Simple Average Method
The simple average method is a statistical technique for calculating the mean of a set of numbers. It is also known as the arithmetic mean and is the most commonly used method for finding the central tendency of a set of data. In this method, we add up all the values in a set of data and divide the sum by the total number of values. This gives us the average value of the set of data.

For example, let's say we want to find the average score of a student in a class. We have the following scores: 80, 90, 70, 85, and 95. To find the average, we add up all the scores and divide the sum by the total number of scores:

(80 + 90 + 70 + 85 + 95) ÷ 5 = 84

Therefore, the average score of the student in the class is 84.

Weighted Average Method
The weighted average method is a statistical technique that takes into account the relative importance or weight of each value in a set of data when calculating the mean. This method is used when the values in a set of data are not equally important.

In this method, we multiply each value in the set of data by its corresponding weight, add up all the products, and divide the sum by the total weight of all the values.

For example, let's say we want to find the average score of a student in a class, but the midterm exam is worth 40% of the total grade and the final exam is worth 60%. We have the following scores:

Midterm exam score: 80, weight: 40%
Final exam score: 90, weight: 60%

To find the weighted average, we multiply each score by its corresponding weight and add up the products:

(80 × 0.40) + (90 × 0.60) = 84

Therefore, the weighted average score of the student in the class is 84.

Key Differences

- Simple average method is used when all values in a set of data are equally important, whereas weighted average method is used when the values in a set of data are not equally important.
- In simple average method, all values are given the same weight, whereas in weighted average method, each value is multiplied by its corresponding weight.
- Simple average method assigns equal importance to all values, whereas weighted average method assigns different weights to the values based on their importance.

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.