Estimating $\sum_{k=1}^N a_kb_k$ given the means $\bar a_k,\bar b_k$ and determining the error I need to calculate the following expression:
$$\sum_{k=1}^N a_k b_k$$
I know the average values of $a_k$ , defined as $\overline {a_k} = {\sum_{k=1}^N a_k \over N } $ and $b_k$ , defined as $\overline {b_k} = {\sum_{k=1}^N b_k \over N } $.
I don't know the standard deviation but one extra information that I have is that with some accuracy, I can say that all the population $k=,..,N$ are in one of the three different states and I know that what fraction are in each states. In terms of numbers, it means that $a_k$ can only have 3 values. I don't know those values, but I know that for instance, 80% of N have the first value, $a_1$, 19% have the value of $a_2$ and 1% the value of $a_3$. The same kind of information is provided for $b_k$
If only knowing these quantities, I have to make some approximation, I would like to know how much error I am producing with that approximation. $N$ is relatively big. 
Any help is appreciated. :)
Narj

Mathematically, the largest the product can be (assuming all then numbers are positive and "reasonable size") is $N^2\overline{a_k}\overline{b_k}$ when all but one of the $a_k,b_k$ are zero.  The smallest it can be is zero if $a_k$ can be zero.  In an engineering sense, you probably have some information on how much variation is reasonable, which will allow much better assessment.
