# Descriptive statistics on time series data

My task is to summarize the descriptive statistics of time series data ( mean, SD , standard error ). It is fairly straightforward for a stationary series. But How do we find out the mean, standard deviation, standard error of the non stationary time series data?

I have been reading about the stationary series and how to make a time series stationary by differencing the series. If we take the mean of the differenced series, will that be a reflector of mean of the original data.

Any inputs on this?

Well, if a time series not is stationary, it does not have a well-defined mean or variance. Non-stationary means (among others) that the marginal distribution of $$X_t$$ depends on $$t$$. So (the distribution of) each $$X_t$$ (could) have a mean, variance, etc, but how do you estimate it based on only one observation?

As to your question If we take the mean of the differenced series, will that be a reflector of mean of the original data? the answer is NO. If your series is a random walk, it has no mean, but the differenced series has a mean of zero!

So your question how to do descriptive statistics on a non-stationary time-series is a tough one, and difficult to answer in generality, as will depend on context. But

1. Plots ...
2. describing the non-stationarity. Do you see
1. increasing level, constant variability?
2. increasing level, increasing variability?
3. does it seem stationary after differencing once?
4. does it seem stationary after differencing twice?
5. ...
3. autocorrelation function of suitable differenced series ...
4. ...

In short, generally non-stationary series need some serious modeling as a help for interpretation. Maybe we can say more if you give some context.