# Is it possible to take the mean of a time series when it is not stationary?

Many guides, when determining the mean of a time series $$X_{t}$$, assumes that $$X_{t}$$ must be stationary before calculating $$\hat{\mu}=\frac{1}{T}\sum_{i=1}^{T} x_{i}$$.

But consider the following example: I want to get the mean number of daily COVID cases over the last year. Assume $$X_{t}$$ is a process chronicling daily covid cases for a year and that $$X_{t} \sim N(\mu,\sigma^{2})$$. (This distribution is probably incorrect in the real world, but stay with me for this example).

If modeled this way, $$\mu$$ may vary from month to month depending on regulations and world events. Using differencing, $$X_t$$ cannot be converted to a stationary process, so I cannot follow the methods outlined in textbooks like here.

But at the end of the day, I just want the mean number of covid cases in a year. Is it really so bad to just get the sample mean $$\bar{x}$$ and report that? I feel like I may be getting conceptually confused here, so could use some guidance on where I may be misunderstanding the problem.

• "This distribution is probably incorrect" No, the word "totally" is not spelled P R O B A B L Y. ;) Aug 8 '21 at 19:36

(Also, your notation for the sample mean is unusual here. The sample mean here would be denoted as $$\bar{x}_T = \tfrac{1}{T} \sum_{i=1}^T x_i$$. We only use the notation $$\hat{\mu}$$ when we are using the sample mean as an estimator of a parameter $$\mu$$, which is apparently not what you want to do.)