Expected value of several average numbers

Question

I have a series of numbers representing average values of an event occurs everyday. Is it possible to find the expected value or limit of the event based on the daily mean values?

Thanks,

Answer 1

You could see if the 'cumulative' averages seem to converge. Suppose you have $k$ daily averages $\bar x_1, \dots, \bar x_k,$ where the number of values averaged each day is about the same. Then the cumulative average up to day $k$ is $$\bar X_k = \frac{1}{k}\sum_{i=1}^k \bar x_i.$$

If the numbers per day are remarkably different, it might be better to use do a cumulative weighted average up to each day: $$ \tilde X_k = \frac{\sum_{i=1}^k n_i\bar x_i}{\sum_{i=1}^k n_i}.$$

Either way, you could plot the cumulative averages against day to see if there seems to be convergence.

Example: A thousand daily averages are distributed $\mathsf{Norm}(\mu=100, \sigma=20)$ and sample sizes are about equal. You might expect the cumulative averages to converge to 100. Some authors call this kind of plot a 'trace'. (R code is provided in case you're interested.)

 x = rnorm(1000, 100, 20)      # 1000 daily averages (fake data)
 n = 1:1000                    # days so far: 1, 2, ..., 1000
 a = cumsum(x)/n               # 1000 cumulative averages
 plot(n,a, type="l", lwd=2, ylim=c(80,120))
 abline(h=100, col="green3")   # reference line

If this is for monitoring a long-term continuing process, then maybe just do noving cumulative averages starting at 30 or 90 days ago, instead of accumulating everything from the beginning of time.

Note: Your question is somewhat vague. If you can't interpret my Answer to use in your specific situation, please explain your difficulty in a Comment.