I am observing a phenomena, to simplify it, we say the acceleration of objects, $Oi$, for which I measure the acceleration over the time with a frequency of $100$HZ.
Let's imagine I did $50$ experiments, producing $50$ datasets. A dataset, $Di$, contains the acceleration of the object $Oi$ over the time e.g. $2$ seconds. For each dataset $Di$, I can calculate the Mean, $M_i$, and the Standard Deviation, $SDi$. Therefore, each object can be described with $Oi(M_i,SDi)$
Moreover, I compute the combined mean as in the following:
1) The Combined Mean : $Mc= Sum(N_i*M_i)/Sum(N_i)$ where $N_i$ is the cardinality of the dataset $D_i$, per $i=1.50$
2) The Combined standard deviation, $SDc$. \begin{align} \text{Deviance} &= \sum_{i=1}^g n-1_i s_i^2 + \sum_{i=1}^g n_i (\bar{x_i} - \bar{x})^2 \\ \text{SD} &= \sqrt{\frac{1}{N_{\text{total}}-1}\: \text{deviance}} \end{align} The result of the above computation is to build a acceleration model $F$ which can be described with the combined mean $Mc$ and combined $SDc$, $F(Mc, SDc)$.
I would like to build a filter that uses $F(Mc,SDc)$ as a model to select those experiments among others that have a complete different distribution and therefore, mean and standard deviation.
The first question is: What is the best statistical test to apply in order to compare $F(Mc,SDc)$ and $O(Mj, SDj)$? The second question is: How I should phrase the Null Hypthesis and the Alternative Hypothesis? Additional info: $F(Mc,SDc)$ can be approximate to a Gaussian but for the others, $O(Mj, SDj)$, no info.
Please any help on this would be very appreciated.
Best Regards, Carlo