Say I have 2 sets of numbers with an equal number of values:
a,b,c
d,e,f
Is it possible for mean(a,b,c,d,e,f) to be significantly lower than mean(mean(a,b,c),mean(d,e,f))? I have 2 large data sets where this is happening and I need to know whether it's possible or I made an error.
It is not possible (for the arithmetic mean).
Take the two tuples $x = (x_1,...,x_n),y=(y_1,...,y_n)$, then the question amounts to if :
$$\frac{\sum_{i=1}^nx_i+\sum_{i=1}^ny_i}{2n} = \frac{\frac{\sum_{i=1}^nx_i}{n}+\frac{\sum_{i=1}^ny_i}{n}}{2}$$
This is clearly the case by simple calculations for fractions.
It is possible if the two series are of unequal size. Say, the average of $(0,0,1)$ is $1/3$, while the averages of $(0,0)$ and $(1)$ are $0$ and $1$, respectively, and their average is $1/2$. However, if the series are of the same size, it cannot happen (as proven in the other answer here, by @linkja).