Simple Average Method, calculating moving average and moving variance, how can I say if stationary or not? Lets say I have a sample data (here is just 10 numbers, in real I have about 10000 measurement results). Then, I want to check if the data is stationary or not using Simple Average Method. For example, my data set of size N = 10:
X = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0};

I calculated averages (window = 3):
SAM = {2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0}

with this formula:
(1+2+3)/3 = 2
(2+3+4)/3 = 3
(3+4+5)/3 = 4
...

and put them into SAM table above. Then, calculated the differences beetwen my averages, SAM[i+1] = SAM[i], and I got a differences table: 1 1 1 1 1 1 1 from which I see that the difference between means (averages) is constant (its always 1).
Can I assume that with this simple test my data X = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0}; is stationary? 
 A: If your first differences are constant then your data are not stationary, as the mean is increaing over time. Your first differences are indeed stationary with mean 1 variance 0. 
With time series data, one of the most critical quesions is how to make the data stationary (one could argue this is the whole point of time series analysis). In practice this includes identifying trend, seasonality/cyclicality, stochastic drift, and autocorrelation. This will require more than the moving average can provide on its own.
However, you can probably use the moving average profitably if you want to get a rough confirmation that there  is no trend or periodicity. In this case, you are using the moving average as a smoothing device. You can simply regress your data vs time and see if the best fit line has a large slope, if not, then you don't have a strong linear trend. Also, if you don't notice any increases in spead about the line or any periodcity (ocillating values or tight clumps of data followed by disperse clouds of data) then you have confirmed that the first/second order trend and periodicity are not present to a large degree.
You will need more sophisticated tools to get any more quantitative. This is essentially a time series analysis, which is a whole field of statistics. A large part of that field is devoted to establishing and testing stationarity; therfore, I can't do it justice in this short space suffice it to say that yoru question has been heavily studies by researchers in time series analysis. See this for some basic background.
