent the average of the day Temperature and its uncertainty? Shouldn't we simply calculate its uncertainty as the standard deviation of the data? After all, every data is accurate.

• Is this a fair way to summarize your question? "How do I report the mean temperature for a day, while also giving a sense of the variation of the temperature throughout the day, given that the variation in temperature is much larger than the uncertainty of the thermometer?" If so, one option is to report the mean temperature and the high and low temperatures for the day. You could include measurement uncertainty on those values if you wanted. You generally don't want to use the standard deviation for data that you don't think is (at least approximately) Gaussian distributed. Commented Apr 18, 2022 at 13:06

From your data, you estimate the mean $$\mu$$ and the standard deviation of the mean $$\sigma_{\mu}$$ and report the result as $$\mu \pm \sigma_{\mu}$$. See my answer to Uncertainty in repetitive measurements for details.
Let's start with the perfect measurement. In this case the average temperature of a day would an exactly known quantity, and its uncertainty is zero. In contrast, the estimator $$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (T_i - \bar T)^2}$$ is obviously non-zero, if we have a temperature change during the day. This shows that $$s$$ is not the uncertainty of the average value $$\bar T$$.