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I have a 5*5 grid (as in the figure). At each point there are 5 dishes of varying numbers of bacteria. I'd like to write a formula which calculates the average of these 5 and then divides by the global average of all 125 dishes, ie:

$$\displaystyle C_{i}=\frac{\frac{1}{m}\sum^m_{j=1}c_{ij}}{\frac{1}{mn}\sum_i\sum_jc_{ij}}$$

where $n$ is the number of zones and $m$ is the total number of dishes in each zone.

Is this formula correct?

enter image description here

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I don't think proof-strategy is a good tag for this question. I've tried algebra-precalculus, since the questions seems to be about a relatively simple formula. And I've added average. Of course, if anyone can think of more appropriate tags, feel free to retag the question. – Martin Sleziak Oct 6 '12 at 7:11
up vote 1 down vote accepted

Assuming that in $c_{ij}$ the first index tells you which point you're at ($i = 1\dots 25$) and the second index tells you which dish you're looking at ($j=1 \dots 5)$ and $m=5$ and $n=25$ then yes, your formula is correct.

Regarding your comment, there is a subtle notational issue. The $i$ on the left hand side and in the numerator is a "real" index. The $j$s and the $i$s in the denominator are all "dummy" indices - they're only there because we need something to sum over. You could call them whatever you want. If you replace the $i$s in the denominator with $k$s instead (which is much better practice) then it no longer looks like you can cancel anything.

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Thank you very much! I was wondering because at first glance it looks like you could cancel the numerator and end up with $\frac{n}{\sum_i^n c_{ij}}$ and obviously that would make no sense. Could you clarify why this wouldn't be the case? Much appreciated. – HCAI Oct 5 '12 at 12:30
@user1134241 I edited my answer. – Chris Taylor Oct 5 '12 at 12:50

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