TITLE: Why is an average of an average usually incorrect?
QUESTION [119 upvotes]: Can someone explain why taking an average of an average usually results in a wrong answer?  Is there ever a case where the average of the average can be used correctly?  
As an example, let's say that an assessment is given to three schools and I want to find out the average score for all three schools combined and the average score per school.  When I attempt to add the three individual scores and divide by three I get a number that is very close (+/- 1 percent) to the actual overall average.

REPLY [31 votes]: Thomas Andrews already answered the question, but I'd like to present a more analytical solution to the problem.
The average of averages is only equal to the average of all values in two cases:

if the number of elements of all groups is the same; or
the trivial case when all the group averages are zero

Here's why this is so.
Consider two sets $X = \{x_1, x_2, ..., x_n\}$ and $Y = \{y_1, y_2, ..., y_m\}$ and their averages:
$$ \bar{x} = \frac{\sum_{i=1}^{n}{x_i}}{n} \,,\, 
   \bar{y} = \frac{\sum_{i=1}^{m}{y_i}}{m}
$$
The average of the averages is:
$$ average(\bar{x}, \bar{y}) 
  = \frac{\frac{\sum_{i=1}^{n}{x_i}}{n} + \frac{\sum_{i=1}^{m}{y_i}}{m}}{2}
  = \frac{\sum_{i=1}^{n}{x_i}}{2n} + \frac{\sum_{i=1}^{m}{y_i}}{2m}
$$
Now consider the whole group 
$Z = \{x_1, x_2, ..., x_n, y_1, y_2, ..., y_m\}$ and its average:
$$ \bar{z} = \frac{\sum_{i=1}^{n}{x_i} + \sum_{i=1}^{m}{y_i}}{n + m}$$
For the general case, we can see that these averages are different:
$$  \frac{\sum_{i=1}^{n}{x_i}}{2n} + \frac{\sum_{i=1}^{m}{y_i}}{2m} 
\ne \frac{\sum_{i=1}^{n}{x_i} + \sum_{i=1}^{m}{y_i}}{n + m} 
$$
This answers the first OP question, as to why the average of averages usually gives the wrong answer.
However, if we make $n = m$, we have:
$$  \frac{\sum_{i=1}^{n}{x_i}}{2n} + \frac{\sum_{i=1}^{m}{y_i}}{2n} 
  = \frac{\sum_{i=1}^{n}{x_i} + \sum_{i=1}^{n}{y_i}}{2n} 
$$
This is why the average of averages is equal to the average of the whole group when the groups have the same size.
The second case is trivial: $\bar{x} = \bar{y} = average(\bar{x}, \bar{y}) = 0$.
Note that the above reasoning can be extended for any number of groups.